Marquette University, USA
In the set of idempotents of a locally inverse semigroup we can define naturally another binary operation . For let be the unique element in the sandwich set . Algebras isomorphic to are called pseudosemilattices and they form a variety of idempotent binary algebras. Usually, pseudosemilattices are not semigroups. In fact, a pseudosemilattice which is also a semigroup is a normal band.
For an -variety of locally inverse semigroups we can consider (up to isomorphisms) the class of all pseudosemilattices . Auinger showed that is always a variety and that the map from the lattice of -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
. 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
, some results about the lattice
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.