semilattices - MathStructures

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=====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]]  |yes |
^[[Definable principal congruences]]  |yes |
^[[Equationally def. pr. cong.]]  |no |
^[[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--53[[MRreview]]

====Subclasses====
[[Semilattices with zero]] 

[[Semilattices with identity]] 

====Superclasses====
[[Bands]]

[[Commutative semigroups]] 

[[Partial semilattices]] 


====References====

[(Ln19xx>
)]
Trace: • division_algebras • dunn_monoids • finite_algebras
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