Equations on Semidirect Products of Commutative Semigroups

In this paper; we study equations on semidirect products of commutative semigroups. Let Comq,r denote the pseudovariety of all finite semigroups that satisfy the equations xy = yx and xr + q = xr. The pseudovariety Com1,1 is the pseudovariety of all finite semilattices. We consider the product pseudovariety Comq,r * Comq',r' generated by all semidirect products of the form S * T with S &#x2208 Comq,r and T &#x2208 Comq',r'. We give an algorithm to decide when an equation holds in Comq,r * Comq',r'. Finite complete sets of equations are described for all the products Comq,r * Comq',r' which provide polynomial time algorithms to test membership. Our results imply finite complete sets of equations for Gcom * Com1,1 and (Com&#x2229 A) * Com1,1 (among others). Here; Gcom denotes the pseudovariety of all finite commutative groups; Com the pseudovariety of all finite commutative semigroups and A the pseudovariety of all finite aperiodic semigroups

Locally countable pseudovarieties

The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety V (of finite semigroups) for which all pro-V semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressable in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular J-classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality $1\le x^n$ is locally countable if and only if $n=1$

Trees, Congruences and Varieties of Finite Semigroups

A classification scheme for regular languages or finite semigroups was proposed by Pin through tree hierarchies, a scheme related to the concatenation product, an operation on languages, and to the Schützenberger product, an operation on semigroups. Starting with a variety of finite semigroups (or pseudovariety of semigroups) V, a pseudovariety of semigroups &#x25CAu(V) is associated to each tree u. In this paper, starting with the congruence &#x03B3A generating a locally finite pseudovariety of semigroups V for the finite alphabet A, we construct a congruence &#x2261u (&#x03B3A) in such a way to generate &#x25CAu(V) for A. We give partial results on the problem of comparing the congruences &#x2261u (&#x03B3A) or the pseudovarieties &#x25CAu(V). We also propose case studies of associating trees to semidirect or two-sided semidirect products of locally finite pseudovarieties

Locally countable pseudovarieties

The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety V (of finite semigroups) for which all pro-V semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressible in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular J -classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality 1 6 x n is locally countable if and only if n = 1

Trees, congruences and varieties of finite semigroups

AbstractA classification scheme for regular languages or finite semigroups was proposed by Pin through tree hierarchies, a scheme related to the concatenation product, an operation on languages, and to the Schützenberger product, an operation on semigroups. Starting with a variety of finite semigroups (or pseudovariety of semigroups) V, a pseudovariety of semigroups ♦u(V) is associated to each tree u. In this paper, starting with the congruence γA generating a locally finite pseudovariety of semigroups V for the finite alphabet A, we construct a congruence u (γA) in such a way to generate ♦u(V) for A. We give partial results on the problem of comparing the congruences u (γA) or the pseudovarieties ♦u(V). We also propose case studies of associating trees to semidirect or two-sided semidirect products of locally finite pseudovarieties

Semigroup C*-algebras

We give an overview of some recent developments in semigroup C*-algebras.Comment: to appear as a chapter in the Book "K-Theory for Group C*-Algebras and Semigroup C*-Algebras" jointly authored with Joachim Cuntz, Siegfried Echterhoff and Guoliang Y

Tameness of pseudovariety joins involving R

2000 Mathematics Subject Classification: 20M07 (primary); 20M05, 20M35, 68Q70 (secondary).In this paper, we establish several decidability results for pseudovariety joins of the form VvW, where V is a subpseudovariety of J or the pseudovariety R. Here, J (resp. R) denotes the pseudovariety of all J-trivial (resp. R-trivial) semigroups. In particular, we show that the pseudovariety VvW is (completely) kappa-tame when V is a subpseudovariety of J with decidable kappa-word problem and W is (completely) kappa-tame. Moreover, if W is a kappa-tame pseudovariety which satisfies the pseudoidentity x_1...x_ry^{\omega+1}zt^\omega = x_1... x_ryzt^\omega, then we prove that RvW is also kappa-tame. In particular the joins RvAb, RvG, RvOCR, and RvCR are decidable.União Europeia (UE). Fundo Europeu de Desenvolvimento Regional (FEDER) - POCTI/32817/MAT/2000.International Association for the Promotion of Co-operation with Scientists from the New Independent States (NIS) of the Former Soviet Union (INTAS) - project 99-1224.Fundação para a Ciência e a Tecnologia (FCT)

Matrix semigroups over semirings

The multiplicative semigroup $M_n(F)$ of $n\times n$ matrices over a field $F$ is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form $M_n(S)$, where $S$ is a semiring, and the subsemigroups $UT_n(S)$ and $U_n(S)$ of $M_n(S)$ consisting of upper triangular and unitriangular matrices. Our main interest is in the case where $S$ is an idempotent semifield, where we also consider the subsemigroups $UT_n(S^*)$ and $U_n(S^*)$ consisting of those matrices of $UT_n(S)$ and $U_n(S)$ having all elements on and above the leading diagonal non-zero. Our guiding examples of such $S$ are the 2-element Boolean semiring $\mathbb{B}$ and the tropical semiring $\mathbb{T}$. In the first case, $M_n(\mathbb{B})$ is isomorphic to the semigroup of binary relations on an $n$-element set, and in the second, $M_n(\mathbb{T})$ is the semigroup of $n\times n$ tropical matrices. Il'in has proved that for any semiring $R$ and $n>2$, the semigroup $M_n(R)$ is regular if and only if $R$ is a regular ring. We therefore base our investigations for $M_n(S)$ and its subsemigroups on the analogous but weaker concept of being Fountain (formerly, weakly abundant). These notions are determined by the existence and behaviour of idempotent left and right identities for elements, lying in particular equivalence classes. We show that certain subsemigroups of $M_n(S)$, including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We give a detailed study of a family of Fountain semigroups arising in this way that has particularly interesting and unusual properties.Comment: 50 page

Two algebraic approaches to variants of the concatenation product

AbstractWe extend an existing approach of the bideterministic concatenation product of languages aiming at the study of three other variants: unambiguous, left deterministic and right deterministic. Such an approach is based on monoid expansions. The proofs are purely algebraic and use another approach, based on properties on the kernel category of a monoid relational morphism, without going through the languages. This gives a unified fashion to deal with all these variants and allows us to better understand the connections between these two approaches. Finally, we show that local finiteness of an M-variety is transferred to the M-varieties corresponding to these variants and apply the general results to the M-variety of idempotent and commutative monoids