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

Piecewise testable tree languages

This paper presents a decidable characterization of tree languages that can be defined by a boolean combination of Sigma_1 sentences. This is a tree extension of the Simon theorem, which says that a string language can be defined by a boolean combination of Sigma_1 sentences if and only if its syntactic monoid is J-trivial

On shuffle products, acyclic automata and piecewise-testable languages

We show that the shuffle L \unicode{x29E2} F of a piecewise-testable language $L$ and a finite language $F$ is piecewise-testable. The proof relies on a classic but little-used automata-theoretic characterization of piecewise-testable languages. We also discuss some mild generalizations of the main result, and provide bounds on the piecewise complexity of L \unicode{x29E2} F

Wreath Products of Forest Algebras, with Applications to Tree Logics

We use the recently developed theory of forest algebras to find algebraic characterizations of the languages of unranked trees and forests definable in various logics. These include the temporal logics CTL and EF, and first-order logic over the ancestor relation. While the characterizations are in general non-effective, we are able to use them to formulate necessary conditions for definability and provide new proofs that a number of languages are not definable in these logics

Syntactic Monoids in a Category

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is an entropic variety of algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in case the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids