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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]]
 +)]
 +
 +