# Action lattices

HomePage | RecentChanges | Preferences

### Definition

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

### Morphisms

Let A and B be action lattices. A morphism from A to B is a function h : AB that is a homomorphism: h(xy) = h(x)∨h(y)  and  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 open Quasiequational theory undecidable First-order theory undecidable Locally finite no Residual size unbounded Congruence distributive yes Congruence modular yes Congruence n-permutable yes, n = 2 Congruence regular no Congruence e-regular yes Congruence uniform no Congruence extension property no Definable principal congruences no Equationally definable principal congruences no 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

### Subclasses

[Commutative action lattices]?

### Superclasses

Action algebras
Residuated lattices

HomePage | RecentChanges | Preferences