Ordered sets: Retracts and connections

Abstract

The purpose of this paper is to introduce the concept of a connection between partially ordered sets, as a natural generalization of the concept of a Galois connection [12,13]. In the present theory, retraction operators (idempotent maps) will play the role of closure operators. We shall show how all the salient features of the theory of Galois connections are preserved, while the scope and applicability of the theory are significantly enhanced. The basic definitions are as Jollows: We use the term ordered set to mean partially-ordered set. A map AB from an ordered set A to an/ordered set B is any order-preserving function from A to B. The composite A-Bg Cof two yaps we write fg, in the order of composition from left to right. A retraction A-B is a map for which there exists a left inverse (called a corerracftion) A LB, such that gf is the identity map eE on the ordered set B. If A-B is a retraction, then B is a retract of the ordered set A. The other composite fg is then an idempotent map on the ordered set A, a retraction operator on A. A connection between two ordered sets A, B is a pair of maps