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Join-semidistributive lattices

Abbreviation: JsdLat

Definition

A join-semidistributive lattice is a lattice $\mathbf{L}=\langle L,\vee ,\wedge \rangle$ that satisfies

the join-semidistributive law SD$_{\vee}$: $x\vee y=x\vee z\Longrightarrow x\vee y=x\vee(y\wedge z)$

Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be join-semidistributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

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

Examples

Example 1: $D[d]=\langle D\cup\{d'\},\vee ,\wedge\rangle$, where $D$ is any distributive lattice and $d$ is an element in it that is split into two elements $d,d'$ using Alan Day's doubling construction.

Properties

Classtype quasivariety undecidable yes yes no no no no no no unbounded

Finite members

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