Some pseudovariety joins involvinglocally trivial semigroups

In this paper, we introduce commutative, archimedean, nonpotent Γ-semigroup, Putcha Γ-semigroup and-Γ-semigroup. We prove that a finitely generated, commutative, nonpotent, archimedian Γ-semigroup is power joined. We also prove that if is a commutative, nonpotent, archimedian Γ-semigroup, then, is power joined if and only if every finitely generalized sub-Γ-semigroup of is archimedian. Thereafer, we obtain some equivalent conditions. Then, we prove that an archimedian nonpotent Γ-semigroup is power joined if and only if the structure Γgroup = / of is periodic for some ∈ ; equivalently, it is true for all ∈. We also show that every medial Γsemigroup is a left and right Putcha Γ-semigroup and that every-Γ-semigroup is a right and left Putcha Γsemigroup.

International Journal of Algebra and Computation, 2000

This paper is concerned with the structure of semigroups of implicit operations on various subpseudovarieties V of [Formula: see text] ∩ [Formula: see text], where [Formula: see text] and [Formula: see text] are the pseudovarieties of all semigroups S in which each regular [Formula: see text] -class is, respectively, a rectangular group and a group, and where [Formula: see text] is the pseudovariety of semigroups locally in [Formula: see text] As an application, we give a characterization of the variety of languages recognized by semigroups in V and derive some join decompositions of pseudovarieties.

Communications in Algebra, 2004

In this paper, we show that σ-reducibility is preserved under joins with K, where K is the pseudovariety of semigroups in which idempotents are left zeros. Reducibility of joins with D, the pseudovariety of semigroups in which idempotents are right zeros, is also considered. In this case, we were able to prove that σ-reducibility is preserved for joins with pseudovarieties verifying a certain property of cancellation. As an example involving the semidirect product, we prove that Sl * K is κ-tame, where Sl stands for the pseudovariety of semilattices. plicit operation x → x ω−1. A much more difficult case is the pseudovariety Sl * D, whose κ-tameness was announced by Teixeira in collaboration with the author [11]. Notice that the pseudovarieties K, D [9] and Sl are tame. Let σ be an implicit signature. Proving the σ-tameness of a pseudovariety V involves proving two properties: that the σ-word problem for V is decidable, and that V is σ-reducible. This paper is concerned with the σ-reducibility of pseudovariety joins of the form V ∨ W, where V is one of the pseudovarieties K, D, N or LI, where N and LI are respectively the pseudovarieties of all nilpotent and locally trivial semigroups. These joins were studied by the author in [10], where they were computed for several pseudovarieties V. Here, we prove that σ-reducibility is preserved for joins under K or N; that is, if W is a σ-reducible pseudovariety, then K ∨ W and N ∨ W are also σreducible. We deduce in particular that K ∨ W and N ∨ W are κ-tame for every κ-tame pseudovariety W. The study of the σ-reducibility of joins with D is much more difficult

2015

This paper concerned with basic concepts and some results on (idempotent) semigroup satisfying the identities of three variables. The motivation of taking three for the number of variables has come from the fact that many important identities on idempotent semigroups are written by three or fewer independent variables. We consider the semigroup satisfying the property abc = ac and prove that it is left semi-normal and right quasi-normal. Again an idempotent semigroup with an identity aba = ab and aba = ba (ab = a, ab = b) is always a semilattices and normal. An idempotent semigroup is normal if and only if it is both left quasi-normal and right quasi-normal. If a semigroup is rectangular then it is left and right semi-regular.

Ciric2 Abstract. We study completely …-regular semigroups admitting a de- composition into a semilattice of æn-simple semigroups, and describe them in terms of properties of their idempotents. In the general case, semi- groups admitting a decomposition into a semilattice of æn-simple semi- groups were characterized by M. ´ Ciric and S. Bogdanovic in (3) (see The- orem 1 below), in terms of paths of length n in the graph corresponding to the relation ¡!, and in terms of principal filters and n-radicals. Here we prove that in the completely …-regular case, it suces to consider only those paths of length n starting and/or ending with and idempotent, as well as principal filters and n-radicals generated by idempotents.

Semigroup Forum, 1978

Semigroup Forum, 2013

An algebraic structure is finitely related (has finite degree) if its term functions are determined by some finite set of finitary relations. We show that the following finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semigroups. Further we provide the first example of a semigroup that is not finitely related: the 6-element Brandt monoid.

Semigroup Forum, 2001

We define pq-varieties as classes of finite semigroups closed under the formation of subsemigroups and finitary direct products. Different examples of pq-varieties arise naturally in the study of semigroups of mappings. We consider for these pq-varieties two typical problems of the theory of pseudovarieties.

We show that a regular semigroups satisfying certain conditions in the containing semigroup is closed. As immediate corollaries, we have got that the special semigroup amalgam U = [{S, S }; U ; {i, α | U }] within the class of left [right] quasi-normal orthodox semigroups, R[L]-unipotent semigroups and left[right] Clifford semigroups is embeddable in a left [right] quasi-normal orthodox semigroup, R[L]-unipotent semigroup and left[right] Clifford semigroup respectively. Finally we have shown that the class of all semigroups satisfying the identity xyz = xz and the class of all semigroups satisfying the identity xy = xyx[yx = xyx] are closed within the class of all semigroups satisfying the identities xyz = xz and xy = xyx[yx = xyx] respectively.

International Journal of Algebra and Computation, 2012

In this paper we prove that, if V is a κ-tame pseudovariety which satisfies the pseudoidentity xyω+1z = xyz, then the pseudovariety join LSl ∨ V is also κ-tame. Here, LSl denotes the pseudovariety of local semilattices and κ denotes the implicit signature consisting of the multiplication and the (ω – 1)-power. As a consequence, we deduce that LSl ∨ V is decidable. In particular the joins LSl ∨ Ab, LSl ∨ G, LSl ∨ OCR and LSl ∨ CR are decidable.