Benjamin Steinberg

Möbius functions and semigroup representation theory

This paper explores several applications of Mobius functions to the representation theory of finite semigroups. We extend Solomons approach to the semigroup algebra of a finite semilattice via Mobius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix algebras over group rings. We also extend work of Bidigare, Hanlon, Rockmore and Brown on calculating eigenvalues of random walks associated to certain classes of finite semigroups; again Mobius functions play an important role.

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 Cernys conjecture for an important class of automata.

ON A CLASS OF AUTOMATA GROUPS GENERALIZING LAMPLIGHTER GROUPS

We study automata groups generated by reset automata. Every lamplighter group ℤ/nℤ wr ℤ can be generated by such an automaton, and in general these automata groups are similar in nature to lamplighters: they are amenable locally-finite-by-cyclic groups; under mild decidable conditions, the semigroups generated by such automata are free. Parabolic subgroups and fractal properties are considered.

On the Decidability of Iterated Semidirect Products with Applications to Complexity

The notion of hyperdecidability has been introduced by the rst author as a tool to prove decidability of semidirect products of pseudovarieties of semigroups. In this paper we consider some stronger notions which lead to improved decidability results allowing us in turn to establish the decidability of some iterated semidirect products. Roughly speaking, the decidability of a semidirect product follows from a mild, commonly veriied property of the rst factor plus the stronger property for all the other factors. A key role in this study is played by intermediate free semigroups (relatively free objects of expanded type lying between relatively free and relatively free proonite objects). As an application of the main results, the decidability of the Krohn-Rhodes (group) complexity is shown to follow from non-algorithmic abstract properties likely to be satissed by the pseudovariety of all nite aperiodic semigroups, thereby suggesting a new approach to answer (aarmatively) the question as to whether complexity is decidable.

On Pointlike Sets and Joins of Pseudovarieties

In this paper we give a criterion for a subset of a semigroup to be pointlike for the join or wreath product of two pseudovarieties. As a consequence we obtain a criterion for membership in the join of two pseudovarieties. Applications include pointlikes for J ∨ G.

On the irreducible representations of a finite semigroup

Work of Clifford, Munn and Ponizovskii parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Reess theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovskii result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.

Finite state automata: A geometric approach

Recently, finite state automata, via the advent of hyperbolic and automatic groups, have become a powerful tool in geometric group theory. This paper develops a geometric approach to automata theory, analogous to various techniques used in combinatorial group theory, to solve various problems on the overlap between group theory and monoid theory. For instance, we give a geometric algorithm for computing the closure of a rational language in the profinite topology of a free group. We introduce some geometric notions for automata and show that certain important classes of monoids can be described in terms of the geometry of their Cayley graphs. A long standing open question, to which the answer was only known in the simplest of cases (and even then was non-trivial), is whether it is true, for a pseudovariety of groups H, that a J -trivial co-extension of a group in H must divide a semidirect product of a J -trivial monoid and a group in H. We show the answer is affirmative if H is closed under extension, and may be negative otherwise.

Synchronizing groups and automata

Pin showed that every p-state automaton (p a prime) containing a cyclic permutation and a non-permutation has a synchronizing word of length at most (p - 1)2. In this paper, we consider permutation automata with the property that adding any nonpermutation will lead to a synchronizing word and establish bounds on the lengths of such synchronizing words. In particular, we show that permutation groups whose permutation character over the rationals splits into a sum of only two irreducible characters have the desired property.

INEVITABLE GRAPHS AND PROFINITE TOPOLOGIES: SOME SOLUTIONS TO ALGORITHMIC PROBLEMS IN MONOID AND AUTOMATA THEORY, STEMMING FROM GROUP THEORY

This paper deals with several algorithmic problems in monoid and automata theory arising from group theory. For H a pseudovariety of groups, we give a characterization of the regular elements of the H-kernel of a finite monoid. In particular, we show that if the extension problem for partial one-to-one maps for H is decidable, then so is the set of regular elements of the H-kernel. The extension problem for partial one-to-one maps for H asks if there is an algorithm to determine, given a finite set X and a set S of partial one-to-one maps on X, whether there is a finite set Y containing X so that each of the maps of S can be extended to permutations of Y in such a manner that the group generated by these permutations is in H. This problem is decidable for the pseudovariety of p-groups and nilpotent groups. We explore some other examples here. We also show that if the above problem is decidable, then so is the membership problem for JⓜH. Some applications to the membership problem for J*H are given. Finally, we show that certain pseudovarieties of groups, including the pseudovarieties of p-groups for p prime, are hyperdecidable. The techniques used here lay the groundwork for several future results on problems of this nature.

Syntactic and Global Semigroup Theory: A Synthesis Approach

This paper is the culmination of a series of work integrating syntactic and global semigroup theoretical approaches for the purpose of calculating semidirect products of pseudovarieties of semigroups. We introduce various abstract and algorithmic properties that a pseudovariety of semigroups might possibly satisfy. The main theorem states that given a finite collection of pseudovarieties, each satisfying certain properties of the sort alluded to above, any iterated semidirect product of these pseudovarieties is decidable. In particular, the pseudovariety G of finite groups satisfies these properties. J. Rhodes has announced a proof, in collaboration with J. McCammond, that the pseudovariety A of finite aperiodic semigroups satisfies these properties as well. Thus, our main theorem would imply the decidability of the complexity of a finite semigroup. Their work, in light of our main theorem, would imply the decidability of the complexity of a finite semigroup.