Semilattices

Abbreviation: Slat

Definition

A semilattice is a structure $\mathbf{S}=\langle S,\cdot \rangle$, where $\cdot$ is an infix binary operation, called the semilattice operation, such that

$\cdot$ is associative: $(xy)z=x(yz)$

$\cdot$ is commutative: $xy=yx$

$\cdot$ is idempotent: $xx=x$

Remark: This definition shows that semilattices form a variety.

Definition

A join-semilattice is a structure $\mathbf{S}=\langle S,\vee \rangle$, where $\vee$ is an infix binary operation, called the $\emph{join}$, such that

$\leq$ is a partial order, where $x\leq y\Longleftrightarrow x\vee y=y$

$x\vee y$ is the least upper bound of $\{x,y\}$.

Definition

A meet-semilattice is a structure $\mathbf{S}=\langle S,\wedge \rangle$, where $\wedge$ is an infix binary operation, called the $\emph{meet}$, such that

$\leq$ is a partial order, where $x\leq y\Longleftrightarrow x\wedge y=x$

$x\wedge y$ is the greatest lower bound of $\{x,y\}$.

Morphisms

Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:Sarrow T$ that is a homomorphism:

$h(xy)=h(x)h(y)$

Examples

Example 1: $\langle \mathcal{P}_\omega(X)-\{\emptyset\},\cup\rangle$, the set of finite nonempty subsets of a set $X$, with union, is the free join-semilattice with singleton subsets of $X$ as generators.

Properties

Classtype variety decidable in polynomial time decidable undecidable yes 2 no no yes no no no yes yes yes

\end{table}

Finite members

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

These results follow from the paper below and the observation that semilattices with $n$ elements are in 1-1 correspondence to lattices with $n+1$ elements.

Jobst Heitzig,J\”urgen Reinhold,Counting finite lattices, Algebra Universalis, 482002,43–53MRreview