Nonassociative relation algebras

Abbreviation: NA

Definition

A nonassociative relation algebra is a structure A = (A,∨,0, ∧,1,¬,o,^∪,e) such that

(A,∨,0, ∧,1,¬) is a Boolean algebra,
e is an identity for o:   xoe = x  and  eox = x,
o is join-preserving:   (x∨y)oz = (xoz)∨(yoz)
^∪ is an involution:   x^∪∪ = x  and  (xoy)^∪ z = y^∪ox^∪
∪ is join-preserving:   (x∨y)^∪ z = x^∪∨y^∪
o is residuated:   x^∪o(¬(xoy)) ≤ ¬y.

Remark:

Morphisms

Let A and B be relation algebras. A morphism from A to B is a function h : A → B that is a Boolean homomorphism and preserves o, ^∪, e: h(xoy) = h(x)oh(y)  and  h(x^∪) = h(x)^∪  and  h(e) = e.

Some results

Examples

Properties

Finite members

Subclasses

Superclasses