Action algebras

Abbreviation: Act

Definition

An action algebra is a structure A = (A,∨,0,·,1,^*,\,/) of type (2,0,2,0,1,2,2) such that
(A,∨,0,·,1,^*) is a Kleene algebra,
\ is the left residual of ·:   y ≤ x\z  ⇔   xy ≤ z, and
/ is the right residual of ·:   x ≤ z/y  ⇔   xy ≤ z.

Remark:

Morphisms

Let A and B be action algebras. A morphism from A to B is a function h : A→B 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^*) = h(x)^*  and  h(0) = 0  and  h(1) = 1.

Some results

Examples

Properties

Finite members

Subclasses

Superclasses