On Constantive Simple and Order-Primal Algebras

Abstract

A finite algebra A ¼ ðA; F A Þ is said to be order-primal if its clone of all term operations is the set of all operations defined on A which preserve a given partial order e on A. In this paper we study algebraic properties of order-primal algebras for connected ordered sets (A; e). Such order-primal algebras are constantive, simple and have no non-identical automorphisms. We show that in this case F A cannot have only unary fundamental operations or only one at least binary fundamental operation. We prove several properties of the varieties and the quasi-varieties generated by constantive and simple algebras and apply these properties to order-primal algebras. Further, we use the properties of order-primal algebras to formulate new primality criteria for finite algebras. (2000): 08B05, 06A11, 06A06. Mathematics Subject Classification