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Action latticesAction lattices
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.
| 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 |