365 research outputs found

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

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    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

    An effective characterization of the alternation hierarchy in two-variable logic

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    We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities. This implies decidability of the individual levels. More generally we show that the two-sided semidirect product of a decidable variety with the variety J is decidable

    Convolution, Separation and Concurrency

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    A notion of convolution is presented in the context of formal power series together with lifting constructions characterising algebras of such series, which usually are quantales. A number of examples underpin the universality of these constructions, the most prominent ones being separation logics, where convolution is separating conjunction in an assertion quantale; interval logics, where convolution is the chop operation; and stream interval functions, where convolution is used for analysing the trajectories of dynamical or real-time systems. A Hoare logic is constructed in a generic fashion on the power series quantale, which applies to each of these examples. In many cases, commutative notions of convolution have natural interpretations as concurrency operations.Comment: 39 page

    Eilenberg Theorems for Free

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    Eilenberg-type correspondences, relating varieties of languages (e.g. of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. Numerous such correspondences are known in the literature. We demonstrate that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main contribution is a variety theorem that covers e.g. Wilke's and Pin's work on ∞\infty-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two previous categorical approaches of Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new results, including an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words

    Effective dimension of finite semigroups

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    In this paper we discuss various aspects of the problem of determining the minimal dimension of an injective linear representation of a finite semigroup over a field. We outline some general techniques and results, and apply them to numerous examples.Comment: To appear in J. Pure Appl. Al

    M\"obius Functions and Semigroup Representation Theory II: Character formulas and multiplicities

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    We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of M\"obius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idempotent semigroups and semigroups with basic algebras. Using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers. These include the random walks on chambers of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and Diaconis. Applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups, generalizing and simplifying earlier results of Solomon for the rook monoid.Comment: Some minor typos corrected and references update

    Application of verification techniques to inverse monoids

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    The word problem for inverse monoids generated by a set GammaGamma subject to relations of the form e=fe=f, where ee and ff are both idempotents in the free inverse monoid generated by GammaGamma, is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node xx to a node yy that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. Finally, some results on free partially commutative inverse monoids are presented
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