Commutative residuated partially ordered semigroups

Abbreviation: CRPoSgrp


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}: $xy=yx$

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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.


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 $x\le y\Longrightarrow h(x)\le h(y)$.


A \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$


Example 1:

Basic results


Finite members


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)= &\\



[[Commutative residuated lattice-ordered semigroups]] expanded type


[[Residuated partially ordered semigroups]] same type
[[Commutative partially ordered semigroups]] reduced type


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