# Differences

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residuated_partially_ordered_monoids [2010/07/29 15:46]
127.0.0.1 external edit
residuated_partially_ordered_monoids [2019/02/24 14:15] (current)
jipsen
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-=====Name of class=====+=====Residuated partially ordered monoids=====
-Abbreviation: **Abbr**+Abbreviation: **RpoMon**
====Definition==== ====Definition====
-A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +A \emph{residuated partially ordered monoid} (or \emph{rpo-monoid}) is a structure$\mathbf{A}=\langle A,\le,\cdot,1,\backslash,/\rangle$such that -...\rangle$ such that+
-$\langle A,...\rangle$ is a [[name of class]]+$\langle A,\le\rangle$ is a [[partially ordered set]],
-$op_1$ is (name of property):  $axiom_1$+$\langle A,\cdot,1\rangle$ is a [[monoid]] and
-$op_2$ is ...:  $...$+$\backslash$ is the left residual of $\cdot$:  $x\cdot y\le z\iff y\le x\backslash z$
-Remark: This is a template. +$/$ is the right residual of $\cdot$:  $x\cdot y\le z\iff x\le z/y$.
-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== ==Morphisms==
-Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:  +Let $\mathbf{A}$ and $\mathbf{B}$ be residuated po-monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is an order-preserving homomorphism:
-$h(x ... y)=h(x) ... h(y)$ +$x\le y\implies h(x)\le h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \backslash y)=h(x) \backslash h(y)$, $h(x / y)=h(x) / h(y)$.
- +
-====Definition==== +
-A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle + -...\rangle$ such that +
- +
-$...$ is ...:  $axiom$ +
-   +
-$...$ is ...:  $axiom$+
====Examples==== ====Examples====
-Example 1:
====Basic results==== ====Basic results====
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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. 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]]                        |(value, see description) [(Ln19xx)]  |+^[[Classtype]]                        |order variety [(Ln19xx)]  |
^[[Equational theory]]                | | ^[[Equational theory]]                | |
^[[Quasiequational theory]]           | | ^[[Quasiequational theory]]           | |
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====Subclasses==== ====Subclasses====
-  [[...]] subvariety
-  [[...]] expansion+[[Commutative residuated partially ordered monoids]]
+
+[[Involutive residuated partially ordered monoids]]
+
+[[Residuated lattices]]
====Superclasses==== ====Superclasses====
-  [[...]] supervariety
-  [[...]] subreduct+[[Partially ordered monoids]]
+
+[[Residuated partially ordered semigroups]]