# Differences

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ordered_semilattices [2016/11/26 19:08] (current)
jipsen created
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+=====Ordered semilattices=====
+
+Abbreviation: **OSlat**
+
+====Definition====
+An \emph{ordered semilattice} is a [[ordered semigroup]] $\mathbf{A}=\langle A,\cdot,\le\rangle$ that is
+
+\emph{commutative}:  $x\cdot y = y\cdot x$ and
+
+\emph{idempotent}:  $x\cdot x = x$
+
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be ordered semigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism:
+$h(x \cdot y)=h(x) \cdot h(y)$,
+$x\le y\Longrightarrow h(x)\le h(y)$.
+
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+
+^[[Classtype]]                        |universal  |
+^[[Equational theory]]                | |
+^[[Quasiequational theory]]           | |
+^[[First-order theory]]               | |
+^[[Locally finite]]                   | |
+^[[Residual size]]                    | |
+^[[Congruence distributive]]          | |
+^[[Congruence modular]]               | |
+^[[Congruence $n$-permutable]]        | |
+^[[Congruence regular]]               | |
+^[[Congruence uniform]]               | |
+^[[Congruence extension property]]    | |
+^[[Definable principal congruences]]  | |
+^[[Equationally def. pr. cong.]]      | |
+^[[Amalgamation property]]            | |
+^[[Strong amalgamation property]]     | |
+^[[Epimorphisms are surjective]]      | |
+
+====Finite members====
+$\begin{array}{rr} +f(1)=&1\\ +f(2)=&2\\ +f(3)=&5\\ +f(4)=&14\\ +f(5)=&42\\ +f(6)=&132\\ +f(7)=&\\ +f(8)=&\\ +\end{array}$
+
+This sequence is the Catalan numbers http://oeis.org/A000108
+
+====Subclasses====
+
+
+====Superclasses====
+[[Commutative ordered semigroups]]
+
+
+====References====
+
+[(Lastname19xx>
+F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
+)]
+
+

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