Residuation in commutative ordered monoids with minimal zero
James Raftery, Clint J. van Alten · Reports on Mathematical Logic · 2000
A commutative pomonoid is a structure A = 〈A;⊕, 0;≤〉, whose reduct 〈A;⊕, 0〉 is a commutative monoid where ≤ is a partial order of A for which ⊕ is isotone in both of its arguments. We call A residuated provided that for any x, y ∈ A there is a least z ∈ A such that x ≤ z ⊕ y, in which case this z is denoted by x . − y and the binary operation . − on A is called residuation. In particular, such structures A satisfy x ≤ y ⇔ x . − y ≤ 0. The abstract study of such pomonoids was inspired by the ideal lattices of commutative unital rings, with ideal multiplication, reversed set inclusion and the ring itself in the roles of ⊕, ≤ and 0. Here, although (unary) inverses are absent, residuation supplies a binary operation abstracting division. More recently, residuated commutative pomonoids in their full generality have received attention as natural models for fragments of lin-