Strictly Order Primal Algebras

Partial orders and the clones of functions preserving them have been thoroughly studied in recent years. The topic of this papers is strict orders which are irreflexive, asymmetric and transitive subrelations of partial orders. We call an algebra A = (A,Ω) strictly order primal if for some strict order (A,<) the term functions are precisely the functions which preserve this strict order. Our approach has some parallels to the theory of order primal algebras [8], [2]. We present new examples of congruence distributive varieties and of strict orders without near unanimity operations. Then we give a series of new examples showing that there are varieties which are (n + 1)-permutable but not n-permutable. Furthermore the dual category of strict chains is described by the methods from B. Davey and H. Werner [3]. Throughout we use the notations of Grätzer [4] and assume a knowledge of Davey-Werner [3] for the last section.