Abstract
A monoid K is the internal Zappa–Szép product of two submonoids, if every element of K admits a unique factorisation as the product of one element of each of the submonoids in a given order. This definition yields actions of the submonoids on each other, which we show to be structure preserving. We prove that K is a Garside monoid if and only if both of the submonoids are Garside monoids. In this case, these factors are parabolic submonoids of K and the Garside structure of K can be described in terms of the Garside structures of the factors. We give explicit isomorphisms between the lattice structures of K and the product of the lattice structures on the factors that respect the Garside normal forms. In particular, we obtain explicit natural bijections between the normal form language of K and the product of the normal form languages of its factors.
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1 Introduction
The notion of Zappa–Szép products generalises those of direct and semidirect products; the key property is that every element of the Zappa–Szép product can be written uniquely as a product of two elements, one from each factor, in any given order.
For instance, a group K is the (internal) Zappa–Szép product of two subgroups G and H, written \(K=G\bowtie H\), if for every \(k\in K\) there exist unique elements \(g\in G\) and \(h\in H\) such that \(g h = k\), or equivalently, if \(K=GH\) and \(G\cap H=\{\mathbf {1}\}\). As taking inverses is an anti-isomorphism, one also has \(K=HG\) and one obtains unique elements \(g'\in G\) and \(h'\in H\) such that \(h' g' = k\). However, in general neither \(g=g'\) nor \(h=h'\) need to hold. The special case that one of the factors, say G, is a normal subgroup yields a semidirect product \(G\rtimes H\); if both factors are normal, one obtains the direct product \(G\times H\).
Note that if we consider the case of monoids, the symmetry under swapping the factors is not automatic, that is, \(K=GH\) need not imply \(K=HG\).
Zappa–Szép products have been studied for various categories of algebraic objects by many authors; see for instance [1–6, 11–20]. There are subtle differences in definitions between some of these references, for instance regarding the symmetry under swapping of factors.
In the context of Garside monoids, Zappa–Szép products were studied by Picantin [14]; he used the term crossed products. Given a family \(M_1,\ldots ,M_\ell \) of Garside monoids and a family of maps \(\Theta _{i,j}:M_i\times M_j\rightarrow M_j\) that satisfy certain compatibility conditions, Picantin constructs a Garside structure on the set \(M_1\times \cdots \times M_\ell \), that is, he considers external products. (He also uses this construction for decomposing Garside monoids, namely to show that every Garside monoid is the iterated crossed product of Garside monoids that have a cyclic centre.)
The construction in [14] is, however, very technical. The difficulty comes, firstly, from the compatibility conditions for the maps \(\Theta _{i,j}\), which are needed to ensure that the way in which the factors \(M_i\) are made to interact is consistent and the crossed product is well-defined; and, secondly, from the fact that crossed products on more than two factors are considered. In the setting of [14], both considering external products (which require explicit compatibility conditions) and considering products involving more than two factors is necessary, so these complications are unavoidable.
Our situation is different: We are primarily interested in decomposing a given Garside monoid into simpler components that are also Garside monoids. In this situation, the way in which the potential factors interact is defined by the ambient monoid and there is no need for explicit compatibility conditions. It is therefore natural for us to consider internal Zappa–Szép products:
Definition 1
Given a monoid K with two submonoids G and H, say that K is the (internal) Zappa–Szép product of G and H, written \(K=G\bowtie H\), if for every \(k\in K\) there exist unique \(g_1, g_2\in G\) and \(h_1, h_2 \in H\) such that \(g_1 h_1 = k = h_2 g_2\).
We will say that \(g_1 h_1\) is the G H-decomposition of k and that \(h_2 g_2\) is its H G-decomposition.
It is obvious from this definition that forming internal Zappa–Szép products is commutative (that is, \(K=G\bowtie H\) if and only if \(K=H\bowtie G\)). It is, however, not associative, that is \(K = (F \bowtie G) \bowtie H\), meaning that there exists a submonoid \(K'\) such that \(K = K' \bowtie H\) and \(K' = F \bowtie G\), does not imply \(K = F \bowtie (G\bowtie H)\); see Example 36. As our construction can easily be applied iteratively, we can restrict to the case of two factors.
Picantin shows that, for each ordering of the factors in a crossed product, an element of the product can be written uniquely as a product of elements of the factors in that order [14, Proposition 3.6]. Thus, every crossed product is also an internal Zappa–Szép product in the sense of Definition 1, so in particular, every monoid that can be decomposed as a crossed product can also be decomposed as an internal Zappa–Szép product. A priori, decomposability as an internal Zappa–Szép product could be weaker (cf. Example 11), although it turns out that the notions are equivalent in the case of Garside monoids; cf. Corollary 33.
In the context of general (i.e., non Garside) modoids and categories, both internal and external Zappa–Szép products are discussed in detail in [3]; see also [4, 5]. [3] defines both internal and external Zappa–Szép products of monoids, and proves that both notions are equivalent [3, Lemma 3.9]. (The author calls the external version “not attractive”.)
It must be emphasised, however, that external Zappa–Szép products in the sense of [3] are not the same as crossed products in the sense of [14]: Example 11 gives a monoid that is the (internal and external) Zappa–Szép product of two submonoids, but that cannot be decomposed as a crossed product. In particular, it cannot be concluded from [3, Lemma 3.9] that the notions of internal Zappa–Szép products (Definition 1) and crossed products coincide. It follows with the results from [14] and Lemma 32 that they do coincide in the situation of Garside monoids (cf. Corollary 33), but this is not obvious a priori.
The main contributions of this paper to the concept of Zappa–Szép products of Garside monoids which was introduced in [14] are the following:
-
1.
We develop a notion of internal Zappa–Szép products of Garside monoids which is less technical than that of crossed products and more suitable for analysing decompositions of Garside monoids: Our definition of an internal Zappa–Szép product assumes only the existence of unique decompositions of elements; from this definition we derive natural maps which we prove to be bijective actions preserving algebraic structures in certain situations. Our approach in this regard is analogous to the one in [3]. The definition of crossed products in [14] features similar maps, but it imposes various properties of the latter as requirements.
While decomposability as a crossed product and decomposability as an internal Zappa–Szép product turn out to be equivalent in the case of Garside monoids, this is not at all obvious from the definitions. Indeed, this equivalence suggests that the definitions of crossed products and internal Zappa–Szép products are in some sense optimal, as they describe a structure that is obtained naturally by both, constructing and deconstructing Garside monoids as products of submonoids.
-
2.
Our techniques allow us to relate the regular language of normal forms in a Zappa–Szép product to the languages of normal forms in its factors; there are no similar results in [14]: We give explicit translations, in both directions, between an automaton accepting the regular language of normal forms in a Zappa–Szép product on the one hand and a pair of automata accepting the regular languages of normal forms in the factors on the other hand. We also give explicit algorithms transforming normal forms.
The structure of the paper is as follows: in Sect. 2 we recall the main concepts used in the paper and fix notation. In Sect. 3 we define actions of the factors of a Zappa–Szép product on each other and analyse their properties. Section 4 is devoted to the case that either the Zappa–Szép product of two monoids is a Garside monoid or that both of the factors are; we will show that both conditions are equivalent. Finally, in Sect. 5 we consider the situation where the Garside elements of the factors and of the product are chosen in a compatible way; we show that in this case the regular language of normal forms in the product can be described effectively in terms of those of the factors.
2 Background
In order to fix notation, we briefly recall the main concepts used in the paper. For details we refer to [7–9].
Let M be a monoid and \(\mathbf {1}\) its identity. The monoid M is called left-cancellative if for any \(x,y,y'\) in M, the equality \(xy = xy'\) implies \(y = y'\). Similarly, M is called right-cancellative if for any \(x,y,y'\) in M, the equality \(yx = y'x\) implies \(y = y'\).
For \(x,y\in M\), we say that x is a left-divisor or prefix of y, writing \(x \,{\preccurlyeq }_M\, y\), if there exists an element \(u\in M\) such that \(y = xu\). If the monoid is obvious, we simply write \(x \,{\preccurlyeq }\, y\) to reduce clutter. Similarly, we say that x is a right-divisor or suffix of y, writing \(y\,{\succcurlyeq }_M\, x\) or \(y\,{\succcurlyeq }\, x\), if there exists \(u\in M\) such that \(y = ux\). Moreover, we say that x is a factor of y, writing , if there exist elements \(u,v\in M\) such that \(y=uxv\). If M does not contain any non-trivial invertible elements, then the relation \({\preccurlyeq }\) is a partial order if M is left-cancellative, and the relation \({\succcurlyeq }\) is a partial order if M is right-cancellative.
An element \(a\in M{\mathbin {{\backslash }}}\{\mathbf {1}\}\) is called an atom if whenever \(a = uv\) for \(u,v\in M\), either \(u = \mathbf {1}\) or \(v = \mathbf {1}\) holds. The existence of atoms implies that M does not contain any non-trivial invertible elements. The monoid M is said to be atomic if it is generated by its set \(\mathcal {A}\) of atoms and if for every element \(x\in M\) there is an upper bound on the length of decompositions of x as a product of atoms, that is, if \(||x||_{\mathcal {A}} := \sup \{ k\in \mathbb {N}: x=a_1\cdots a_k \text { with }a_1,\ldots ,a_k\in \mathcal {A}\} < \infty \).
An element \(d\in M\) is called balanced, if the set of its left-divisors is equal to the set of its right-divisors. In this case, we write \({{\mathrm{Div}}}(d)\) for the set of (left- and right-) divisors of d.
Definition 2
A quasi-Garside structure is a pair \((M, \Delta )\) where M is a monoid and \(\Delta \) is an element of M such that
-
(a)
M is cancellative and atomic,
-
(b)
the prefix and suffix relations are lattice orders, that is, for any pair of elements there exist unique least common upper bounds and unique greatest common lower bounds with respect to \({\preccurlyeq }\) respectively \({\succcurlyeq }\),
-
(c)
\(\Delta \) is balanced, and
-
(d)
M is generated by the divisors of \(\Delta \).
If the set of divisors of \(\Delta \) is finite then we say that \((M, \Delta )\) is a Garside structure.
A monoid M is a (quasi)-Garside monoid if there exists a (quasi)-Garside element \(\Delta \in M\) such that \((M, \Delta )\) is a (quasi)-Garside structure.
Remark
If M is a Garside monoid then the choice of Garside element is not unique. Indeed, if \(\Delta \) is a Garside element then \(\Delta ^\ell \) is also a Garside element for all \(\ell \in \mathbb {N}\).
If \((M,\Delta )\) is a quasi-Garside structure in the above sense, then in the terminology of [7], the set \({{\mathrm{Div}}}(\Delta )\) forms a bounded Garside family for the monoid M. The elements of \({{\mathrm{Div}}}(\Delta )\) are called simple elements. (Note that the set of simple elements depends on the choice of the Garside element.)
