Natural partial order induced by a commutative, associative and idempotent function

Abstract

We study the natural partial order of commutative, associative, idempotent functions, covering as a special case t-norms, t-conorms, uninorms and nullnorms on bounded lattices. Unlike other approaches known in the literature, where the order induced by a given function is defined with respect to the original order on the given bounded lattice, we use original ideas of Hartwig, Nambooripad and Mitsch and define the natural partial order on an arbitrary set without connection to any order. We show that the natural partial order induced by any commutative, associative, idempotent function F corresponds to a meet semi-lattice and that F(x,y) can be expressed by the meet of x and y with respect to the natural partial order. This result can be used for an easy verification of the associativity of a commutative, idempotent function. Further, in some special cases we show the necessary and the sufficient conditions for such a function to be non-decreasing in each coordinate. We show that the natural meet semi-lattice connects the ordinal sum in the sense of Clifford and the ordinal sum in the sense of Birkhoff. An example of an ordinal sum of functions (which need not to be idempotent) on a horizontal sum lattice is also introduced.