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commutative_residuated_partially_ordered_semigroups [2010/07/29 15:46] (current)
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 +=====Commutative residuated partially ordered semigroups=====
 +
 +Abbreviation: **CRPoSgrp**
 +
 +====Definition====
 +A \emph{commutative residuated partially ordered semigroup} is a [[residuated partially ordered semigroup]] $\mathbf{A}=\langle A, \cdot, \to, \le\rangle$ such that
 +
 +$\cdot$ is \emph{commutative}:  $xy=yx$
 +
 +Remark: This is a template.
 +If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page.
 +
 +It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be commutative residuated partially ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism:
 +$h(x \cdot y)=h(x) \cdot h(y)$,
 +$h(x \to y)=h(x) \to h(y)$, and $x\le y\Longrightarrow h(x)\le h(y)$.
 +
 +====Definition====
 +A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
 +...\rangle$ such that
 +
 +$...$ is ...:  $axiom$
 +  
 +$...$ is ...:  $axiom$
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.
 +
 +^[[Classtype]]                        |quasivariety  |
 +^[[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====
 +
 +$\begin{array}{lr}
 +  f(1)= &1\\
 +  f(2)= &\\
 +  f(3)= &\\
 +  f(4)= &\\
 +  f(5)= &\\
 +\end{array}$    
 +$\begin{array}{lr}
 +  f(6)= &\\
 +  f(7)= &\\
 +  f(8)= &\\
 +  f(9)= &\\
 +  f(10)= &\\
 +\end{array}$
 +
 +
 +====Subclasses====
 +  [[Commutative residuated lattice-ordered semigroups]] expanded type
 +
 +
 +====Superclasses====
 +  [[Residuated partially ordered semigroups]] same type
 +
 +  [[Commutative partially ordered semigroups]] reduced type
 +
 +
 +====References====
 +
 +[(Lastname19xx>
 +F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
 +)]
 +
 +