The factorization and representation of lattices

Author:
George Markowsky

Journal:
Trans. Amer. Math. Soc. **203** (1975), 185-200

MSC:
Primary 06A20

DOI:
https://doi.org/10.1090/S0002-9947-1975-0360386-3

MathSciNet review:
0360386

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Abstract | References | Similar Articles | Additional Information

Abstract: For a complete lattice $L$, in which every element is a join of completely join-irreducibles and a meet of completely meet-irreducibles (we say $L$ is a *jm-lattice*) we define the *poset of irreducibles $P(L)$* to be the poset (of height one) $J(L) \cup M(L)(J(L)$ is the set of completely join-irreducibles and $M(L)$ is the set of completely meet-irreducibles) ordered as follows: $a{ < _{P(L)}}b$ if and only if $a \in J(L),b \in M(L)$, and $a \nleqslant { _L}b$. For a jm-lattice $L$, the automorphism groups of $L$ and $P(L)$ are isomorphic, $L$ can be reconstructed from $P(L)$, and the irreducible factorization of $L$ can be gotten from the components of $P(L)$. In fact, we can give a simple characterization of the center of a jm-lattice in terms of its separators (or unions of connected components of $P(L)$). Thus $P(L)$ extends many of the properties of the poset of join-irreducibles of a finite distributive lattice to the class of all jm-lattices. We characterize those posets of height 1 which are $P(L)$ for some jm-lattice $L$. We also characterize those posets of height 1 which are $P(L)$ for a completely distributive jm-lattice, as well as those posets which are $P(L)$ for some geometric lattice $L$. More generally, if $L$ is a complete lattice, many of the above arguments apply if we use “join-spanning” and “meet-spanning” subsets of $L$, instead of $J(L)$ and $M(L)$. If $L$ is an arbitrary lattice, the same arguments apply to “join-generating” and “meet-generating” subsets of $L$.

- Garrett Birkhoff,
*Lattice theory*, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR**0227053** - Henry H. Crapo and Gian-Carlo Rota,
*On the foundations of combinatorial theory: Combinatorial geometries*, Preliminary edition, The M.I.T. Press, Cambridge, Mass.-London, 1970. MR**0290980**
P. Crawley and R. P. Dilworth, - George Markowsky,
*Some combinatorial aspects of lattice theory*, Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973) Dept. Math., Univ. Houston, Houston, Tex., 1973, pp. 36–68. MR**0396352**
---,

*Algebraic theory of lattices*, Prentice-Hall, Englewood Cliffs, N. J., 1973.

*Combinatorial aspects of lattice theory with applications to the enumeration of free distributive lattices*, Ph. D. Thesis, Harvard University, Cambridge, Mass., June 1973.

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Keywords:
Poset of irreducibles,
completely join-irreducible,
Galois connection,
irreducible factorization,
representations,
group of automorphism,
geometric lattice,
poset of join-irreducibles,
join-spanning set,
distributive lattice,
separators,
center

Article copyright:
© Copyright 1975
American Mathematical Society