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 : 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/y) = h(x)/h(y) and h(x*) = h(x)* and h(0) = 0 and h(1) = 1.
|Congruence n-permutable||yes, n = 2|
|Congruence extension property||no|
|Definable principal congruences||no|
|Equationally definable principal congruences||no|
|Strong amalgamation property|
|Epimorphisms are surjective|