On a Product of Finite Monoids

In this paper, for each positive integer m, we associate with a finite monoid S0 and m finite commutative monoids S1,…, Sm, a product &#x25CAm(Sm,…, S1, S0). We give a representation of the free objects in the pseudovariety &#x25CAm(Wm,…, W1, W0) generated by these (m + 1)-ary products where Si &#x2208 Wi for all 0 &#x2264 i &#x2264 m. We then give, in particular, a criterion to determine when an identity holds in &#x25CAm(J1,…, J1, J1) with the help of a version of the Ehrenfeucht-Fraïssé game (J1 denotes the pseudovariety of all semilattice monoids). The union &#x222Am>0&#x25CAm (J1,…, J1, J1) turns out to be the second level of the Straubing’s dot-depth hierarchy of aperiodic monoids

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

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