The structure of the endomorphism monoid of a stable basis algebra A is described. It is shown to be an abundant monoid; the subsemigroup of endomorphisms of finite rank has a regular semigroup of left quotients. This is a revised version of the article that appeared in Algebra Universalis 51 (2004), 257− −285. The authors are grateful to Professor George Bergman for correcting their original d...

We examine the computational complexity of some problems from algebraic automata theory and from the field of communication complexity: testing Green’s relations (relations that are fundamental in monoid theory), checking the property of a finite monoid to have only Abelian subgroups, and determining the deterministic communication complexity of a regular language. By well-known algebraizations...

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the maximal syntactic complexity of languages in that subclass, taken as a function of the state complexity n of these languages. We study the syntactic complexity of six subclasses of star-free languages. We find a tight upper bound of (n−...

Balanced grammars are a generalization of parenthesis grammars in two directions. First, several kind of parentheses are allowed. Next, the set of right-hand sides of productions may be an infinite regular language. XML-grammars are a special kind of balanced grammars. This paper studies balanced grammars and their languages. It is shown that there exists a unique minimal balanced grammar equiv...

This paper provides an algebraic characterization of the Strictly Piecewise class of languages studied by Rogers et al. 2010. These language are a natural subclass of the Piecewise Testable languages (Simon 1975) and are relevant to natural language. The algebraic characterization highlights a similarity between the Strictly Piecewise and Strictly Local languages, and also leads to a procedure ...

We prove that for every finite monoid M , there exists a finite language L such that M divides the syntactic monoid of L∗. Moreover, one can choose for L a full finite prefix code. The same result for finite group has already been proved by Schützenberger. This result is crucial in the proof of the J.-F. Perrot’s theorem that the only variety closed under star operation is the variety of all ra...

We investigate the duality between algebraic and coalgebraic recognition of languages to derive a generalization of the local version of Eilenberg’s theorem. This theorem states that the lattice of all boolean algebras of regular languages over an alphabetΣ closed under derivatives is isomorphic to the lattice of all pseudovarieties ofΣ-generated monoids. By applying our method to different cat...

The problem of deciding whether a monoid that is given through a nite presentation is a regular monoid as well as two closely related decision problems are considered. Since these problems are undecidable in general, they are investigated for certain restricted classes of nite presentations. Some new decidability results as well as some new undecidability results are obtained, which show that f...

The aim of this paper is to study semigroups possessing $E$-regular elements, where an element $a$ of a semigroup $S$ is {em $E$-regular} if $a$ has an inverse $a^circ$ such that $aa^circ,a^circ a$ lie in $ Esubseteq E(S)$. Where $S$ possesses `enough' (in a precisely defined way) $E$-regular elements, analogues of Green's lemmas and even of Green's theorem hold, where Green's relations ${mathc...