Lattices

Abbreviation: Lat

Definition

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

Basic results

Properties

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)

Diagrams of lattices of size 2 to 7

Subclasses

Superclasses

References


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