209 research outputs found
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
Computing relative abelian kernels of finite monoids
Let H be a pseudovariety of
abelian groups corresponding to a recursive supernatural number.
In this note we explain how a concrete implementation of an algorithm to
compute the kernel of a finite monoid relative to H can be achieved.
The case of the pseudovariety Ab of all finite abelian groups was
already treated by the second author and plays an important role here, where we
will be interested in the proper subpseudovarieties of Ab. Our work
relies on an algorithm obtained by Steinberg
Sound and complete axiomatizations of coalgebraic language equivalence
Coalgebras provide a uniform framework to study dynamical systems, including
several types of automata. In this paper, we make use of the coalgebraic view
on systems to investigate, in a uniform way, under which conditions calculi
that are sound and complete with respect to behavioral equivalence can be
extended to a coarser coalgebraic language equivalence, which arises from a
generalised powerset construction that determinises coalgebras. We show that
soundness and completeness are established by proving that expressions modulo
axioms of a calculus form the rational fixpoint of the given type functor. Our
main result is that the rational fixpoint of the functor , where is a
monad describing the branching of the systems (e.g. non-determinism, weights,
probability etc.), has as a quotient the rational fixpoint of the
"determinised" type functor , a lifting of to the category of
-algebras. We apply our framework to the concrete example of weighted
automata, for which we present a new sound and complete calculus for weighted
language equivalence. As a special case, we obtain non-deterministic automata,
where we recover Rabinovich's sound and complete calculus for language
equivalence.Comment: Corrected version of published journal articl
Solvable monoids with commuting idempotents
International Journal of Algebra and Computation, 15, nÂş 3 (2005), p. 547-570The notion of Abelian kernel of a nite monoid extends the notion of derived
subgroup of a nite group. In this line, an extension of the notion of solvable group
to monoids is quite natural: they are the monoids such that the chain of Abelian
kernels ends with the submonoid generated by the idempotents. We prove in this paper that the nite idempotent commuting monoids satisfying this property are precisely those whose subgroups are solvable
Effective dimension of finite semigroups
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
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; andÄŚernĂ˝'s conjecture for an important class of automata
Relative abelian kernels of some classes of transformation monoids
We consider members of some well studied classes of finite transformation
monoids and give descriptions of their abelian kernels relative to
decidable pseudovarieties of abelian groups
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