Definition 3
A monoid M is conical if for all \(x,y \in M\), \(x y = \mathbf {1}\) implies that \(x=\mathbf {1}=y\). In particular, all Garside monoids are conical.
Notation 4
If M is a left-cancellative atomic monoid, then least common upper bounds and greatest common lower bounds are unique if they exist. In this situation, we will write \(x \vee y\) for the \({\preccurlyeq }\)-least common upper bound of \(x,y\in M\) if it exists, and we write \(x \wedge y\) for their \({\preccurlyeq }\)-greatest common lower bound if it exists. If \(x,y\in M\) admit a \({\preccurlyeq }\)-least common upper bound, we define \(x\backslash y\) as the unique element of M satisfying \(x(x\backslash y)=x\vee y\).
Similarly, if M is a right-cancellative atomic monoid, we will write \(x \mathbin {\widetilde{\vee }}y\) and \(x \mathbin {\widetilde{\wedge }}y\) for the \({\succcurlyeq }\)-least common upper bound respectively the \({\succcurlyeq }\)-greatest common lower bound of x and y if they exist, and if x and y admit a \({\succcurlyeq }\)-least common upper bound, we define y / x as the unique element of M satisfying \((y/x)x=x\mathbin {\widetilde{\vee }}y\).
Lemma 5
([8, Lemme 1.7]) If M is a Garside monoid, then one has
for any \(a,b,c\in M\).
If \((M,\Delta )\) is a Garside structure, we write \(\mathcal {D}_M\) for the set of simple elements \({{\mathrm{Div}}}(\Delta )\), and we define the set of proper simple elements as \(\mathcal {D}^{\!{}^{\circ }\!}_M = \mathcal {D}_M {\mathbin {{\backslash }}} \{ \mathbf {1}, \Delta \}\), where \(\mathbf {1}\) is the identity element of M. To avoid clutter, we will usually drop the subscript if there is no danger of confusion. For \(x\in \mathcal {D}\), there exists a unique element \(\partial x=\partial _M x\in \mathcal {D}\) such that \(x \partial x=\Delta \). Clearly, \(\partial x\in \mathcal {D}^{\!{}^{\circ }\!}\) iff \(x\in \mathcal {D}^{\!{}^{\circ }\!}\). Moreover, for \(x\in M\), we define \(\Delta _x := \Delta _x^{M} := \bigvee \{ y\backslash x : y \in M \}\) [14].
Given a set X we will write \(X^* = \bigcup _{i = 0}^{\infty } X^i\) for the set of strings of elements of X. We will write \(\varepsilon \) for the empty string and separate the letters of a string with dots, for example we will write \(a \mathbin {.}b \mathbin {.}a \in \{a,b\}^*\).
Given a (quasi)-Garside structure \((M, \Delta )\) we can define the left normal form of an element by repeatedly extracting the \({\preccurlyeq }\)-GCD of the element and \(\Delta \). More precisely, the normal form of \(x \in M\) is the unique word \(\mathrm {NF}(x) = x_1 \mathbin {.}x_2 \mathbin {.}\cdots \mathbin {.}x_\ell \) in \((\mathcal {D}{\mathbin {{\backslash }}}\{\mathbf {1}\})^*\) such that \(x = x_1 x_2 \cdots x_\ell \) and \(x_i = \Delta \wedge x_i x_{i+1} \cdots x_\ell \) for \(i=1,\ldots ,\ell \), or equivalently, \(\partial x_{i-1} \wedge x_i=\mathbf {1}\) for \(i=2,\ldots ,\ell \). We write \(x_1|x_2|\cdots |x_\ell \) for the word \(x_1 \mathbin {.}x_2 \mathbin {.}\cdots \mathbin {.}x_\ell \) together with the proposition that this word is in normal form.
If \(x_1|x_2|\cdots |x_\ell \) is the normal form of \(x\in M\), we define the infimum of x as \(\inf (x) = \max \{ i\in \{1,\ldots ,\ell \} : x_i = \Delta \}\), the supremum of x as \(\sup (x) = \ell \), and the canonical length of x as \(\mathrm {cl}(x) = \sup (x)-\inf (x)\). Note that \(\inf (x)\) is the largest integer i such that \(\Delta ^i\,{\preccurlyeq }\, x\) holds, and \(\sup (x)\) is the smallest integer i such that \(x\,{\preccurlyeq }\, \Delta ^i\) holds.
Let \(\mathcal {L}\) be the language on the set \(\mathcal {D}^{\!{}^{\circ }\!}\) of proper simple elements consisting of all words in normal form, and write \(\mathcal {L}^{(n)}\) for the subset consisting of words of length n:
We also define
Definition 6
([10, Definition 2.2]) Let M be a Garside monoid with set of atoms \(\mathcal {A}\), let \(\delta \) be a balanced simple element of M, and let \(M_\delta \) be the submonoid of M generated by \(\{ a\in \mathcal {A}: a \,{\preccurlyeq }\,\delta \}\).
\(M_\delta \) is a parabolic submonoid of M, if \(\{ x \in M : x\,{\preccurlyeq }\, \delta \} = \mathcal {D}\cap M_\delta \) holds.
Proposition 7
([10, Lemma 2.1]) If M is a Garside monoid and \(\delta \) is a balanced simple element of M such that \(M_\delta \) is a parabolic submonoid of M, then \(M_\delta \) is a sublattice of M for both \({\preccurlyeq }\) and \({\succcurlyeq }\) that is closed under the operations \(\backslash \) and / . In particular, \(M_\delta \) is a Garside monoid with Garside element \(\delta \).
Remark
If \(M_\delta \) is a parabolic submonoid of M, then for any \(x\in M_\delta \), the left normal form of x in the Garside monoid \(M_\delta \) coincides with its left normal form of x in the Garside monoid M.
Definition 8
([14]) For a Garside monoid M with set of atoms \(\mathcal {A}\) we define the quasi-centre of M as \(\mathrm {QZ}:= \mathrm {QZ}_M := \{ u\in M : \mathcal {A}u = u \mathcal {A}\}\), and we say that M is \(\Delta \)-pure if \(\Delta _a=\Delta _b\) holds for any \(a,b\in \mathcal {A}\).
Proposition 9
([14]) Let M be a Garside monoid with set of atoms \(\mathcal {A}\).
-
(a)
For any \(x\in M\) and \(c\in \mathrm {QZ}\), one has .
-
(b)
For any \(a,b\in \mathcal {A}\), one has either \(\Delta _a = \Delta _b\) or \(\Delta _a\wedge \Delta _b=\mathbf {1}\).
-
(c)
If \(c_1, c_2\in \mathrm {QZ}\), then \(c_1\wedge c_2\in \mathrm {QZ}\).
-
(d)
For any \(x\in M\), one has \(\Delta _x = \bigwedge ( \mathrm {QZ}\cap xM )\). In particular, \(\Delta _x\in \mathrm {QZ}\) and \(x \,{\preccurlyeq }\, \Delta _x\).
-
(e)
For any \(x,y\in M\), one has \(\Delta _{x\vee y} = \Delta _x\vee \Delta _y\).
-
(f)
\(\mathrm {QZ}\) is a free abelian monoid with basis \(\{ \Delta _a : a\in \mathcal {A}\}\).
Proof
The claims hold by [14, Lemma 1.7, Lemma 2.9, Lemma 2.11, Proposition 2.12, Lemma 2.14, Proposition 2.15]. \(\square \)
Remark
The results from [14] used in the proof of Proposition 9 do not depend on the notion of crossed products.
We will only consider the prefix lattice, but the left-right symmetry of our definitions means that analogous results hold for the suffix ordering and the right normal form; cf. Lemma 12.
3 Actions on the factors of Zappa–Szép products
In the situation of Definition 1, the process of rewriting the GH-decomposition of an element into its HG-decomposition, or vice versa, defines a left-action and a right-action of H on G, as well as a left-action and a right-action of G on H; this section is devoted to analysing these actions.
Our treatment is analogous to [3], except that Definition 1 is symmetric under swapping the two factors of a Zappa–Szép product, whereas in [3], the factors play different roles. The situation of Definition 1 is equivalent to requiring both \(K=G\bowtie H\) and \(K=H\bowtie G\) in the terminology of [3].
Definition 10
Converting HG-decompositions into GH-decompositions, and vice versa, gives us the following maps:
such that \(h g = (h \vartriangleright g)(h \vartriangleleft g)\) and \(g h = (g \blacktriangleright h)(g \blacktriangleleft h)\).
These definitions correspond to the following commutative diagrams:
Remark
(Comparison with crossed products [14])
-
1.
Let G and H be cancellative, conical monoids with finitely many atoms, and let \(\Theta _{1,2}: G\times H\rightarrow H\) and \(\Theta _{2,1}: H\times G\rightarrow G\) satisfy the following conditions:
-
(CP-a)
For any \(g\in G\) and any \(h\in H\), the maps \(h'\mapsto \Theta _{1,2}(g,h')\) and \(g' \mapsto \Theta _{2,1}(h,g')\) are bijections.
-
(CP-b)
For any \(g,g_1,g_2\in G\) and \(h,h_1,h_2\in H\), one has
$$\begin{aligned}&\Theta _{1,2}(g_1 g_2,h) = \Theta _{1,2}(g_2, \Theta _{1,2}(g_1,h)) \\&\Theta _{2,1}(h_1 h_2,g) = \Theta _{2,1}(h_2, \Theta _{2,1}(h_1,g)) \\&\Theta _{1,2}(g,h_1 h_2) = \Theta _{1,2}(g, h_1) \Theta _{1,2}(\Theta _{2,1}(h_1,g),h_2) \\&\Theta _{2,1}(h,g_1 g_2) = \Theta _{2,1}(h, g_1) \Theta _{2,1}(\Theta _{1,2}(g_1,h),g_2). \end{aligned}$$
The crossed product \(G\bowtie _\Theta H\) of G and H with respect to \(\Theta _{1,2}\) and \(\Theta _{2,1}\) is defined [14] as the quotient of the free product of G and H by the relations corresponding to the following commutative diagrams:
By [14, Proposition 3.6], every element of \(G\bowtie _\Theta H\) admits a unique GH-decomposition and a unique HG-decomposition, that is, \(G\bowtie _\Theta H\) is the internal Zappa–Szép product of its submonoids G and H, and maps \(\vartriangleright \), \(\vartriangleleft \), \(\blacktriangleright \), \(\blacktriangleleft \) as in Definition 10 exist.
-
(CP-a)
-
2.
Conversely, we shall see that in an internal Zappa–Szép product \(G\bowtie H\) where the factors G and H are cancellative, conical monoids with finitely many atoms, and if moreover common multiples exist in G and H, the maps
$$\begin{aligned} g' \mapsto h \vartriangleright g' \qquad g' \mapsto g' \blacktriangleleft h \qquad h' \mapsto h' \vartriangleleft g \qquad h' \mapsto g \blacktriangleright h' \end{aligned}$$are bijections for all \(g\in G\) and \(h\in H\) that satisfy properties analogous to (CP-b) above (cf. Lemma 13, Lemma 20, Lemma 21, Lemma 25). In this case, it is possible to construct suitable maps \(\Theta _{1,2}\) and \(\Theta _{2,1}\) from the maps \(\vartriangleright \), \(\vartriangleleft \), \(\blacktriangleright \), \(\blacktriangleleft \), and one can identify \(G\bowtie H\) with the crossed product \(G\bowtie _\Theta H\); cf. Corollary 33.
