E-varieties of locally inverse semigroups and varieties of pseudosemilattices








L. A. Oliveira
University of Porto, Portugal


Marquette University, USA










In the set of idempotents of a locally inverse semigroup $S$ we can define naturally another binary operation $\wedge$. For $e,f\in E(S)$ let $e\wedge f$ be the unique element in the sandwich set $S(f,e)$. Algebras isomorphic to $(E(S),\wedge )$ are called pseudosemilattices and they form a variety $\bf PS$ of idempotent binary algebras. Usually, pseudosemilattices are not semigroups. In fact, a pseudosemilattice which is also a semigroup is a normal band.

For an $e$-variety $\bf V$ of locally inverse semigroups we can consider (up to isomorphisms) the class of all pseudosemilattices ${\bf E_V}=\{(E(S),\wedge )\vert S\in {\bf V}\}$. Auinger showed that ${\bf E_V}$ is always a variety and that the map $\varphi: {\cal L}_e({\bf LI})\rightarrow {\cal L}({\bf PS}), {\bf V}\mapsto {\bf E_V}$ from the lattice of $e$-varieties of locally inverse semigroups to the lattice of varieties of pseudosemilattices is a complete surjective homomorphism.

We will talk about the structure of the lattice ${\cal L}({\bf PS})$. In particular, we will try to adress questions like the joint of varieties, covers, and the cardinality of intervals of this lattice. Due to the homomorphism $\varphi$, some results about the lattice ${\cal L}_e({\bf LI})$ will also be presented.






Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.

 

 Return to the top of the page