On varieties of meet automata

AbstractEilenberg’s variety theorem gives a bijective correspondence between varieties of languages and varieties of finite monoids. The second author gave a similar relation between conjunctive varieties of languages and varieties of semiring homomorphisms. In this paper, we add a third component to this result by considering varieties of meet automata. We consider three significant classes of languages, two of them consisting of reversible languages. We present conditions on meet automata and identities for semiring homomorphisms for their characterization

On Varieties of Automata Enriched with an Algebraic Structure (Extended Abstract)

Eilenberg correspondence, based on the concept of syntactic monoids, relates varieties of regular languages with pseudovarieties of finite monoids. Various modifications of this correspondence related more general classes of regular languages with classes of more complex algebraic objects. Such generalized varieties also have natural counterparts formed by classes of finite automata equipped with a certain additional algebraic structure. In this survey, we overview several variants of such varieties of enriched automata.Comment: In Proceedings AFL 2014, arXiv:1405.527

Varieties of Languages in a Category

Eilenberg's variety theorem, a centerpiece of algebraic automata theory, establishes a bijective correspondence between varieties of languages and pseudovarieties of monoids. In the present paper this result is generalized to an abstract pair of algebraic categories: we introduce varieties of languages in a category C, and prove that they correspond to pseudovarieties of monoids in a closed monoidal category D, provided that C and D are dual on the level of finite objects. By suitable choices of these categories our result uniformly covers Eilenberg's theorem and three variants due to Pin, Polak and Reutenauer, respectively, and yields new Eilenberg-type correspondences

Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory

In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and \v{C}ern\'y's conjecture for an important class of automata

F-sets and finite automata

The classical notion of a k-automatic subset of the natural numbers is here extended to that of an F-automatic subset of an arbitrary finitely generated abelian group $\Gamma$ equipped with an arbitrary endomorphism F. This is applied to the isotrivial positive characteristic Mordell-Lang context where F is the Frobenius action on a commutative algebraic group G over a finite field, and $\Gamma$ is a finitely generated F-invariant subgroup of G. It is shown that the F-subsets of $\Gamma$ introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X is a closed subvariety then X intersect $\Gamma$ is F-automatic. Derksen's notion of a k-normal subset of the natural numbers is also here extended to the above abstract setting, and it is shown that F-subsets are F-normal. In particular, the X intersect $\Gamma$ appearing in the Mordell-Lang problem are F-normal. This generalises Derksen's Skolem-Mahler-Lech theorem to the Mordell-Lang context.Comment: The final section is revised following an error discovered by Christopher Hawthorne; it is no longer claimed that an F-normal subset has a finite symmetric difference with an F-subset. The main theorems of the paper remain unchange