However this need not be possible in general: Example 11 constructs a monoid that can be decomposed as the internal Zappa–Szép product of two cancellative, conical monoids with finitely many atoms that do not admit common multiples, but that cannot be written as a crossed product of these factors.
While there are connections between the notions of crossed products and internal Zappa–Szép products, the two notions are not equivalent. Moreover, there are conceptual differences: Internal Zappa–Szép products model the decomposition of a given monoid into components, whereas crossed products are a specific construction by which given monoids can be composed into a new monoid.
Example 11
Consider the monoid K defined by the monoid presentation
and the submonoids G and H of K generated by \(\{a_1,b_1\}\) respectively \(\{a_2,b_2\}\).
The relations of K preserve the string over the alphabet \(\{a,b\}\) obtained from a word in the atoms of K by ignoring generator subscripts, in particular the length of the word, as well as the number of atoms contained in G respectively H. Indeed, any two words in the atoms of K that yield the same string over the alphabet \(\{a,b\}\) and involve the the same number of atoms contained in G (and thus the the same number of atoms contained in H) are equivalent. Thus, each element x of K can be identified with a pair consisting of a word \(w_x\) of length \(\ell _x\) over the alphabet \(\{a,b\}\) and an integer \(g_x\in \{0,\ldots ,\ell _x\}\) giving the number of atoms contained in G; one has \(x\in G\) iff \(g_x=\ell _x\) and \(x\in H\) iff \(g_x=0\). Hence, the number of elements of word length \(\ell \) in G, and also in H, is \(2^\ell \), and the number of elements of word length \(\ell \) in K is \((\ell +1)\cdot 2^\ell \). In particular, the submonoids G and H are free of rank 2, and each element of K admits a unique GH-decomposition and a unique HG-decomposition, so one has \(K=G\bowtie H\), where G and H are cancellative, conical monoids with finitely many atoms. Note that neither in G nor in H do common multiples of elements exist.
From the commutative diagrams
we see that one has for any \(u\in \{a,b\}\), any \(g\in \{a_1,b_1\}\), and any \(h\in \{a_2,b_2\}\) the following identities:
That is, the maps \(g \mapsto u_2 \vartriangleright g\), \(h \mapsto h \vartriangleleft u_1\), \(h \mapsto u_1 \blacktriangleright h\), and \(g \mapsto g \blacktriangleleft u_2\) are not injective in this case. Moreover, as the commutative diagrams in (1) are all the commutative diagrams involving two pairs of atoms, it is not possible to define maps \(\Theta _{1,2}\) and \(\Theta _{2,1}\) with the properties (CP-a) and (CP-b), so K cannot be identified with a crossed product of G and H.
3.1 Basic properties
We start by noting some basic properties of the maps \(\vartriangleright \), \(\vartriangleleft \), \(\blacktriangleright \), \(\blacktriangleleft \).
Lemma 12
Consider the set of propositions built out of monoid operations, logical operations, quantifiers over G, H and K, and the operations \(\vartriangleright \), \(\vartriangleleft \), \(\blacktriangleright \), \(\blacktriangleleft \). We can define two transformations of this set as follows.
-
\(\sigma :\) Swap \(G \longleftrightarrow H\), \(\vartriangleright \longleftrightarrow \blacktriangleright \) and \(\vartriangleleft \longleftrightarrow \blacktriangleleft \).
-
\(\tau :\) Replace the monoids with their opposites, reversing all monoid expressions and all triangle operations:
-
\(G \longrightarrow G^\mathrm {op}\), \(H \longrightarrow H^\mathrm {op}\) and \(K\longrightarrow K^\mathrm {op}\).
-
\(x \cdot y \longrightarrow y^\mathrm {op}\cdot x^\mathrm {op}\).
-
\(h \vartriangleright g \longleftrightarrow g^\mathrm {op}\blacktriangleleft h^\mathrm {op}\) and \(h \vartriangleleft g \longleftrightarrow g^\mathrm {op}\blacktriangleright h^\mathrm {op}\).
-
Then for any proposition E we have \(E \Longleftrightarrow \sigma (E) \Longleftrightarrow \tau (E)\).
Proof
The equivalence of E and \(\sigma (E)\) is clear; if you swap the roles of G and H then you swap the definitions of the triangle operations.
To see that E is equivalent to \(\tau (E)\), first observe that \(K = G \bowtie H\) if and only if \(K^\mathrm {op}= G^\mathrm {op}\bowtie H^\mathrm {op}\). The anti-isomorphisms between K and \(K^\mathrm {op}\) transforms the equality \(hg = (h \vartriangleright g)(h\vartriangleleft g)\) to
but taking \(H^\mathrm {op}G^\mathrm {op}\)-decompositions gives us
The uniqueness of \(H^\mathrm {op}G^\mathrm {op}\)-decompositions means we have the following equalities.
If E is simply an equality between two monoid-triangle expressions, i.e. E is \(x = y\), then by (2) \(\tau (x) = x^\mathrm {op}\) and \(\tau (y) = y^\mathrm {op}\) so the equivalence of E and \(\tau (E)\) follows from the fact that \(\mathrm {op}\) is a bijection.
Logical conjunction and disjunction and the universal and existential quantifiers are unchanged by \(\tau \), e.g. \(\tau (A \wedge B) \equiv \tau (A) \wedge \tau (B)\). Therefore, the equivalence of E and \(\tau (E)\) follows by structural induction. \(\square \)
Lemma 13
([3, Lemma 3.2]) The maps \(\vartriangleright \) and \(\blacktriangleright \) define left actions, and \(\vartriangleleft \) and \(\blacktriangleleft \) define right actions:
Moreover, the actions act on products as follows:
Lemma 14
([3, Corollary 3.3.1]) The identity elements of the submonoids act trivially.
Lemma 15
For all \(g \in G\), \(h \in H\) we have the following logical equivalences.
Proof
Consider the equation \(hg = (h \vartriangleright g)(h \vartriangleleft g)\). If \(g = \mathbf {1}\) then this is an element of H, so by the uniqueness of GH-decompositions \(h \vartriangleright g = \mathbf {1}\). Similarly, if \(h \vartriangleright g = \mathbf {1}\) then this is also an element of H, so by the uniqueness of HG-decompositions \(g = \mathbf {1}\).
The remaining equivalences follow by Lemma 12. \(\square \)
Lemma 16
For all \(g \in G\) and \(h \in H\):
Proof
Rewriting a GH-decomposition as a HG- and then back as a GH-decomposition, we have
By the uniqueness of GH-decompositions we can the deduce the first two equations. The second two can be shown in the same way. \(\square \)
3.2 Actions and the monoid structure
Lemma 17
Suppose that \(K = G \bowtie H\) and that H is conical. Then, for all \(x,y \in K\), \(xy \in G\) implies that \(x \in G\) and \(y \in G\).
Proof
Let \(g = xy \in G\). Suppose that we have the following GH-decompositions of x and y:
Now
whence by the uniqueness of the GH-decomposition of g, we have the following:
So, as H is conical, \(h_x \vartriangleleft g_y = \mathbf {1}= h_y\) and so, by Lemma 15, \(h_x = \mathbf {1}\). Hence \(x = g_x \in G\) and \(y = g_y \in G\).
Lemma 18
Suppose that \(K = G \bowtie H\) and that H is conical. For all \(h \in H\) we have that if \(a \in G\) is an atom then \(h \vartriangleright a\) is an atom.
Proof
Suppose that \(h \vartriangleright a = x y\), that is, \(ha = xy h'\) where \(h' = h \vartriangleleft a\). By Lemma 17, \(x, y \in G\), so we may apply Lemma 13 to the action of \(h'\) on xy, which gives us
Now if a is an atom, we have that either \((x \blacktriangleleft (y \blacktriangleright h')) = \mathbf {1}\) or \((y \blacktriangleleft h') = \mathbf {1}\). So, by Lemma 15, either \(x = \mathbf {1}\) or \(y = \mathbf {1}\) holds. \(\square \)
Lemma 19
If \(h \vartriangleleft g = h\) then one has \(h \vartriangleright (g^\ell ) = (h \vartriangleright g)^\ell \) for all \(\ell \in \mathbb {N}\).
Proof
Using induction on \(\ell \), we obtain
Lemma 20
Suppose that \(K = G \bowtie H\), that G is left-cancellative and that common multiples with respect to the prefix order exist in G for every pair of elements. Then \(\vartriangleright \) acts by injections.
Proof
Suppose that \(h \vartriangleright g_1 = h \vartriangleright g_2\); we have to show that \(g_1 = g_2\). Let \(g = h \vartriangleright g_1\). There exist \(h_1, h_2 \in H\) such that
Let \(g_1 \bar{g}_1 = g_2 \bar{g}_2\) be a common multiple of \(g_1\) and \(g_2\) in G. We have
so uniqueness of GH-decompositions and the left-cancellativity of G give us the following:
So \(h_1 \bar{g}_1 = h_2 \bar{g}_2\), and uniqueness of HG-decompositions then yields \(h_1 = h_2\). Substituting this into (4) gives \(h g_1 = h g_2\), and using the uniqueness of HG-decompositions again, we obtain \(g_1 = g_2\). \(\square \)
Remark
Example 11 shows that the condition that common multiples with respect to the prefix order exist in G cannot be dropped.
Lemma 21
Suppose that \(K = G \bowtie H\), that G is atomic and that \(\vartriangleright \) acts surjectively on the set of atoms. Then \(\vartriangleright \) acts surjectively on the whole of G.
Proof
We need to show that given any \(h \in H\) and \(g \in G\) there exists \(g' \in G\) such that \(h \vartriangleright g' = g\). As G is atomic we may proceed by induction on the length of the longest decomposition of g as a product of atoms.
Suppose that \(g = a g_1\) where a is an atom of G. As \(\vartriangleright \) acts surjectively on the set of atoms, there exists an atom \(b \in G\) such that \(h \vartriangleright b = a\). The longest atomic decomposition of \(g_1\) must be at least one shorter than that for g, so by induction there exists \(g_1' \in G\) such that
Now
So \(g' = b g_1'\) is the required element of G. \(\square \)
Lemma 22
(cf. [3, Lemma 3.12(viii)]) Suppose that \(K = G \bowtie H\), that G and H are left-cancellative and that \(\vartriangleright \) acts by injections. Then K is left-cancellative.
