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semilattices [2010/07/29 18:30] (current)
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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.
 +
 +
 +====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 $\{x,y\}$.
 +==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.
 +
 +
 +====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]]  | |
 +^[[Equationally def. pr. cong.]]  | |
 +^[[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]]
 +
 +[[Search for finite semilattices]]
 +
 +====Subclasses====
 +[[One-element algebras]]
 +
 +[[Semilattices with zero]]
 +
 +[[Semilattices with identity]]
 +
 +====Superclasses====
 +[[Commutative semigroups]]
 +
 +[[Partial semilattices]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]