### Table of Contents

## Semilattices

Abbreviation: **Slat**

### Definition

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

##### Morphisms

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

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

### Definition

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

$\leq $ is a partial order,

$x\leq y\implies x\vee z\leq y\vee z$ and $z\vee x\leq z\vee y$,

$x\le x\vee y$ and $y\leq x\vee y$,

$x\vee x\leq x$.

This definition shows that semilattices form a partially-ordered variety.

### Definition

A \emph{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 \emph{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 $\{x,y\}$.

### 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.

### Basic results

### Properties

Classtype | variety |
---|---|

Equational theory | decidable in polynomial time |

Quasiequational theory | decidable |

First-order theory | undecidable |

Locally finite | yes |

Residual size | 2 |

Congruence distributive | no |

Congruence modular | no |

Congruence meet-semidistributive | yes |

Congruence n-permutable | no |

Congruence regular | no |

Congruence uniform | no |

Congruence extension property | |

Definable principal congruences | yes |

Equationally def. pr. cong. | yes |

Amalgamation property | yes |

Strong amalgamation property | yes |

Epimorphisms are surjective | 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,\emph{Counting finite lattices}, Algebra Universalis, \textbf{48}2002,43–53MRreview