[Home]Action lattices

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Abbreviation: ActLat


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.


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.

Some results



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

Finite members

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


[Commutative action lattices]?


Action algebras
Residuated lattices

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Last edited November 7, 2004 4:15 pm (diff)