FG1 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.