Interassociativity
by
Matthew Gould
Vanderbilt University
Coauthors: Sheri J. Boyd, Amy Nelson

Let (S, ·) be a semigroup. An interassociate of (S, ·) is a semigroup (S, *) such that the algebra (S, ·, *) satisfies the equations x ·(y * z) = (x ·y) *z and x*(y ·z) = (x * y)·z. As an obvious example, for fixed a in S the binary operation x*y = xay defines an interassociate of (S, ·); interassociates of this form are called variants of (S, ·).

We develop a method for constructing interassociates by means of translations. This method is exhaustive if (S, *) is a Clifford semigroup (a.k.a. semilattice of groups), and for these semigroups the interassociates are characterized in terms of inflations; in the special case when (S, ·) is a semilattice, every interassociate is an inflation of a P-ideal of (S, ·). (A P-ideal is an ideal whose intersection with each principal ideal is principal.) Interassociates of completely simple and completely 0-simple semigroups are also described; it turns out that every interassociate of a completely simple semigroup is itself completely simple.

It is easily proved that every interassociate of a monoid is a variant thereof, whence it follows that a monoid is isomorphic to each of its interassociates if and only if the monoid is a group. It is also true, but not so easily proved, that if a commutative semigroup containing an idempotent is isomorphic to each of its interassociates, then it is a group. The following regular semigroups have the property of being isomorphic to each interassociate: rectangular bands (semigroups that can be expressed, up to isomorphism, as the direct product of a left zero semigroup and a right zero semigroup), left groups (direct product of a group and a left zero semigroup), and right groups (dually defined). Conversely, we prove that if a regular semigroup (or, more generally, a weakly reductive, globally idempotent semigroup containing an idempotent) is isomorphic to each of its interassociates, then it is a rectangular band, a left group, or a right group; this is analogous to a result of Hickey on variants. It is easily seen that a rectangular band has no interassociates other than itself; we prove that this property characterizes rectangular bands.

Date received: August 5, 1999

Copyright © 1999 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 # cadj-26.