Geometrical and logical invariants of algebras and varieties
by
Boris Plotkin
Hebrew University, Jerusalem

We consider algebras in a variety of algebras £. A well known invariant of every algebra H in £ is its elementary theory Th(H). Two algebras H1 and H2 are elementary equivalent if Th(H1) = Th(H2) (the notion defined by A.Tarski). We introduce a more strong notion of logically geometrical equivalence of two algebras (LG-equivalence). This LG-equivalence implies elementary equivalence, but not vice versa. In the talk we consider problems related to the notion of LG-equivalence of algebras. In particular, let us mention the following one:

Let £ be an arbitrary variety of algebras, W = W(X) a free algebra in this variety with the finite set X. We say that this algebra W is LG-separable in £, if any other algebra H, LG-equivalent to W, is isomorphic to W. It is proved that this property holds for free semigroups and free inverse semigroups. A study of other interesting cases is in progress.

We consider also other problems related to geometrical and logical invariants of algebras and varieties.

Date received: February 24, 2008