Semigroup varieties closed for certain extensions
Abstract
Let F: S $\to F(S)$ associate the semigroup F(S) to any given semigroup S. We say that the subvariety W of the variety S of all semigroups is closed for F if for every $S \in$ W we have F(S) $\in$ W. The F-closed semigroup varieties then form a complete lattice $FL\sb{v}$(S), but not necessarily a complete sublattice of the lattice $L\sb{v}$(S) of semigroup varieties. This thesis consists of two parts. In the first part we view F as the object mapping of a functor F: C$\sb{\bf S} \to$ C$\sb{\bf S}$, where C$\sb{\bf S}$ is the category whose objects are semigroups and whose morphisms are onto homomorphisms. A sufficient condition for F is found so that $FL\sb{v}$(S) is a complete sublattice of $L\sb{v}$(S). Two well-known functors satisfy this condition: the functor B, where B(S) is the Bruck extension of S for every semigroup S, and the functor I, where I(S) is the idempotent generated extension of S. We find an antichain of the cardinality of the continuum and a chain isomorphic to the chain of real numbers in $BL\sb{v}$(S), and we also give some interesting properties of the smallest variety closed for the Bruck extension. The second part of the thesis is about the power extension P, where for every semigroup S, P(S) is the power semigroup of S. The complete lattice $PL\sb{v}$(S) is not a complete sublattice of $L\sb{v}$(S). We proved that $PL\sb{v}$(S) is isomorphic to an interval of $L\sb{v}$(S) with greatest element the variety Var$(x\sb1 x\sb2 x\sb1$ = 0 = $x\sbsp{1}{2}).$ For each semigroup variety V, PV is the semigroup variety determined by the permutation identities which hold in V. Therefore, in order to study $PL\sb{v}$(S), we need to investigate which permutation identities can be consequences of a given set of identities. As a consequence of these investigations we find, for every positive integer $n \ge$ 2, a class of varieties minimal for not satisfying any permutation identity in n variables.
This paper has been withdrawn.