## Implicative lattices

Abbreviation: ImpLat

### Definition

An implicative lattice is a structure $\mathbf{A}=\langle A,\vee,\wedge,\to\rangle$ such that

$\langle A,\vee,\wedge\rangle$ is a distributive lattices $\to$ is an implication:

$x\to(y\vee z) = (x\to y)\vee(x\to z)$

$x\to(y\wedge z) = (x\to y)\wedge(x\to z)$

$(x\vee y)\to z = (x\to z)\wedge(y\to z)$

$(x\wedge y)\to z = (x\to z)\vee(y\to z)$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be involutive lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\vee y)=h(x)\wedge h(y)$, $h(x\to y)=h(x)\to h(y)$

Nestor G. Martinez,H. A. Priestley,On Priestley spaces of lattice-ordered algebraic structures, Order, 151998,297–323MRreview

Nestor G. Martinez,A simplified duality for implicative lattices and $l$-groups, Studia Logica, 561996,185–204MRreview

Example 1:

### Properties

Classtype variety no unbounded yes yes

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$