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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Hypercodes, right convex languages and their syntactic monoids

Author: G. Thierrin
Journal: Proc. Amer. Math. Soc. 83 (1981), 255-258
MSC: Primary 20M35; Secondary 68F05
MathSciNet review: 624909
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Abstract: If $ {X^ * }$ is the free monoid generated by the alphabet $ X$, then any subset $ L$ of $ {X^ * }$ is called a language over $ X$. If $ {P_L}$ is the principal congruence determined by $ L$, then the quotient monoid $ {\text{syn}}(L) = {X^ * }/{P_L}$ is called the syntactic monoid of $ L$. A hypercode over $ X$ is any set of nonemtpy words that are noncomparable with respect to the embedding order of $ {X^ * }$. If $ H$ is a hypercode, then the language $ \tilde H = \{ x\vert x \in {X^ * }$ and $ a \leqslant x$ for some $ a \in H\} $ is a right convex ideal of $ {X^ * }$. The syntactic monoid $ {\text{syn}}(\tilde H)$ can be characterized as a monoid with a disjunctive $ \mu $-zero. The two particular interesting cases when $ {\text{syn}}(\tilde H)$ is a nil monoid and when $ {\text{syn}}(\tilde H)$ is a semillatice are also characterized.

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Keywords: Monoid, language, syntactic monoid, embedding order, hypercode, convex, nil monoid, semilattice
Article copyright: © Copyright 1981 American Mathematical Society

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