How finite is a three-element unary algebra?
by
Jennifer Hyndman
University of Northern British Columbia
Coauthors: Jane Pitkethly (La Trobe University)

Within the class of three-element unary algebras there is a tight connection between several nice finiteness properties. In 1989, Bestsennyi introduced three special three-element unary algebras, V, L, and D.

Theorem

Let M be a three-element unary algebra. Then the following are equivalent.

1. M does not have V, L or D as a reduct;
2. the quasi-equational theory of M is finitely based;
3. M has finite rank;
4. M has enough algebraic operations;
5. M is quasi-injective.

Bestsennyi  has already proven that conditions (1) and (2) are equivalent, and Lampe, McNulty and Willard  have shown that (4) implies (3) in general.

Although the properties of finite rank and enough algebraic operations come from duality theory, the equivalences in this theorem are independent of dualisability. There are several three-element unary algebras that have enough algebraic operations, but are not dualisable. For those algebras that are dualisable, known results from duality theory were used to establish the equivalences.

Date received: December 30, 2001