Proof
Suppose we have \(x,y_1,y_2 \in K\) such that \(x y_1 = x y_2\). We have GH-decompositions \(x = g_x h_x\), \(y_1 = g_1 h_1\) and \(y_2 = g_2 h_2\). As
uniqueness of GH-decompositions implies
Left-cancellativity of G implies that \(h_x \vartriangleright g_1 = h_x \vartriangleright g_2\). So, as \(\vartriangleright \) acts injectively \(g_1 = g_2\). Left-cancellativity of H then implies that \(h_1 = h_2\), whence \(y_1 = y_2\). \(\square \)
3.3 Submonoids acting by bijections
We will see in Sect. 4 that in the situations we are interested in the submonoids act on each other by bijections. We analyse this special case in the remainder of this section.
Definition 23
Suppose that \(K = G \bowtie H\). We say that the submonoids act on each other by bijections, if for all \(h\in H\) the maps
and for all \(g\in G\) the maps
are bijections.
In this case, we denote the inverses of these maps as follows:
Obviously, there are no elements \(g^{-1}\) and \(h^{-1}\); this is just a notational convenience.
Lemma 24
If \(K = G \bowtie H\) and the submonoids act on each other by bijections then, for all \(g \in G\) and \(h \in H\), one has
Proof
Suppose \(g' = h^{-1} \vartriangleright g\). So \(h \vartriangleright g' = g\) and \(h g' = g h'\) for some \(h' \in H\). From this we see that \(h \vartriangleleft g' = h'\) and \(g \blacktriangleright h' = h\). Substituting for \(g'\) in the former and rearranging the latter we have \(h \vartriangleleft (h^{-1} \vartriangleright g) = h' = g^{-1} \blacktriangleright h\). Hence the first equation holds.
The remaining equations are shown in an analogous fashion. \(\square \)
Lemma 25
If \(K = G \bowtie H\) and the submonoids act on each other by bijections then, for all \(g, g_1, g_2 \in G\) and \(h, h_1, h_2 \in H\), the following identities hold:
Proof
The first set of equations clearly hold as \(\vartriangleright \), \(\vartriangleleft \), \(\blacktriangleright \) and \(\blacktriangleleft \) define actions.
Now consider the first of the second set of equations. If we apply h we have the following.
Hence, as required, \(h^{-1} \vartriangleright g_1 g_2 = (h^{-1} \vartriangleright g_1) \left( (g_1^{-1} \blacktriangleright h)^{-1} \vartriangleright g_2\right) \).
The other equations can be shown in the same way. \(\square \)
Proposition 26
Suppose \(K = G \bowtie H\), that G and H are cancellative, and that G and H act on each other by bijections.
Then for all \(g \in G\) and \(h \in H\), the left actions are isomorphisms of the prefix order and the right actions are isomorphisms of the suffix order:
Proof
Lemma 13 implies that these maps are poset morphisms: For example, if \(g_1 \,{\preccurlyeq }_G g'\) then there exists \(g_2\) such that \(g' = g_1 g_2\), whence we obtain \(h \vartriangleright g' = (h \vartriangleright g_1) \left( (h \vartriangleleft g_1) \vartriangleright g_2\right) \) and so \(h \vartriangleright g_1 \,{\preccurlyeq }_G h \vartriangleright g'\).
Similarly, Lemma 25 implies that the inverses of these maps are poset morphisms: For example, if \(g_1 \,{\preccurlyeq }_G g'\) then there exists \(g_2\) such that \(g' = g_1 g_2\). So \(h^{-1} \vartriangleright g' = (h^{-1} \vartriangleright g_1) \left( (g_1^{-1} \blacktriangleright h)^{-1} \vartriangleright g_2\right) \) and thus \(h^{-1} \vartriangleright g_1\, {\preccurlyeq }_G h^{-1} \vartriangleright g'\).\(\square \)
Lemma 27
Suppose \(K = G \bowtie H\), that G and H are cancellative, and that G and H act on each other by bijections.
Then for all \(g_1,g_2 \in G\) such that \(g_1\vee g_2\) exists in G and all \(h \in H\) one has the following:
Proof
For any \(h'\in H\), one has \(h'\vartriangleright (g_1\vee g_2) = (h'\vartriangleright g_1)\vee (h'\vartriangleright g_2)\) by Proposition 26. On the other hand, using Lemma 13, one has \(h'\vartriangleright (g_1\vee g_2) = h'\vartriangleright \big (g_1(g_1\backslash g_2)\big ) = (h'\vartriangleright g_1)\big ((h'\vartriangleleft g_1)\vartriangleright (g_1\backslash g_2)\big )\). As G is cancellative, these imply \((h'\vartriangleright g_1)\backslash (h'\vartriangleright g_2) = (h'\vartriangleleft g_1)\vartriangleright (g_1\backslash g_2)\). The first claim follows setting \(h'=h\vartriangleleft g_1^{-1}\) and simplifying \((h\vartriangleleft g_1^{-1})\vartriangleright g_1 = g_1\blacktriangleleft h^{-1}\) using Lemma 24.
The second claim is shown in the same way, using Lemma 25 instead of Lemma 13 and Lemma 16 instead of Lemma 24. \(\square \)
Lemma 28
Suppose \(K = G \bowtie H\), that G and H are cancellative and conical, and that G and H act on each other by injections.
Then \((K,\,{\preccurlyeq }_K)\) and \((K,{\succcurlyeq }_K)\) are posets, and the restrictions of \({\preccurlyeq }_K\) and \({\succcurlyeq }_K\) to \(G\times G\) and \(H\times H\) coincide with \({\preccurlyeq }_G\), \({\succcurlyeq }_G\), \({\preccurlyeq }_H\) and \({\succcurlyeq }_H\) respectively:
Proof
By Lemma 22 and Lemma 12, the monoid K is cancellative, hence \({\preccurlyeq }\) and \({\succcurlyeq }\) define partial orders.
Lemma 17 implies that G and H are closed under , which implies the rest of the claim. For instance, if \(g_1 \,{\preccurlyeq }_K\, g_2\) holds for \(g_1,g_2\in G\), then there exists \(x\in K\) such that \(g_1 x = g_2\). As G is closed under , one has \(x\in G\) and thus \(g_1 \,{\preccurlyeq }_G\, g_2\). Conversely, \(g_1 \,{\preccurlyeq }_G\, g_2\) trivially implies \(g_1 \,{\preccurlyeq }_K\, g_2\). \(\square \)
In the situation of Lemma 28, we will in the following just write \({\preccurlyeq }\) and \({\succcurlyeq }\) instead of \({\preccurlyeq }_K\), \({\succcurlyeq }_K\), \({\preccurlyeq }_G\), \({\succcurlyeq }_G\), \({\preccurlyeq }_H\) and \({\succcurlyeq }_H\).
Proposition 29
Suppose \(K = G \bowtie H\), that G and H are cancellative and conical, and that G and H act on each other by bijections.
Then for all \(g \in G\) and \(h\in H\), we have
where \(h' = g^{-1} \blacktriangleright h\) and \(g' = h^{-1} \vartriangleright g\). Moreover, if \(g_1,g_2\in G\) and \(h_1,h_2\in H\) satisfy \(g_1\vee h_1 =g_2\vee h_2\) or \(g_1\mathbin {\widetilde{\vee }}h_1 =g_2\mathbin {\widetilde{\vee }}h_2\), then \(g_1=g_2\) and \(h_1=h_2\).
Proof
We are in the situation of Lemma 28.
First we will show that \(gh' = hg'\). As \(h' = g^{-1} \blacktriangleright h\), we have \(g \blacktriangleright h' = h\) and, using Lemma 24, \( gh' = h (g \blacktriangleleft h') = h (g \blacktriangleleft (g^{-1} \blacktriangleright h)) = h (h^{-1} \vartriangleright g) = h g' \). Therefore, \(gh' = hg'\) is a common upper bound of g and h with respect to \({\preccurlyeq }\).
Now assume we have \(g_1,g_2\in G\) and \(h_1,h_2\in H\) such that \(gh_1g_1=hg_2h_2\) is a common upper bound of g and h with respect to \({\preccurlyeq }\). As we have
uniqueness of GH-decompositions implies that \(g (h_1\vartriangleright g_1) = (h \vartriangleright g_2)\). Acting by \(h^{-1}\) on both sides of this equality and applying Lemma 25, we obtain \(g' = h^{-1} \vartriangleright g \,{\preccurlyeq }\, g_2\), and thus \(h g' \,{\preccurlyeq }\, h g_2 \,{\preccurlyeq }\, h g_2 h_2\).
Finally, let \(g_1,g_2\in G\) and \(h_1,h_2\in H\) satisfy \(g_1\vee h_1 =g_2\vee h_2 = x\in K\). By the first part of the proposition, we have \(x=g_1 (g_1^{-1} \blacktriangleright h_1) = g_2 (g_2^{-1} \blacktriangleright h_2)\). Uniqueness of GH-decompositions implies \(g_1=g_2\) and \(g_1^{-1} \blacktriangleright h_1 = g_2^{-1} \blacktriangleright h_2\), and this in turn implies \(h_1=h_2\), as the action of G on H is by bijections.
The claims for \(\mathbin {\widetilde{\vee }}\) are analogous. \(\square \)
Remark
A result for crossed products that is analogous to the following Theorem 30 is [14, Proposition 3.12]; in the light of Corollary 33, indeed Theorem 30 follows from [14, Proposition 3.12] if K is a Garside monoid.
Theorem 30
Suppose \(K = G \bowtie H\), that G and H are cancellative and conical, and that G and H act on each other by bijections.
The map \(G \times H \rightarrow K\) given by \((g,h) \mapsto g \vee h\) is a poset isomorphism \((G, \,{\preccurlyeq }_G)\times (H,\, {\preccurlyeq }_H) \rightarrow (K, \,{\preccurlyeq }_K)\).
Similarly, the map \(G \times H \rightarrow K\) given by \((g,h) \mapsto g \mathbin {\widetilde{\vee }}h\) is a poset isomorphism \((G, {\succcurlyeq }_G)\times (H, {\succcurlyeq }_H) \rightarrow (K, {\succcurlyeq }_K)\).
Proof
We are in the situation of Lemma 28, so we will drop the subscripts of the partial orders.
By Proposition 29, we can write any \(x \in K\) in a unique way as \(x = g_1 \vee h_2\), where \(g_1 h_1\) and \(h_2 g_2\) are the GH-, respectively, HG-decompositions of x. Hence the map \((g,h) \mapsto g \vee h\) is a bijection.
Claim
If \(g_1, g_2 \in G\) and \(h_1, h_2 \in H\) are such that \(g_1 \,{\preccurlyeq }\, g_2\) and \(h_1 \,{\preccurlyeq }\, h_2\) then \(g_1 \vee h_1 \,{\preccurlyeq }\, g_2 \vee h_2\).
