Dualizability And Graph Algebras

. We characterize the finite graph algebras which are dualizable. Indeed, a finite graph algebra is dualizable if and only if each connected component of the underlying graph is either complete or bipartite complete (or a single point). 1. Introduction This paper serves two purposes: it provides a characterization of finite graph algebras which are dualizable, and it elaborates some techniques which promise to be useful in establishing that various finite algebras are not dualizable. Those techniques are also applied herein to sharpen some existing nondualizability results. It turns out that a finite graph algebra is dualizable if and only if each connected component of the underlying graph is either complete or complete bipartite (or a single point). This in turn is known to be equivalent to the graph algebra having a finitely axiomatizable equational theory. Graph algebras were introduced by C. Shallon in her dissertation [26] as a general framework for constructing finite algebras ..

DEFINABLE PRINCIPAL CONGRUENCES AND SOLVABILITY

Abstract. We prove that in a locally finite variety that has definable principal congruences (DPC), solvable congruences are nilpotent, and strongly solvable congruences are strongly abelian. As a corollary of the arguments we obtain that in a congruence modular variety with DPC, every solvable algebra can be decomposed as a direct product of nilpotent algebras of prime power size. 1