Some Results on the Lattice of Varieties of Involution Semigroups
by
Igor Dolinka
Institute of Mathematics, University of Novi Sad, Novi Sad, Yugoslavia

An involution semigroup is a pair (S, ^* ), where S is a semigroup, while ^* is an involutiorial antiautomorphism of S. In other words, the identities (xy) ^* =y ^* x ^* and (x ^* ) ^* =x are satisfied. The study of involution semigroups from the standpoint of the theory of varieties was initiated by Nordahl and Scheiblich in 1978, and since then, the topic was growing, especially with the contributions of Fajtlowicz, Adair, Petrich and many others.

One of the standard ways for producing involution semigroups from ordinary ones is the construction of the 0-direct sum of a semigroup with its dual. We discuss the equational properties of the involution in these sums. Suprisingly, only two identities (and always the same ones) suffice to give a complete relative axiomatization for such involution semigroups over the equational theory of the underlying semigroup.

Further, inspired by the results of Fajtlowicz (who gave a complete list of minimal varieties of involution semigroups), we present the part of the lattice of varieties of involution semigroups generated by its minimal members.

Finally, we focus our attention to the lattice of varieties of involution bands. We describe some lower parts of this lattice, generalizing a result of Adair from 1982. In particular, we obtain the full description of all varieties of normal bands with involution (there are 14 such varieties, while the variety of ordinary normal bands has only 8 subvarieties). It turns out that the lattice of varieties of involution bands has a much more complex structure than the corresponding lattice for ordinary bands. For example, in contrast with the latter one, the lattice of varieties of involution bands has no finite width. Also, we give a characterization of the subdirectly irreducible bands with involution, and point out explicitely the subdirectly irreducible normal bands with involution.

Date received: May 5, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-17.