A coordinatisation of implicative (semi-)lattices by Baer semigroups

SynopsisWe show that if L is a bounded (semi-)lattice and f is a range-closed idempotent residuated mapping on L then f is multiplicative if and only if it is decreasing. We then use this to prove that a bounded (semi-)lattice is implicative if and only if it can be coordinatised by a weakly multiplicative (right) Baer semigroup.

Galois Connections and Pair Algebras

Unless further restricted, P, Q, and R denote arbitrary partially ordered sets whose order relations are all written “≦” .An isotone mapping ϕ: P → Q is said to be residuated if there is an isotone mapping ψ: Q → P such that(RM 1) xϕψ ≧ x for all x i n P;(RM 2) yψϕ ≦ for all y in Q.Let Q* denote the partially ordered set with order relation dual to that of Q.(A) The following conditions are equivalent:(i) ϕ: P → Q* is a Galois connection;(ii) ϕ: P → Q is a residuated mapping;(iii) Max{z ∈ P: zy ≦ y} exists for all y in Q and is equal to yψ.Since ψ is uniquely determined by ϕ, it will be denoted by ϕ+.