Mathematical Structures: Semilattices

# Semilattices

Difference (from prior major revision) (no other diffs)

Changed: 81,82c81
 $\left\langle \mathbb{N},+\right\rangle$, the natural numbers, with additition.
 $\left\langle \mathcal{P}_\omega(X)-\{\emptyset\},\cup\right\rangle$, the set of finite nonempty subsets of a set $X$, with union, is the free join-semilattice with singleton subsets of $X$ as generators.

http://mathcs.chapman.edu/structuresold/files/Semilattices.pdf
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\begin{document}
\textbf{\Large Semilattices}

\abbreviation{Slat}
\begin{definition}
A \emph{semilattice} is a structure $\mathbf{S}=\left\langle S,\cdot \right\rangle$, 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.

\end{definition}
\begin{definition}
A \emph{join-semilattice} is a structure $\mathbf{S}=\left\langle S,\vee \right\rangle$, 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 $\{x,y\}$.
\end{definition}
\begin{definition}
A \emph{meet-semilattice} is a structure $\mathbf{S}=\left\langle S,\wedge \right\rangle$, 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 $\{x,y\}$.
\end{definition}
\begin{morphisms}
Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:

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

\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
$\left\langle \mathcal{P}_\omega(X)-\{\emptyset\},\cup\right\rangle$, the set of finite nonempty subsets of a set $X$, with union, is the free join-semilattice with singleton subsets of $X$ as generators.
\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & decidable in polynomial time\\\hline
Quasiequational theory & decidable\\\hline
First-order theory & undecidable\\\hline
Locally finite & yes\\\hline
Residual size & 2\\\hline
Congruence distributive & no\\\hline
Congruence modular & no\\\hline
Congruence meet-semidistributive & yes\\\hline
Congruence n-permutable & no\\\hline
Congruence regular & no\\\hline
Congruence uniform & no\\\hline
Congruence extension property & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
\end{tabular}
\end{properties}
\end{table}\begin{finite_members} $f(n)=$ number of members of size $n$.

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

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\href{http://www.ams.org/mathscinet-getitem?mr=1 930 032}{MRreview}

\href{Search_for_finite_semilattices.pdf}{Search for finite semilattices}
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\

\href{One-element_algebras.pdf}{One-element algebras}

\href{Semilattices_with_zero.pdf}{Semilattices with zero}

\href{Semilattices_with_identity.pdf}{Semilattices with identity}

\end{subclasses}
\begin{superclasses}\

\href{Commutative_semigroups.pdf}{Commutative semigroups}

\href{Partial_semilattices.pdf}{Partial semilattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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