Varieties of Restriction Semigroups and Varieties of Categories

The variety of restriction semigroups may be most simply described as that generated from inverse semigroups (S, ·, −1) by forgetting the inverse operation and retaining the two operations x+ = xx−1 and x* = x−1x. The subvariety B of strictrestriction semigroups is that generated by the Brandt semigroups. At the top of its lattice of subvarieties are the two intervals [B2, B2M = B] and [B0, B0M]. Here, B2and B0 are, respectively, generated by the five-element Brandt semigroup and that obtained by removing one of its nonidempotents. The other two varieties are their joins with the variety of all monoids. It is shown here that the interval [B2, B] is isomorphic to the lattice of varieties of categories, as introduced by Tilson in a seminal paper on this topic. Important concepts, such as the local and global varieties associated with monoids, are readily identified under this isomorphism. Two of Tilson\u27s major theorems have natural interpretations and application to the interval [B2, B] and, with modification, to the interval [B0, B0M] that lies below it. Further exploration may lead to applications in the reverse direction

On the power pseudovariety PCS\mathbf{PCS}

Some new semantic and syntactic characterizations of the members of the power pseudovariety $\mathbf{PCS}$ are obtained. This leads in particular to new algorithms for deciding membership in $\mathbf{PCS}$

Semigroups whose idempotents form a subsemigroup

We prove that every semigroup S in which the idempotents form a subsemigroup has an E-unitary cover with the same property. Furthermore, if S is E-dense or orthodox, then its cover can be chosen with the same property. Then we describe the structure of E-unitary dense semigroups. Our results generalize Fountain's results on semigroups in which the idempotents commute, and are analogous to those of Birget, Margolis and Rhodes, and of Jones and Szendrei on finite E-semigroups. ––– Nous montrons que tout semigroupe S dont les idempotents forment un sous-semigroupe admet un revêtement E-unitaire avec la même propriété. De plus, si S est E-dense ou orthodoxe, alors son revêtement peut être choisi de même. Enfin, nous décrivons la structure des semigroupes E-unitaires denses. Nos résultats généralisent ceux de Fountain sur les semigroupes dont les idempotents commutent, et sont analogues à ceux de Birget, Margolis et Rhodes et de Jones et Szendrei sur les E-semigroupes finis. ––– Prova-se que todo o semigrupo S cujos idempotentes formam um subsemigrupo admite uma cobertura E-unitária com a mesma propriedade. Além disso, se S é E-denso ou regular, então a sua cobertura pode ser escolhida como sendo do mesmo tipo. Enfim, descreve-se a estrutura dos semigrupos finitos E-unitários densos. Estes resultados estendem os de Fountain sobre semigrupos cujos idempotentes comutam, e os de Birget, Margolis e Rhodes, e Jones e Szendrei sobre E-semigrupos finitos