Pointlike reducibility of pseudovarieties of the form V*D

In this paper, we investigate the reducibility property of semidirect products of the form V *D relatively to (pointlike) systems of equations of the form x1 =...= xn, where D denotes the pseudovariety of definite semigroups. We establish a connection between pointlike reducibility of V*D and the pointlike reducibility of the pseudovariety V. In particular, for the canonical signature consisting of the multiplication and the (omega-1)-power, we show that V*D is pointlike-reducible when V is pointlike-reducible.European Regional Development Fund, through the programme COMPETEThis work was supported by the European Regional Development Fund, through the programme COMPETE, and by the Portuguese Government through FCT – Fundação para a Ciência e a Tecnologia, under the project PEst-C/MAT/UI0013/2014

Complete reducibility of pseudovarieties

The notion of reducibility for a pseudovariety has been introduced as an abstract property which may be used to prove decidability results for various pseudovariety constructions. This paper is a survey of recent results establishing this and the stronger property of complete reducibility for specific pseudovarieties.FCT through the Centro de Matemática da Universidade do Minho and Centro de Matemática da Universidade do Port

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)

Reducibility of joins involving some locally trivial pseudovarieties

In this paper, we show that sigma-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 sigma-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 k-tame, where Sl stands for the pseudovariety of semilattices.FCT through the Centro de Matemática da Universidade do MinhoEuropean Community Fund FEDE

Tameness of joins involving the pseudovariety of local semilattices

In this paper we prove that, if V is a kappa-tame pseudovariety which satisfies the pseudoidentity xy^{\omega+1}z=xyz, then the pseudovariety join LSl v V is also kappa-tame. Here, LSl denotes the pseudovariety of local semilattices and kappa denotes the implicit signature consisting of the multiplication and the (omega-1)-power. As a consequence, we deduce that LSl v V is decidable. In particular the joins LSl v Ab, LSl v G, LSl v OCR and LSl v CR are decidable.European Science Foundation (ESF) through the programme ``Automata: from Mathematics to Applications (AutoMathA)''Fundação para a Ciência e a Tecnologia (FCT) under the project PEst-C/MAT/UI0013/2011.FCT through the project PTDC/MAT/65481/2006, which was partly funded by the European Community Fund FEDEREuropean Regional Development Fund, through the programme COMPET