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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]] | ||

+ | )] | ||

+ | |||

+ | |||

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