Connections and Polarities

Galois connections - also called residuated/residual maps - have been studied and used extensively in both mathematics and computer science. In the 1980s, Galois connections were generalized to connections, and in the 1990s, a counterpart to Galois connections, called Lagois connections, were discovered. Lagois connections, as the name suggests, are similar to Galois connections; however, the “movement” from arbitrary points to image points - which is done via the composite maps of the connections - is in the same “direction” with respect to the order relations in both partially ordered sets. This similarly directed movement seems to be characteristic of (many) computer science applications. In this chapter we present these concepts in their “discovery” order; we also give properties and examples. We begin with a pre-Galois connection concept, called a polarity.