If \(g_1 \,{\preccurlyeq }\, g_2\) and \(h_1 \,{\preccurlyeq }\, h_2\), then there are \(g_3 \in G\) and \(h_3 \in H\) such that \(g_1 g_3 = g_2\) and \(h_1 h_3 = h_2\). Now consider the following, where \(g_1 \vee h_1 = g_1 h_1' = h_1 g_1'\):
By the uniqueness of GH- and HG-decompositions and Proposition 29, we thus have \((g_1 \vee h_1) \left( (h_1'^{-1} \vartriangleright g_3) \vee (g_1'^{-1} \blacktriangleright h_3)\right) = g_1 g_3 \vee h_1 h_3 = g_2 \vee h_2\). Hence \(g_1 \vee h_1 \,{\preccurlyeq }\, g_2 \vee h_2\) and so the claim holds.
Claim
If \(g_1, g_2 \in G\) and \(h_1, h_2 \in H\) are such that \(g_1 \vee h_1 \,{\preccurlyeq }\, g_2 \vee h_2\) then \(g_1 \,{\preccurlyeq }\, g_2\) and \(h_1 \,{\preccurlyeq }\, h_2\).
If \(g_1 \vee h_1 \,{\preccurlyeq }\, g_2 \vee h_2\), there are \(g_3 \in G\) and \(h_3 \in H\) such that \((g_1 \vee h_1)(g_3 \vee h_3) = g_2 \vee h_2\).
Now consider the following.
Therefore \((g_1 \vee h_1)(g_3 \vee h_3) = g_1 \left( (g_1^{-1} \blacktriangleright h_1) \vartriangleright g_3\right) \vee h_1 \left( (h_1^{-1} \vartriangleright g_1) \blacktriangleright h_3\right) \). So \(g_2 = g_1 \left( (g_1^{-1} \blacktriangleright h_1) \vartriangleright g_3\right) \) and \(h_2 = h_1 \left( (h_1^{-1} \vartriangleright g_1) \blacktriangleright h_3\right) \). Hence \(g_1 \,{\preccurlyeq }\, g_2\) and \(h_1 \,{\preccurlyeq }\, h_2\) and so the claim holds.
We have shown that the map \((g,h)\mapsto g\vee h\) is invertible and that both this map and its inverse preserve the ordering. Therefore it is an isomorphism between the respective posets.
The claim for the map \((g,h)\mapsto g\mathbin {\widetilde{\vee }}h\) is shown analogously. \(\square \)
Lemma 31
Suppose \(K = G \bowtie H\), that G and H are cancellative and conical, and that G and H act on each other by bijections.
Then, for all \(g_1,g_2 \in G\) such that \(g_1\vee g_2\in G\) exists and for all \(h_1,h_2 \in H\) such that \(h_1\vee h_2\in H\) exists, the elements \(g_1 \vee h_1\) and \(g_2\vee h_2\) of K admit a \({\preccurlyeq }\)-least common upper bound in K, and one has
Proof
Let \(g'=g_1\backslash g_2\) and \(h'=h_1\backslash h_2\), so \(g_1\vee g_2 = g_1 g'\) and \(h_1\vee h_2 = h_1 h'\). Using Theorem 30, Proposition 29, Lemma 24 and Lemma 25, we obtain
and likewise
Thus, as K is cancellative, applying Proposition 29 yields
\(\square \)
4 Actions in the case of Garside monoids
In this section, we analyse the actions of the factors of a Zappa–Szép product on one another in the case that the product is a Garside monoid, or that both of the factors are Garside monoids. Using these results, we prove that a Zappa–Szép product \(K=G\bowtie H\) of monoids is a Garside monoid if and only if both G and H are Garside monoids.
Lemma 32
If \(K = G \bowtie H\) is a Garside monoid then the submonoids act on each other by bijections.
Proof
We will first show that the maps are injective. So suppose that \(h \vartriangleright g_1 = h \vartriangleright g_2 = g\); we need to show that \(g_1 = g_2\).
Let \(h_1 = h \vartriangleleft g_1\) and \(h_2 = h \vartriangleleft g_2\). So we have
First consider the case when \(g_1 \wedge g_2 = \mathbf {1}\). Taking the GCD of the two elements in (5) gives
Uniqueness of GH-decompositions implies \(g = \mathbf {1}\) (and \(h = h_1 \wedge h_2\)), and uniqueness of the HG-decompositions in (5) then implies \(g_1 = \mathbf {1}= g_2\).
Now suppose that \(g_1 \wedge g_2 \ne \mathbf {1}\), so
for some \(\bar{g}_1, \bar{g}_2 \in G\) with \(\bar{g}_1 \wedge \bar{g}_2 = \mathbf {1}\).
We can now apply the formula for the action on a product from Lemma 13:
Cancellativity of K means that \( h' \vartriangleright \bar{g}_1 = h' \vartriangleright \bar{g}_2 \), where \(h' = h \vartriangleleft (g_1 \wedge g_2)\). As \(\bar{g}_1 \wedge \bar{g}_2 = \mathbf {1}\) we can apply the first case to deduce that \(\bar{g}_1 = \bar{g}_2\) and so \(g_1 = g_2\).
Similar arguments show that the other maps are injective, so it remains to show that the maps are surjective.
First note that, by Lemma 18, the maps take atoms to atoms. So, as the sets of atoms are finite, the maps are bijections on these sets. The surjectivity of the maps then follows from Lemma 21 and Lemma 12. \(\square \)
Remark
An analogous result for crossed products is [14, Lemma 3.2].
Corollary 33
If K is a Garside monoid with submonoids G and H, one has \(K = G\bowtie H\) if and only if K can be written as a crossed product of the monoids G and H.
Proof
If K is a crossed product of G and H, then every element of K has a unique GH-decomposition and a unique HG-decomposition by [14, Proposition 3.6], so one has \(K=G\bowtie H\).
If \(K = G\bowtie H\), then Lemma 32 implies that G and H act on each other by bijections and we can define
for \(g\in G\) and \(h\in H\). The maps \(\Theta _{1,2}\) and \(\Theta _{2,1}\) satisfy (CP-a) and (CP-b) by Lemma 25, so K is a crossed product of G and H. \(\square \)
Remark
Corollary 33 says that, in the context of Garside monoids, the notion of crossed products in the sense of [14] and the notion of (external or internal) Zappa–Szép-products in the sense of [3] are equivalent. Example 11 shows that this is not true in general.
Theorem 34
If \(K=G\bowtie H\) is a Garside monoid then G and H are parabolic submonoids of K. In particular, G and H are Garside monoids.
Proof
Let \(d_G d_H\) and \(e_H e_G\) be the GH-, respectively, HG-decompositions of \(\Delta \).
Suppose that \(x \in \mathcal {D}\cap G\) is a simple element which lies in G. Now, as x is a simple element, there is \(\partial x \in K\) such that \(x \partial x = \Delta \). Let gh be the GH-decomposition of \(\partial x\). So we have
As \(x \in G\), the uniqueness of GH-decompositions means that
Hence x is a prefix of \(d_G\). Since \(d_G\) is a simple element and a member of the submonoid G, we have that \(d_G\) is the \({\preccurlyeq }\)-LCM of the intersection of \(\mathcal {D}\) and G. A similar argument shows that \(e_G\) is the \({\succcurlyeq }\)-LCM of the same set.
Now observe that \(e_G \in \mathcal {D}\cap G\) and hence \(e_G \,{\preccurlyeq }\, d_G\). We also have that \(d_G \in \mathcal {D}\cap G\) and so \(e_G \,{\succcurlyeq }\, d_G\). Together these imply that \(d_G = e_G\).
If x is a prefix of \(d_G\) then it is a simple element and, by Lemma 17, an element of G. Therefore x is an element of \(\mathcal {D}\cap G\). So, by (6), x is a suffix of \(e_G = d_G\). Similarly, every suffix of \(d_G\) is also a prefix of \(d_G\). Therefore \(d_G\) is a balanced element with \({{\mathrm{Div}}}(d_G) = \mathcal {D}\cap G\).
Every element of K can be written as a product of atoms, so, by Lemma 17, G is generated by the atoms of K which lie in G. Every atom in G is clearly in \(\mathcal {D}\cap G\), hence G is generated by the divisors of \(d_G\). Therefore, G is a parabolic submonoid and \(d_G\) is a Garside element.
The same argument with the roles of G and H reversed shows that H is also a parabolic submonoid and that \(d_H = e_H\) is a Garside element. \(\square \)
Remark
The proof of Theorem 34 shows that decomposing a Garside element of a Garside monoid \(K=G\bowtie H\) gives Garside elements for G and H. However, not every pair of Garside elements for the submonoids G and H can be produced this way, as Example 36 shows.
Proposition 35
Let \(K=G\bowtie H\) be a Garside monoid and let \(g\in G\). Then \(\Delta _g^{K} = \bigvee \{ x\backslash g : x \in K \} \in G\).
Proof
For \(x=g_1\vee h_1\) with \(g_1\in G\) and \(h_1\in H\), write \(g_1\vee g = g_1g_2\). Then
with \(g_3 = (g_1^{-1}\blacktriangleright h_1)^{-1} \vartriangleright g_2\), that is, \(x\backslash g = g_3\in G\). As x was arbitrary and G is a parabolic submonoid by Theorem 34, we have \(\Delta _g^{K}\in G\). \(\square \)
Example 36
Consider the monoid \(K = \langle a,b,c \,|\, ab=ba, ac=cb, bc=ca\rangle ^+\) with the submonoids \(G_1 = \langle a \rangle ^+\), \(G_2 = \langle b \rangle ^+\), \(G=\langle a,b \rangle ^+\) and \(H = \langle c \rangle ^+\). (That is, \(G \cong \mathbb {N}_0^2\) and \(K \cong \mathbb {N}_0^2 \rtimes \langle c \rangle ^+\) where the action of c on \(\mathbb {N}_0^2\) is given by swapping the coordinates.) Clearly, the monoid K is a Zappa–Szép product of the submonoids G and H. Moreover, K, G and H are Garside monoids whose minimal Garside elements are \(\Delta _K=abc\), \(\Delta _G=ab\), respectively \(\Delta _H=c\).
-
1.
We see that not every pair of Garside elements for G and H can be obtained by decomposing a Garside element of K: The element \(\Delta '_G=a^2b\) is also a Garside element for the monoid G, yet \(\Delta '_G \Delta _H=a^2bc=cab^2\) is not balanced (and not equal to \(\Delta _H\Delta '_G\)) and so cannot be a Garside element for the monoid K.
-
2.
Moreover, although for \(g\in G\) one has \(\Delta _g^{K} = \bigvee \{ x\backslash g : x \in K \} \in G\) by Proposition 35, in general \(\Delta _g^{K} \ne \Delta _g^{G} = \bigvee \{ x\backslash g : x \in G \}\): For \(x\in G\) one has \(x\backslash a=\mathbf {1}\) if \(a\,{\preccurlyeq }\, x\) and \(x\backslash a=a\) otherwise. Thus, \(\Delta _a^{G}=a\). However, \(c\backslash a=b\), so \(\Delta _a^{K}\ne \Delta _a^{G}\), although both are elements of G. (In fact, \(\Delta _a^{K}=ab=\Delta _G\).)
