We show that the Ehrhart h-vector of an integer Gorenstein polytope with a unimodular triangulation satisfies McMullen’s g-theorem; in particular it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” re...

Given a nite or countably in nite set A, let A∗ be the set of all nite sequences of elements of A, including the empty sequence. We call A an alphabet, the elements of A letters, and the elements of A∗ words. By de ning an associative binary operation on two words by concatenating them, we see that A∗ is a monoid under the operation of concatenation, and we call A∗ the free monoid on A. The com...

In formal language theory, studying shortest strings in languages, and variations thereof, can be useful since these strings can serve as small witnesses for properties of the languages, and can also provide bounds for other problems involving languages. For example, the length of the shortest string accepted by a regular language provides a lower bound on the state complexity of the language. ...

We study various complexity properties of suffix-free regular languages. The quotient complexity of a regular language L is the number of left quotients of L; this is the same as the state complexity of L, which is the number of states in a minimal deterministic finite automaton (DFA) accepting L. A regular language L′ is a dialect of a regular language L if it differs only slightly from L (for...

A nitely presented monoid has a decidable word problem if and only if it can be presented by some left-recursive convergent string-rewriting system if and only if it has a recursive cross-section. However, context-free cross-sections or even regular cross-sections do not suuce. This is shown by presenting examples of nitely presented monoids with decidable word problems that do not admit regula...

A coding partition of a set of words partitions this set into classes such that whenever a sequence, of minimal length, has two distinct factorizations, the words of these factorizations belong to the same class. The canonical coding partition is the finest coding partition that partitions the set of words in at most one unambiguous class and other classes that localize the ambiguities in the f...

In this paper, we establish the Černý-Pin conjecture for automata with the property that their transition monoid cannot recognize the language {a, b}∗ab{a, b}∗. For the subclass of automata whose transition monoids have the property that each regular J -class is a subsemigroup, we give a tight bound on lengths of reset words for synchronizing automata thereby answering a question of Volkov.

In a recent paper we gave a counterexample to a longstanding conjecture concerning the characterization of regular languages of level 2 in the Straubing-Thérien concatenation hierarchy of star-free languages. In that paper a new upper bound for the corresponding pseudovariety of monoids was implicitly given. In this paper we show that it is decidable whether a given monoid belongs to the new up...