ACTA issues

Natural dualities: operations and their graphs

Sarah M. Johansen

Acta Sci. Math. (Szeged) 78:1-2(2012), 25-48

Abstract. We investigate the connection between the dualisability of a finite algebra and its graph---the relational structure obtained by replacing each fundamental operation by its graph. We show that if the graph of an algebra is dualisable, then the algebra is also dualisable. The two-element meet semilattice is shown to be a counterexample to the converse. We prove that the graph of every finite algebra with a single unary operation in its type is dualisable. We also show that a duality for each finite directed path, considered as a partial algebra, can be established from a duality for its graph.

AMS Subject Classification (1991): 06D50, 06A06

Keyword(s): natural duality, graph, unar, directed path

Received June 22, 2010, and in revised form March 12, 2011. (Registered under 41/2010.)