-
3.
The example also shows that forming Zappa–Szép products is not associative: We have \(K = (G_1 \bowtie G_2) \bowtie H\), but any parabolic submonoid containing both b and c also must contain \(a\,{\preccurlyeq }\, cb\), so it is not true that \(K = G_1 \bowtie (G_2 \bowtie H)\).
Theorem 37
Suppose that \(K = G \bowtie H\) and that G and H are Garside monoids. Then K is a Garside monoid.
Proof
We write \(\Delta _g\) to mean \(\Delta _g^G\) for \(g\in G\) and \(\Delta _h\) to mean \(\Delta _h^H\) for \(h\in H\).
By Lemma 20 and Lemma 12, the monoids act on each other by injections. Let \(\mathcal {A}_G\) and \(\mathcal {A}_H\) denote the sets of atoms of G respectively H. By Lemma 18, the actions act on the sets \(\mathcal {A}_G\) respectively \(\mathcal {A}_H\), so as these sets are finite, the actions are surjective on the sets of atoms, and thus the actions act surjectively on the whole of the submonoids by Lemma 21.
We are in the situation of Lemma 28. For \(g \in G\) and \(h \in H\), Proposition 29 yields that \(gh' = hg'\) is the \({\preccurlyeq }\)-LCM of g and h in K, where \(h' = g^{-1} \blacktriangleright h\) and \(g' = h^{-1} \vartriangleright g\). Moreover, by Theorem 30, the map \((g,h) \mapsto g \vee h\) is a poset isomorphism, hence \((K,\,{\preccurlyeq })\) is a lattice. Likewise, using the map \((g,h) \mapsto g \mathbin {\widetilde{\vee }}h\), one has that \((K,{\succcurlyeq })\) is a lattice.
As G and H are closed under by Lemma 17, the set of atoms of K is \(\mathcal {A}= \mathcal {A}_G\cup \mathcal {A}_H\). As every element of K has a GH-decomposition and both G and H are atomic, K is generated by \(\mathcal {A}_G\cup \mathcal {A}_H=\mathcal {A}\). Suppose \(k=gh\) with \(g\in G\) and \(h\in H\). By Lemma 18, we can rewrite each expression for k as a product of atoms of K as a GH-decomposition without changing its length. Hence \(||k||_{\mathcal {A}} = ||g||_{\mathcal {A}_G} + ||h||_{\mathcal {A}_H} < \infty \), so K is atomic.
Define \(D_G := \bigvee _{a\in \mathcal {A}_G} \Delta _a = \bigvee \big \{ g\backslash a : g\in G,\, a\in \mathcal {A}_G \big \}\). By Proposition 9, we have that \(D_G = \Delta _{\bigvee \mathcal {A}_G}\) is balanced. Moreover, for any \(h\in H\) we have by Proposition 26, Lemma 27 and Lemma 18
and, likewise, \(h^{-1} \vartriangleright D_G \,{\preccurlyeq }\, D_G\). Hence \(h \vartriangleright D_G = D_G = h^{-1} \vartriangleright D_G\) for any \(h\in H\).
Similarly \(D_H := \bigvee _{a\in \mathcal {A}_H} \Delta _a\) is a balanced element satisfying \(g \blacktriangleright D_H = D_H = g^{-1} \blacktriangleright D_H\) for any \(g\in G\). Now define \(D := D_G G_H\). One has \(D = D_G \vee D_H = D_G \mathbin {\widetilde{\vee }}D_H\) by Proposition 29.
To see that D is balanced, let \(g\in G\) and \(h\in H\) and consider \(g'=h^{-1}\vartriangleright g\) and \(h'=g^{-1}\blacktriangleright h\). Using Proposition 26, Theorem 30, the invariance of \(D_G\) under \(h^{-1}\vartriangleright \cdot \) and the invariance of \(D_H\) under \(g^{-1}\blacktriangleright \cdot \), together with the fact that \(D_G\) and \(D_H\) are balanced, one has
Thus, D is a balanced element of K whose divisors include the generating set \(\mathcal {A}_G\cup \mathcal {A}_H\) of K, so D is a Garside element for K. \(\square \)
Remark
Example 36 shows that the construction of a Garside element for the monoid \(K=G\bowtie H\) in the proof of Theorem 37 is needed; in general \(\Delta _G\Delta _H\) need not be a Garside element for K.
We finish this section by giving a characterisation of Garside monoids that can be decomposed as a Zappa–Szép product.
Definition 38
A Garside monoid is \(\bowtie \)-indecomposable, if it cannot be written as a Zappa–Szép product of two non-trivial submonoids.
Remark
The harder direction of the following Theorem 39 is implied by [14, Proposition 4.5] and Corollary 33. We nevertheless give a direct proof, in order to demonstrate that the result does not depend on the notion of crossed products.
Theorem 39
(cf. [14, Proposition 4.5]) A Garside monoid M is \(\Delta \)-pure if and only if it is \(\bowtie \)-indecomposable.
Proof
We write \(\Delta _x\) to mean \(\Delta _x^M\) for any \(x\in M\).
First assume \(M=G\bowtie H\) with non-trivial monoids G and H. Choose two atoms \(g\in G\) and \(h\in H\). By Proposition 35, we have \(\mathbf {1}\ne g{\preccurlyeq }\Delta _g\in G\) and \(\mathbf {1}\ne h{\preccurlyeq }\Delta _h\in H\). As \(G\cap H=\{\mathbf {1}\}\) by uniqueness of GH-decompositions, this implies that \(\Delta _g\ne \Delta _h\), so M is not \(\Delta \)-pure.
Now assume that M is not \(\Delta \)-pure. By Proposition 9, we can partition the set of atoms of M as \(\mathcal {A}= \mathcal {G}\,\dot{\cup }\,\mathcal {H}\), such that \(\Delta _g=\Delta _{g'}\) for \(g,g'\in \mathcal {G}\) and \(\Delta _g\wedge \Delta _h=\mathbf {1}\) for \(g\in \mathcal {G}\), \(h\in \mathcal {H}\). Let \(G := \langle \mathcal {G}\rangle ^+\), \(H := \langle \mathcal {H}\rangle ^+\), \(D_G := \Delta _g\) for some (hence for all) \(g\in \mathcal {G}\), and \(D_H := \bigvee _{h\in \mathcal {H}}\Delta _h = \Delta _{\bigvee \mathcal {H}}\).
Claim
If \(a\in \mathcal {A}\), then iff \(a\in \mathcal {G}\) and iff \(a\in \mathcal {H}\).
If \(g\in \mathcal {G}\) and \(h\in \mathcal {H}\), then one has \(g{\preccurlyeq }\Delta _g=D_G\) and \(h{\preccurlyeq }\Delta _h{\preccurlyeq }D_H\) by Proposition 9. Conversely, again using Proposition 9, one has \(h\wedge D_G{\preccurlyeq }\Delta _h \wedge \Delta _g=\mathbf {1}\), so \(h \not \!\!{\preccurlyeq }D_G\), and finally, \(g{\preccurlyeq }D_H\) would imply \(\Delta _g{\preccurlyeq }\bigvee _{h\in \mathcal {H}}\Delta _h\) by Proposition 9, but the monoid generated by \(\{ \Delta _a : a \in \mathcal {A}\}\) is free abelian and \(g\in \mathcal {A}{\mathbin {{\backslash }}}\mathcal {H}\). The claim then follows with Proposition 9.
Claim
One has \(D_G\mathcal {G}= \mathcal {G}D_G\), \(D_G\mathcal {H}= \mathcal {H}D_G\), \(D_H\mathcal {G}= \mathcal {G}D_H\), and \(D_H\mathcal {H}= \mathcal {H}D_H\).
By Proposition 9, the elements \(D_G\) and \(D_H\) are quasi-central, so one has \(D_G\mathcal {A}= \mathcal {A}D_G\) and \(D_H\mathcal {A}= \mathcal {A}D_H\). Let \(g\in \mathcal {G}\) and \(h\in \mathcal {H}\). As \(x\backslash h{\preccurlyeq }\Delta _h\) holds for any \(x\in M\) by definition of \(\Delta _h\), assuming \(D_G g = h D_G\) implies \(D_G\backslash h{\preccurlyeq }g\wedge \Delta _h=\mathbf {1}\), so \( h{\preccurlyeq }D_G\), which is a contradiction. Similarly, \(D_H h = g D_H\) would imply \(D_H\backslash g{\preccurlyeq }h\wedge \Delta _g=\mathbf {1}\), so \(g{\preccurlyeq }D_H\), which is a contradiction.
Claim
One has \(g\backslash D_H=D_H\) for any \(g \in G\), and \(h\backslash D_G=D_G\) for any \(h \in H\).
First suppose that \(g\in \mathcal {G}\). Then by the previous claim \(g D_H = D_H g'\) for some atom \(g' \in \mathcal {G}\). As \(g \not \!\!{\preccurlyeq }D_H\), we have \(g \vee D_H = D_H x {\preccurlyeq }D_H g'\) for some non-trivial x. Hence \(x {\preccurlyeq }g'\) and so \(x = g'\) and the first claim follows in this case.
One has \(\mathbf {1}\backslash D_H=D_H\) by definition. If \(\mathbf {1}\ne g \in G\), then write \(g=g_1 g_2 \cdots g_k\) with \(g_1,g_2,\ldots g_k\in \mathcal {G}\). From the previous case and Lemma 5 we obtain \((g_1\cdots g_j)\backslash D_H=g_j\backslash \big ((g_1\cdots g_{j-1})\backslash D_H\big )=g_j\backslash D_H=D_H\) for \(j=1,\ldots ,k\) by induction, hence the first claim is shown.
The second claim is shown in the same way.
Claim
If \(g {\preccurlyeq }D_G\) and \(h {\preccurlyeq }D_H\) are non-trivial, then \(h\backslash g\) is a non-trivial prefix of \(D_G\), and \(g\backslash h\) is a non-trivial prefix of \(D_H\).
Using Lemma 5 and the previous claim, we have \(g\backslash h {\preccurlyeq }g\backslash D_H = D_H\). If \(g\backslash h\) were trivial then \(h{\preccurlyeq }h(h\backslash g) = g {\preccurlyeq }D_G\), which is a contradiction.
The claim for \(h\backslash g\) is shown in the same way.
Claim
If \(g {\preccurlyeq }D_G\) and \(h {\preccurlyeq }D_H\), then \(||g||_\mathcal {A}\le ||h\backslash g||_\mathcal {A}\) and \(||h||_\mathcal {A}\le ||g\backslash h||_\mathcal {A}\).
Consider the first inequality. By Lemma 5 it is enough to consider the case where g is an atom and in this case the result follows from the previous claim.
The argument for the second inequality is analogous.
