ACTA issues

## Kleene algebras: a case-study of clones and dualities from endomorphisms

B. A. Davey, M. Haviar, H. A. Priestley

Acta Sci. Math. (Szeged) 67:1-2(2001), 77-103
2775/2009

 Abstract. A finite endodualisable algebra is always endoprimal, and this fact has led to the discovery of many endoprimal algebras. Recent investigations by the authors have shown that the finite endoprimal algebras in various well-known quasivarieties of algebras having distributive lattice reducts are exactly the finite endodualisable algebras. In the context of semilattices, B. A. Davey and J. G. Pitkethly found examples of finite algebras which are endoprimal but non-endodualisable. The distinction between entailment in the clone-theoretic and duality-theoretic senses was first revealed in the variety $\cal K$ of Kleene algebras. This paper considers the subquasivariety $\cal L$ of $\cal K$ generated by the four-element chain; this contains only fixpoint-free Kleene algebras. The classes of finite endoprimal and endodualisable algebras in $\cal L$ are found. These do not coincide and the way in which this happens leads to a better understanding of the relationship between the two entailment concepts. AMS Subject Classification (1991): 08A35, 08A40, 08C05, 18A40 Keyword(s): Kleene algebra, natural duality, entailment, clone, endodualisable algebra, endoprimal algebra Received November 12, 1999, and in revised form December 6, 2000. (Registered under 2775/2009.)