Uniform decision problems in automatic semigroups

We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which seem to be necessary to make the problem well-defined, the uniform word problem for semigroups described by automatic structures is decidable. Under the same conditions, we show that one can also decide whether the semigroup is completely simple or completely zero-simple; in the case that it is, one can compute a Rees matrix representation for the semigroup, in the form of a Rees matrix together with an automatic structure for its maximal subgroup. On the other hand, we show that it is undecidable in general whether a given element of a given automatic monoid has a right inverse.Comment: 19 page

On the rational subset problem for groups

We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups. We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property. We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decidable for G-automata subsets of M. It follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership.Comment: 19 page

Tropical Roots as Approximations to Eigenvalues of Matrix Polynomials

The tropical roots of txp(x) = max0≤ i≤ℓ ∥Ai∥xi are points at which the maximum is attained for at least two values of i for some x. These roots, which can be computed in only O (ℓ) operations, can be good approximations to the moduli of the eigenvalues of the matrix polynomial P (λ) = Σi=0ℓ λi Ai, in particular when the norms of the matrices Ai vary widely. Our aim is to investigate this observation and its applications. We start by providing annuli defined in terms of the tropical roots of txp (x) that contain the eigenvalues of P (λ). Our localization results yield conditions under which tropical roots offer order of magnitude approximations to the moduli of the eigenvalues of P (λ). Our tropical localization of eigenvalues is less tight than eigenvalue localization results derived from a generalized matrix version of Pellet's theorem but they are easier to interpret. Tropical roots are already used to determine the starting points for matrix polynomial eigensolvers based on scalar polynomial root solvers such as the Ehrlich-Aberth method and our results further justify this choice. Our results provide the basis for analyzing the effect of Gaubert and Sharify's tropical scalings for P (λ) on (a) the conditioning of linearizations of tropically scaled P (λ) and (b) the backward stability of eigensolvers based on linearizations of tropically scaled P (λ). We anticipate that the tropical roots of txp(x), on which the tropical scalings are based will help designing polynomial eigensolvers with better numerical properties than standard algorithms for polynomial eigenvalue problems such as that implemented in the MATLAB function polyeig

ON THE KROHN-RHODES COMPLEXITY OF SEMIGROUPS OF UPPER TRIANGULAR MATRICES

Abstract. We consider the Krohn-Rhodes complexity of certain semigroups of upper triangular matrices over finite fields. We show that for any n> 1 and finite field k, the semigroups of all n × n upper triangular matrices over k and of all n × n unitriangular matrices over k have complexity n − 1. A consequence is that the complexity c> 1 of a finite semigroup places a lower bound of c+1 on the dimension of any faithful triangular representation of that semigroup over a finite field. 1