Monotone clones and the varieties they determine
A clone C on a set P is a family of finitary functions which contains the projections and is closed under composition. A clone C is called monotone if there is some order \(\leq\) on P such that C is the set of all finitary order-preserving functions with respect to this order. A finite nontrivial algebra P is said to be order-primal if the clone of finitary functions on P is monotone. It is proved that the variety generated by an order-primal algebra is congruence-distributive iff it contains at most two non-isomorphic subdirectly irreducible algebras (Theorem 1.7). Further, part (ii) of Theorem 3.3 states that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories then the corresponding ordered sets or their duals generate that same order variety. The authors also show that every finite bounded ordered set of length 3 has a 5-ary order-preserving near unanimity function (see Theorem 4.4).
scientific article; zbMATH DE number 4214084