Abbreviation: CRLSgrp

A \emph{commutative residuated lattice-ordered semigroup} is a residuated lattice-ordered semigroup $\mathbf{A}=\langle A, \vee, \wedge, \cdot, \to\rangle$ 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 lattice-ordered semigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \vee y)=h(x) \vee h(y)$, $h(x \wedge y)=h(x) \wedge h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, and $h(x \to y)=h(x) \to h(y)$.

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:

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.

$\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}$

[[Commutative distributive residuated lattice-ordered semigroups]] subvariety

[[Commutative residuated lattices]] expansion

[[Residuated lattice-ordered semigroups]] supervariety

[[Commutative lattice-ordered semigroups]] subreduct