Claim
For atoms \(g \in \mathcal {G}\) and \(h \in \mathcal {H}\), one has \(h \backslash g \in \mathcal {G}\) and \(g \backslash h \in \mathcal {H}\).
Let \(g \vee D_H = g D_H = D_H g'\) and pick \(h_1, \ldots , h_k \in \mathcal {H}\) such that \(D_H = h_1 h_2 \cdots h_k\) and \(h_1 = h\). Let \(g_1 = g\) and \(g_{i+1} = h_i \backslash g_i\) for \(i=1,\ldots ,k\). By the previous claim we have \(||g_1||_\mathcal {A}\le ||g_2||_\mathcal {A}\le \cdots \le ||g_{k+1}||_\mathcal {A}\) and by Lemma 5 we have \(g_{k+1} = g'\). Now, we have seen that \(g'\) is an atom, that is \(||g'||_\mathcal {A}=1\), hence \(||g_i||_\mathcal {A}=1\) for all i. In particular, \(g_2 = g \backslash h\) is an atom.
The argument for \(h\backslash g\) is analogous.
Claim
The map \(g \mapsto h\backslash g\) for fixed \(h\in \mathcal {H}\) is a bijection on \(\mathcal {G}\) and the map \(h \mapsto g\backslash h\) for fixed \(g\in \mathcal {G}\) is a bijection on \(\mathcal {H}\).
Let \(h\in \mathcal {H}\) and assume \(h\backslash g_1=h\backslash g_2\) for \(g_1,g_2\in \mathcal {G}\). As \(h{\preccurlyeq }D_H\), there exists \(\bar{h}\in H\) such that \(h\bar{h}=D_H\). Moreover, as \(g_1\backslash h{\preccurlyeq }D_H\) and \(g_2\backslash h{\preccurlyeq }D_H\), there exist \(h_1\) and \(h_2\in H\) such that \((g_1\backslash h)h_1=D_H\) respectively \((g_2\backslash h)h_2=D_H\). Finally, let \(g'_1,g'_2\in \mathcal {G}\) be such that \(g_1 D_H=D_H g'_1\) and \(g_2 D_H=D_H g'_2\).
As \(g_1 \not \!\!{\preccurlyeq }D_H\), we have \(D_H \vee g_1 = g_1 D_H = D_H g'_1 = h\big ((h\backslash g_1)\vee \bar{h}\big )\). Likewise, \(D_H \vee g_2 = g_2 D_H = D_H g'_2 = h\big ((h\backslash g_2)\vee \bar{h}\big )\), so \(h\backslash g_1=h\backslash g_2\) implies \(g_1=g_2\) (and \(g'_1=g'_2\)), so the map \(g \mapsto h\backslash g\) is injective. As \(\mathcal {G}\) is finite, the map is a bijection.
The argument for the map \(h \mapsto g\backslash h\) for fixed \(g\in \mathcal {G}\) is analogous.
Claim
For \(x\in M\), there exist \(g_1,g_2\in G\) and \(h_1,h_2\in H\) such that one has \(g_1h_1=x=h_2g_2\), that is, GH-decompositions and HG-decompositions exist.
Given \(x\in M\), consider any expression for x as a product of atoms of M. By the previous claim, we can move all atoms in either \(\mathcal {G}\) or in \(\mathcal {H}\) to the front of the word, using identities of the form \(g(g\backslash h) = h(h\backslash g)\) with \(g,(h\backslash g)\in \mathcal {G}\) and \(h,(g\backslash h)\in \mathcal {H}\).
Claim
GH-decompositions and HG-decompositions are unique.
Consider \(g\in G\) and \(h\in H\) and let \(N := ||g||_{\mathcal {G}}<\infty \). Since \(D_G\mathcal {G}= \mathcal {G}D_G\) holds, and as for \(a\in \mathcal {A}\) we have \(a{\preccurlyeq }D_G\) if and only if \(a\in \mathcal {G}\), one has \(g{\preccurlyeq }D_G^N\). Similarly, \(h{\preccurlyeq }D_H^M\) for some M, and thus \(D_G^N\wedge h{\preccurlyeq }D_G^N\wedge D_H^M=\mathbf {1}\). Writing \(D_G^N = g\overline{g}\), we have \(g{\preccurlyeq }D_G^N\wedge (gh) = g(\overline{g}\wedge h) {\preccurlyeq }g(D_G^N\wedge h) = g\), so \(D_G^N\wedge (gh) = g\). Hence, if \(g_1,g_2\in G\) and \(h_1,h_2\in H\) are such that \(g_1h_1=g_2h_2\), then for \(N := \max \{||g_1||_{\mathcal {G}},||g_2||_{\mathcal {G}}\}\) one has \(g_1 = D_G^N\wedge (g_1h_1) = D_G^N\wedge (g_2h_2) = g_2\) and, by cancellativity of M, then \(h_1=h_2\).
Analogously, \(D_G^N\mathbin {\widetilde{\wedge }}(hg) = g\) yields the uniqueness of HG-decompositions.
Thus, one has \(M=G\bowtie H\). \(\square \)
5 Zappa–Szép Garside structures
We have seen that decomposing a Garside element of a Garside monoid \(K=G\bowtie H\) gives Garside elements for the factors, but that not every pair of Garside elements of the factors can be obtained in this way; cf. Example 36. Clearly one cannot hope to relate the normal form in K of an element to the normal forms in G and H of the factors in its GH- and HG-decompositions, unless the Garside elements of K, G and H are related. In light of this remark we make the following definition:
Definition 40
The tuple \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\), if:
-
(a)
K is a Garside monoid (and hence G and H are also Garside monoids);
-
(b)
\(\Delta _K\), \(\Delta _G\), \(\Delta _H\) are Garside elements for K, G, H, respectively; and
-
(c)
\(\Delta _K = \Delta _G\Delta _H\) holds.
Remark
The proof of Theorem 34 shows that in the situation of Definition 40, one has \( \Delta _G\Delta _H = \Delta _K = \Delta _H\Delta _G\).
5.1 Actions and the lattice structures
A Zappa–Szép Garside structure allows to describe the lattice structure of the product in terms of the lattice structures of the factors. By [14, Proposition 3.12], the lattice of simple elements of the product is the product of the lattices of the simple elements of the factors. To be able to describe normal forms in Sect. 5.2, we need to analyse how the actions of the factors on each other interact with the lattice structures; this is the content of this section.
Theorem 41
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\). For all \(g \in G\) and \(h\in H\), one has
where \({\inf }_K\), \({\inf }_G\), \({\inf }_H\) and \({\sup }_K\), \({\sup }_G\), \({\sup }_H\) denote the infimum respectively the supremum with respect to the Garside structures of K, G and H given by \(\Delta _K\), \(\Delta _G\) and \(\Delta _H\), respectively.
Proof
The number \({\inf }_K(g \vee h)\) is the largest integer \(\ell \) such that \(\Delta _K^\ell {\preccurlyeq }g \vee h\). By Therorem 30, this is equal to the largest integer \(\ell \) such that \(\Delta _G^\ell {\preccurlyeq }g\) and \(\Delta _H^\ell {\preccurlyeq }h\), which is the minimum of \({\inf }_G(g)\) and \({\inf }_H(h)\).
Similarly, \({\sup }_K(g \vee h)\) is the smallest integer \(\ell \) such that \(g \vee h {\preccurlyeq }\Delta _K^\ell \). This is equal to the smallest integer \(\ell \) such that \(g {\preccurlyeq }\Delta _G^\ell \) and \(h {\preccurlyeq }\Delta _H^\ell \), which is the maximum of \({\sup }_G(g)\) and \({\sup }_H(h)\). \(\square \)
Lemma 42
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\). For all \(g \in G\), \(h \in H\), the following identities hold:
Proof
As \(\vartriangleright \) is an action, we can assume that h is a simple element. Suppose \(h \Delta _G = g' h'\), i.e. \(g' = h \vartriangleright \Delta _G\). Observe that \((\widetilde{\partial }_H h) h \Delta _G = \Delta _K\), so \(\Delta _K {\succcurlyeq }g' h'\) and, in particular, \(g'\) is a simple element. Also, \(h \vee \Delta _G = h (h^{-1} \vartriangleright \Delta _G)\in \mathcal {D}\) and thus \(h^{-1} \vartriangleright \Delta _G\in \mathcal {D}\). For \(x = (\Delta _G^{-1} \blacktriangleright h) \partial _G (h^{-1} \vartriangleright \Delta _G)\) we have
Hence \(\Delta _G {\preccurlyeq }h \Delta _G = g' h'\) and so \(\Delta _G {\preccurlyeq }g'\) which implies that \(g' = \Delta _G\). The other identities are shown in the same way. \(\square \)
Corollary 43
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\).
For all \(h \in H\), one has
For all \(g \in G\), one has
Proof
The claim follows from Lemma 42 with Lemma 13 and Lemma 25.\(\square \)
Lemma 44
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\).
For all \(g \in \mathcal {D}_G\) and \(h \in H\), one has
For all \(g \in G\) and \(h \in \mathcal {D}_H\), one has
Proof
Consider the following.
Hence \(\partial _G (h \vartriangleright g) = (h \vartriangleleft g) \vartriangleright \partial _G g\).
For the right action we have the following.
So, by Lemma 42, \((h \vartriangleleft g)(g^{-1} \blacktriangleright \partial _H h) = \Delta _H\), i.e. \(\partial _H (h \vartriangleleft g) = (g^{-1} \blacktriangleright \partial _H h)\). The remaining identities follow with Lemma 12. \(\square \)
Lemma 45
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\). Then, for all \(g \in \mathcal {D}_G\) and \(h \in \mathcal {D}_H\), one has
Proof
Suppose that \(g \vee h = gh' = hg'\). Now, using Lemma 42, we have
and, similarly,
Hence
And thus,
\(\square \)
5.2 Normal forms and Zappa–Szép Garside structures
We will show in this section that, with respect to the Garside elements in a Zappa–Szép Garside structure for \(K=G\bowtie H\), the language of normal form words in the product K can be described in terms of the Cartesian product of the languages of normal form words in the factors G and H.
Recall that we write \(x_1|x_2|\cdots |x_\ell \) to denote a word in (non-trivial) simple elements together with the proposition that this word is in left normal form, that we write \(\overline{\mathcal {L}}\) for the language of words in normal form and \(\mathcal {L}\) for the language given by restricting the alphabet to the set of proper simple elements.
Definition 46
The set of equations (3), from Lemma 13, gives us a natural way to extend the actions on elements to actions on strings of elements. We can define the actions recursively as follows: The actions take the empty string to the empty string, act on length one strings by acting on the element, and act on longer strings by
Likewise, if G and H act on each other by bijections, we can extend the inverse actions to strings by
By Lemma 13 and Lemma 25, these actions on strings of elements commute with the multiplication map \(g_1 \mathbin {.}g_2 \mathbin {.}\cdots \mathbin {.}g_\ell \mapsto g_1 g_2 \cdots g_\ell \).
