Residuated lattices

Abbreviation: RL

Definition

A residuated lattice is a structure L = (L,∨,∧,·,e,\,/) of type (2,2,2,0,2,2) such that

(L,·,e) is a monoid,
(L,∨,∧) is a lattice,
\ is the left residual of ·:   y ≤ x\z  ⇔   xy ≤ z, and
/ is the right residual of ·:   x ≤ z/y  ⇔   xy ≤ z.

Remark:

Morphisms

Let L and M be residuated lattices. A morphism from L to M is a function h : L→M that is a homomorphism: h(x∨y) = h(x)∨h(y)  and  h(x∧y) = h(x)∧h(y)  and  h(x·y) = h(x)·h(y)  and  h(x\y) = h(x)\h(y)  and  h(x/y) = h(x)/h(y)   and  h(e) = e.

Some results

Examples

Properties

Finite members

Subclasses

Superclasses