# Implicative lattices

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### Definition

An implicative lattice is a structure A = (A,∨,∧, → ) such that (A,∨,∧) is a distributive lattice, and  →  is an implication:
x → (yz)  = (x → y)∨(x → z),
x → (yz)  = (x → y)∧(x → z),
(xy) → z  = (x → z)∧(y → z), and
(xy) → z  = (x → z)∨(y → z)

### Morphisms

Let A and B be involutive 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(x) → h(y).

### References

[Nestor G. Martinez, H. A. Priestley, On Priestley spaces of lattice-ordered algebraic structures, Order 15 (1998) 297--323 MRreview]

[Nestor G. Martinez, A simplified duality for implicative lattices and l-groups, Studia Logica 56 (1996) 185--204 MRreview]

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### Subclasses

Goedel algebras
MV-algebras
Lattice-ordered groups

### Superclasses

Distributive lattices

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