Partially ordered set with residuated t-norm

Abstract

We consider properties of partially ordered sets with residuated t-norm and show that 1. If (X;T,0,1) is a bounded partially ordered set with residuated t-norm T, then (X;*,0/sub X/,1/sub X/) is a bounded BCK-algebra with condition (S); 2. Conversely, if (B;*,0/sub B/,1/sub B/) is a bounded BCK-algebra with (S), then (B;T,0,1) is the bounded partially ordered set with residuated t-norm. This means that the class of all bounded partially ordered sets with residuated t-norm coincides with the class of all bounded BCK-algebras with condition (S). Since the class of these algebras forms a variety, the class of partially ordered sets with residuated t-norm is represented by only equations.