# Action algebras

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### 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 : AB that is a homomorphism: h(xy) = h(x)∨h(y)  and  h(x·y) = h(xh(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.

### Properties

 Classtype variety Equational theory Quasiequational theory undecidable First-order theory undecidable Locally finite no Residual size unbounded Congruence distributive Congruence modular Congruence n-permutable Congruence regular Congruence uniform Congruence extension property Definable principal congruences Equationally definable principal congruences Amalgamation property Strong amalgamation property Epimorphisms are surjective

[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  3
[Size 4]?:  16
[Size 5]?:  149
[Size 6]?:  1488

Action lattices

### Superclasses

Kleene algebras
[Residuated join-semilattices]?

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