# Algebra Seminar talk

2005-06-17

Brian **Davey***Natural dualities for semilattice-based algebras*

Abstract:

An algebra is *semilattice based* if included amongst its
fundamental operations is a semilattice operation. Semilattice-based
algebras have become increasingly important in the study of general
algebra. This has been particularly so following the
ground-breaking undecidability results of R. McKenzie (see references).

In this talk I shall report on recent joint work with Marcel Jackson and Rashed Talukder on the dualisability of semilattice-based algebras. The talk will assume no familiarity with the theory of natural dualities (see [Clark and Davey 1998] and [Pitkethly and Davey 2005] and will begin with a gentle introduction to the topic.

It has long been known that every finite lattice-based algebra is dualisable. Indeed, this is one of the reasons that the theory of natural dualities has been so useful in algebraic logic. However, not every semilattice-based algebra is dualisable. For example, [Davey and Pitkethly 2001] proved that a finite pseudocomplemented semilattice is dualisable if and only if it is boolean. We prove that a semilattice-based algebra, all of whose operations are compatible with its semi-lattice operation, is dualisable.

Our theorem yields a large class of dualisable semilattice-based algebras. It has as a corollary the result of [Lampe, McNulty and Willard 2001] that a flat graph algebra is dualisable provided it is entropic.

In quite the opposite direction, we show that, in the class of flat algebras associated with groups, every finite algebra with more than two elements is inherently non-dualisable.

The problem of describing the dualisable finite semilattice-based algebras would appear to be very difficult.

### References

- D. M. Clark and B. A. Davey
- ''Natural Dualities for the Working Algebraist'', Cambridge University Press, Cambridge, 1998.
- B. A. Davey and J. G. Pitkethly
- Dualisability of p-semilattices, ''Algebra Universalis'' 45 (2001), 149-153.
- W. A. Lampe, G. F. McNulty and R. Willard
- Full duality among graph algebras and flat graph algebras, ''Algebra Universalis'' 45 (2001), 311-334.
- R. McKenzie
- The residual bounds of finite algebras, ''Internat. J. Algebra Comput.'' 6 (1996), 1-28.
- R. McKenzie
- The residual bound of a finite algebra is not computable, ''Internat. J. Algebra Comput.'' 6 (1996), 29-48.
- R. McKenzie
- Tarski's finite basis problem is undecidable, ''Internat. J. Algebra Comput.'' 6 (1996), 49-104.
- J. G. Pitkethly and B. A. Davey
- ''Dualisability: Unary Algebras and Beyond'', Advances in Mathematics, Springer, 2005.