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

On logical hierarchies within FO^2-definable languages

We consider the class of languages defined in the 2-variable fragment of the first-order logic of the linear order. Many interesting characterizations of this class are known, as well as the fact that restricting the number of quantifier alternations yields an infinite hierarchy whose levels are varieties of languages (and hence admit an algebraic characterization). Using this algebraic approach, we show that the quantifier alternation hierarchy inside FO^{2}[<] is decidable within one unit. For this purpose, we relate each level of the hierarchy with decidable varieties of languages, which can be defined in terms of iterated deterministic and co-deterministic products. A crucial notion in this process is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle languages of Schwentick, Th\'erien and Vollmer.Comment: arXiv admin note: text overlap with arXiv:0904.289

A coalgebraic perspective on minimization, determinization and behavioural metrics

Coalgebra offers a unified theory of state based systems, including infinite streams, labelled transition systems and deterministic automata. In this paper, we use the coalgebraic view on systems to derive, in a uniform way, abstract procedures for checking behavioural equivalence in coalgebras, which perform (a combination of) minimization and determinization in the system. First, we show that for coalgebras on categories equipped with factorization structures, there exists an abstract procedure for equivalence checking. For instance, when considering epi-mono factorizations in the category of sets and functions, this procedure corresponds to the usual (coalgebraic) minimization algorithm and two states are behaviourally equivalent iff they are mapped to the same state in the minimized coalgebra. Second, motivated by several examples, we consider coalgebras on categories without suitable factorization structures: under certain conditions, it is possible to apply the above procedure after transforming coalgebras with reflections. This transformation can be thought of as some kind of determinization. Finally, we show that the result of the procedure also induces a pseudo-metric on the states, in such a way that the distance between each pair of states is minimized

Visibly Pushdown Languages over Sliding Windows

We investigate the class of visibly pushdown languages in the sliding window model. A sliding window algorithm for a language L receives a stream of symbols and has to decide at each time step whether the suffix of length n belongs to L or not. The window size n is either a fixed number (in the fixed-size model) or can be controlled by an adversary in a limited way (in the variable-size model). The main result of this paper states that for every visibly pushdown language the space complexity in the variable-size sliding window model is either constant, logarithmic or linear in the window size. This extends previous results for regular languages

Varieties of Data Languages

We establish an Eilenberg-type correspondence for data languages, i.e. languages over an infinite alphabet. More precisely, we prove that there is a bijective correspondence between varieties of languages recognized by orbit-finite nominal monoids and pseudovarieties of such monoids. This is the first result of this kind for data languages. Our approach makes use of nominal Stone duality and a recent category theoretic generalization of Birkhoff-type HSP theorems that we instantiate here for the category of nominal sets. In addition, we prove an axiomatic characterization of weak pseudovarieties as those classes of orbit-finite monoids that can be specified by sequences of nominal equations, which provides a nominal version of a classical theorem of Eilenberg and Sch\"utzenberger

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; andČerný&apos;s conjecture for an important class of automata