Proposition 47
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\), and let \(g_1, g_2 \in \mathcal {D}_G\) and \(h_1, h_2 \in \mathcal {D}_H\).
Then one has \(\partial _K(g_1 \vee h_1) \wedge (g_2 \vee h_2) = \mathbf {1}\) if and only if \(\partial _G (h_1^{-1} \vartriangleright g_1) \wedge g_2 = \mathbf {1}\) and \(\partial _H (g_1^{-1} \blacktriangleright h_1) \wedge h_2 = \mathbf {1}\).
If, moreover, \(g_2\ne \mathbf {1}\) and \(h_2\ne \mathbf {1}\), then one has \((g_1 \vee h_1) | (g_2 \vee h_2)\) if and only if \((h_1^{-1} \vartriangleright g_1) | g_2\) and \((g_1^{-1} \blacktriangleright h_1) | h_2\).
Proof
By Theorem 30 and Lemma 45, we have
so \(\partial _K(g_1 \vee h_1) \wedge (g_2 \vee h_2) = \mathbf {1}\) if and only if \(\partial _G (h_1^{-1} \vartriangleright g_1) \wedge g_2 = \mathbf {1}\) and \(\partial _H (g_1^{-1} \blacktriangleright h_1) \wedge h_2 = \mathbf {1}\), so the first claim holds.
The second claim follows, as for simple elements \(s_1,s_2\) of any Garside monoid, one has \(s_1|s_2\) if and only if \(\partial s_1\wedge s_2=\mathbf {1}\) and \(s_2\ne \mathbf {1}\) by definition.\(\square \)
Corollary 48
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\), and let \(g_1, g_2 \in \mathcal {D}_G\) and \(h_1, h_2 \in \mathcal {D}_H\).
Then the following hold:
If, moreover, \(g_2\ne \mathbf {1}\) and \(h_2\ne \mathbf {1}\), then one has the following:
Proof
The equivalences in the first list follow by Lemma 16 and Proposition 47 together with the fact that, for all \(g \in G\) and \(h \in H\), one has \(g h = g \vee (g \blacktriangleright h)\) and \(h g = h \vee (h \vartriangleright g)\). The equivalences in the second list then follow with Lemma 15. \(\square \)
Remark
Proposition 47 and Corollary 48 provide explicit translations, in both directions, between a deterministic finite state automaton accepting the regular language of normal form words over the alphabet \(\mathcal {D}_K^*\) and a pair of deterministic finite state automata accepting the regular languages of normal form words over the alphabets \(\mathcal {D}_G^*\) and \(\mathcal {D}_H^*\), respectively.
Proposition 49
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\).
Given the normal form \(g_1 h_1 | \cdots | g_m h_m \in \overline{\mathcal {L}}_K\) of \(k\in K\) with GH-decomposition \(k=gh\), the following algorithm computes the normal forms \(\mathsf {Word}_G \in \overline{\mathcal {L}}_G\) of g and \(\mathsf {Word}_H \in \overline{\mathcal {L}}_H\) of h.
Proof
By Corollary 48, the returned words are in normal form. \(\square \)
Proposition 50
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\), that \(g_1 \mathbin {.}g_2 \mathbin {.}\cdots \mathbin {.}g_\ell \) is a word in \(\mathcal {D}_G^*\) and that \(h \in H\). Define
For \(i=1,\ldots \ell -1\), one has \(\partial _G g_i \wedge g_{i+1} = \mathbf {1}\) if and only if \(\partial _G g'_i \wedge g'_{i+1} = \mathbf {1}\). In other words, \(g_1 | g_2 | \cdots | g_\ell \) if and only if \(g'_1 | g'_2 | \cdots | g'_\ell \).
Proof
First observe that, for \(i=1,\ldots ,\ell \), we have \(g'_i=\mathbf {1}\) if and only if \(g_i=\mathbf {1}\) by Lemma 15.
Now consider the case when \(\ell = 2\). We have: Hence, by Lemma 15, \(\partial _G g'_1 \wedge g'_2 = \mathbf {1}\) if and only if \(\partial _G g_1 \wedge g_2 = \mathbf {1}\). As \(g'_2=\mathbf {1}\) if and only if \(g_2=\mathbf {1}\), we have \(g'_1 | g'_2\) if and only if \(g_1 | g_2\) as desired.
For the general case, if we let \(h'_i = h \vartriangleleft g_1 g_2 \cdots g_{i-1}\) then we have that \(g'_i \mathbin {.}g'_{i+1} = h'_i \vartriangleright (g_i. g_{i+1})\). So each length 2 subword reduces to the \(\ell = 2\) case. \(\square \)
Corollary 51
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\).
The actions on words fix setwise the languages \(\mathcal {L}_G\) and \(\mathcal {L}_H\).
Proof
This follows from Proposition 50 as, by Lemma 42, the initial power of \(\Delta \) in a word in normal form must be preserved by the actions. \(\square \)
Lemma 52
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\).
If \(g_1, g_2, \ldots , g_\ell \in \mathcal {D}_G\) and \(h, h_1, h_2, \ldots , h_\ell \in \mathcal {D}_H\) with \(\partial _H h \wedge (g_1 \blacktriangleright h_1) = \mathbf {1}\) and \(g_1 h_1 | g_2 h_2 | \cdots | g_\ell h_\ell \) hold, then one has \(h g_1 | h_1 g_2 | \cdots | h_{\ell -1} g_\ell \) and, moreover, \(h g_1 | h_1 g_2 | \cdots | h_{\ell -1} g_\ell | h_\ell \) if \(h_\ell \ne \mathbf {1}\).
Proof
If we let \(h_0 = h\) and \(g_{\ell +1} = \mathbf {1}\) then, by Corollary 48, the hypotheses imply
Moreover, either \(g_i\ne \mathbf {1}\) for \(i=1,\ldots ,\ell \), or \(h_i\ne \mathbf {1}\) for \(i=0,1,\ldots ,\ell \).
Now consider the following.
So, by Proposition 50, we have for \(i=1,\ldots ,\ell -1\) that \(\partial _G g_i \wedge (h_i \vartriangleright g_{i+1}) = \mathbf {1}\) and \(\partial _H (h_{i-1} \vartriangleleft g_i) \wedge h_i = \mathbf {1}\), which, using Corollary 48, implies the claim. \(\square \)
Proposition 53
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\).
Given \(g_1|\cdots |g_m \in \overline{\mathcal {L}}_G\) and \(h_1|\cdots |h_n \in \overline{\mathcal {L}}_H\), the following algorithm computes the normal form of \(g_1 \cdots g_m h_1 \cdots h_n\).
Proof
By Lemma 52, in each iteration the word computed in line 7, or 9, is in normal form.
Remark
Proposition 49 and Proposition 53 provide explicit and effective translations, in both directions, between normal forms in an internal Zappa–Szép product and the normal forms in its factors.
As the existence of effectively computable normal forms is the main motivation for Garside theory, these explicit translations are some of the key results of this paper: Proposition 49 and Proposition 53 make it possible to reduce computational questions according to a decomposition of a Garside monoid as a product of simpler constituents.
Proposition 49 and Proposition 53 are essentially the constructive versions of Theorem 54 and Corollary 55.
Theorem 54
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\).
Then the map \(\phi :\!\overline{\mathcal {L}}_G \times \overline{\mathcal {L}}_H \rightarrow \overline{\mathcal {L}}_K\) given by
is a bijection.
Proof
Clearly \(\mathrm {NF}\big (g_1 g_2 \cdots g_m h_1 h_2 \cdots h_n\big ) \in \overline{\mathcal {L}}_K\), so the map \(\phi \) is well-defined.
As each \(k_1|\cdots |k_\ell \in \overline{\mathcal {L}}_K\) is the image of \(\big (\mathrm {NF}(g),\,\mathrm {NF}(h)\big )\), where g h is the GH-decomposition of \(k_1\cdots k_\ell \in K\), the map \(\phi \) is surjective.
Now assume \(g_1|\cdots |g_m\) and \(g'_1|\cdots |g'_p\) in \(\overline{\mathcal {L}}_G\) and \(h_1|\cdots |h_n\) and \(h'_1|\cdots |h'_q\) in \(\overline{\mathcal {L}}_H\) satisfy \(\phi \big (g_1 | \cdots | g_m ,\, h_1 | \cdots | h_n\big ) = \phi \big (g'_1 | \cdots | g'_p ,\, h'_1 | \cdots | h'_q\big )\). Then one has \(g_1 \cdots g_m h_1 \cdots h_n = g'_1 \cdots g'_p h'_1 \cdots h'_q\) and thus, by uniqueness of GH-decompositions, \(g_1 \cdots g_m = g'_1 \cdots g'_p\) and \(h_1 \cdots h_n = h'_1 \cdots h'_q\). Uniqueness of normal forms then yields \(g_1|\cdots |g_m = g'_1|\cdots |g'_p\) and \(h_1|\cdots |h_n = h'_1|\cdots |h'_q\), so the map \(\phi \) is injective. \(\square \)
Remark
The map \(\phi \) can be realised using the algorithm of Proposition 53 and its inverse, \(\phi ^{-1}\), can be realised using the algorithm from Proposition 49.
Corollary 55
Suppose \((\Delta _K, \Delta _G, \Delta _H)\) is a Zappa–Szép Garside structure for the Zappa–Szép product \(K=G\bowtie H\).
Then the map \(\psi :\!\overline{\mathcal {L}}_G \times \overline{\mathcal {L}}_H \rightarrow \overline{\mathcal {L}}_K\) given by
is a bijection.
Proof
By Proposition 50, \(h'_1 \mathbin {.}h'_2 \mathbin {.}\cdots \mathbin {.}h'_n = (g_1 g_2 \cdots g_m)^{-1} \blacktriangleright (h_1 \mathbin {.}h_2 \mathbin {.}\cdots \mathbin {.}h_n)\) is a word in normal form. So, by Proposition 29, \(\psi (g_1 \mathbin {.}g_2 \mathbin {.}\cdots \mathbin {.}g_m, h_1 \mathbin {.}h_2 \mathbin {.}\cdots \mathbin {.}h_n) = \phi (g_1 \mathbin {.}g_2 \mathbin {.}\cdots \mathbin {.}g_m, (g_1 g_2 \cdots g_m)^{-1} \blacktriangleright (h_1 \mathbin {.}h_2 \mathbin {.}\cdots \mathbin {.}h_n))\). Therefore, as \(\psi \) is a composition of bijections, it is a bijection.\(\square \)
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Both authors acknowledge support under Australian Research Council’s Discovery Projects funding scheme (Project Number DP1094072). Volker Gebhardt acknowledges support under the Spanish Projects MTM2010-19355 and MTM2013-44233-P.
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Gebhardt, V., Tawn, S. Zappa–Szép products of Garside monoids. Math. Z. 282, 341–369 (2016). https://doi.org/10.1007/s00209-015-1542-4
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DOI: https://doi.org/10.1007/s00209-015-1542-4