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Table of Contents

Commutative residuated lattice-ordered semigroups

Abbreviation: CRLSgrp

Definition

A \emph{commutative residuated lattice-ordered semigroup} is a residuated lattice-ordered semigroup A=A,,,, such that

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.

Morphisms

Let A and B be commutative residuated lattice-ordered semigroups. A morphism from A to B is a function h:AB that is a homomorphism: h(xy)=h(x)h(y), h(xy)=h(x)h(y), h(xy)=h(x)h(y), and h(xy)=h(x)h(y).

Definition

A \emph{…} is a structure A=A, of type 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.

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

Superclasses

[[Residuated lattice-ordered semigroups]] supervariety
[[Commutative lattice-ordered semigroups]] subreduct

References