## Lattices

Abbreviation: Lat

### Definition

A \emph{lattice} is a structure $\mathbf{L}=\langle L,\vee ,\wedge \rangle$, where $\vee$ and $\wedge$ are infix binary operations called the \emph{join} and \emph{meet}, such that

$\vee ,\wedge$ are associative: $(x\vee y)\vee z=x\vee (y\vee z)$, $(x\wedge y)\wedge z=x\wedge (y\wedge z)$

$\vee ,\wedge$ are commutative: $x\vee y=y\vee x$, $x\wedge y=y\wedge x$

$\vee ,\wedge$ are absorbtive: $(x\vee y)\wedge x=x$, $(x\wedge y)\vee x=x$.

Remark: It follows that $\vee$ and $\wedge$ are idempotent: $x\vee x=x$, $x\wedge x=x$.

This definition shows that lattices form a variety.

A partial order $\leq$ is definable in any lattice by $x\leq y\Longleftrightarrow x\wedge y=x$, or equivalently by $x\leq y\Longleftrightarrow x\vee y=y$.

##### Morphisms

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

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

### Definition

A \emph{lattice} is a structure $\mathbf{L}=\langle L,\vee ,\wedge \rangle$ of type $\langle 2,2\rangle$ such that

$\langle L,\vee \rangle$ and $\langle L,\wedge \rangle$ are semilattices, and

$\vee ,\wedge$ are absorbtive: $(x\vee y)\wedge x=x$, $(x\wedge y)\vee x=x$

### Definition

A \emph{lattice} is a structure $\mathbf{L}=\langle L,\leq \rangle$ that is a partially ordered set in which all elements $x,y\in L$ have a

least upper bound: $z=x\vee y\Longleftrightarrow x\leq z$, $y\leq z\ \mbox{and}\ \forall w\ (x\leq w$, $y\leq w\Longrightarrow z\leq w)$ and a

greatest lower bound: $z=x\wedge y\Longleftrightarrow z\leq x$, $z\leq y\ \mbox{and}\ \forall w\ (w\leq x$, $w\leq y\Longrightarrow w\leq z)$

### Definition

A \emph{lattice} is a structure $\mathbf{L}=\langle L,\vee ,\wedge ,\leq \rangle$ such that $\langle L,\leq \rangle$ is a partially ordered set and the following quasiequations hold:

$\vee$-left: $x\leq z$ and $y\leq z\ \Longrightarrow x\vee y\leq z$

$\vee$-right: $z\leq x\Longrightarrow z\leq x\vee y$, $\quad z\leq y\Longrightarrow z\leq x\vee y$

$\wedge$-right: $z\leq x$ and $z\leq y\Longrightarrow z\leq x\wedge y$

$\wedge$-left: $x\leq z\Longrightarrow x\wedge y\leq z$, $\quad y\leq z\Longrightarrow x\wedge y\leq z$

Remark: These quasiequations give a cut-free Gentzen system to decide the equational theory of lattices.

### Examples

Example 1: $\langle P(S),\cup ,\cap ,\subseteq \rangle$, the collection of subsets of a sets $S$, ordered by inclusion.

### Properties

Classtype variety decidable in polynomial time decidable undecidable no unbounded yes 1) yes no no no no no no yes yes 2) yes

### Finite members

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= &2 f(5)= &5 f(6)= &15 f(7)= &53 f(8)= &222 f(9)= &1078 f(10)= &5994 f(11)= &37622 f(12)= &262776 f(13)= &2018305 f(14)= &16873364 f(15)= &152233518 f(16)= &1471613387 f(17)= &15150569446 f(18)= &165269824761 f(19)= &1901910625578 \end{array}$3)4)

### Superclasses

1) Nenosuke Funayama,Tadasi Nakayama,\emph{On the distributivity of a lattice of lattice-congruences}, Proc. Imp. Acad. Tokyo, \textbf{18} 1942, 553–554
2) Bjarni J\'onsson,\emph{Universal relational systems}, Math. Scand., \textbf{4} 1956, 193–208
3) Jobst Heitzig, J\“urgen Reinhold, \emph{Counting finite lattices}, Algebra Universalis, \textbf{48}, 2002, 43–53
4) Peter Jipsen, Nathan Lawless, \emph{Generating all finite modular lattices of a given size}, Algebra Universalis, \textbf{74}, 2015, 253–264

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