A right parallelism relation for mappings to posets

Abstract

In this paper, we study mappings f, g : X -> P, where P is a poset and X is a set, under the relation f parallel to g, of right parallelism, f (a) <= f (b) implies g (a) <= g(b), which is reflexive and transitive but not necessarily symmetric. We prove several results of the type: if f has property P and f parallel to g, then g has property P as well, or of the converse type. Doing so permits us to observe several conditions on mappings and/or groupoids (X, *), upon which mappings may act in particular ways, which are of interest in their own right also. The special case f(x) = x with f parallel to g yielding increasing/non-decreasing mappings g: X -> P brings into focus a number of well-known situations seen from a different perspective.