# Differences

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commutative_residuated_partially_ordered_monoids [2019/06/15 06:34]
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commutative_residuated_partially_ordered_monoids [2019/12/12 08:00] (current)
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$\cdot$ is \emph{commutative}:  $xy=yx$ $\cdot$ is \emph{commutative}:  $xy=yx$
-Remark: This is a template. +Remark: These algebras are also known as \emph{lineales}.[(dePaiva2005)]
-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==
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$h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$,
$h(x \to y)=h(x) \to h(y)$, and $x\le y\Longrightarrow h(x)\le h(y)$. $h(x \to y)=h(x) \to h(y)$, 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==== ====Examples====
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$\begin{array}{lr}$\begin{array}{lr}
f(1)= &1\\   f(1)= &1\\
-  f(2)= &\\ +  f(2)= &2\\
-  f(3)= &\\ +  f(3)= &5\\
-  f(4)= &\\ +  f(4)= &24\\
-  f(5)= &\\ +  f(5)= &131\\
-\end{array}$+ f(6)= &1001\\ -$\begin{array}{lr} +
-  f(6)= &\\+
f(7)= &\\   f(7)= &\\
f(8)= &\\   f(8)= &\\
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====References==== ====References====
-[(Lastname19xx+[(dePaiva2005
-F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]  +V. de Paiva, \emph{Lineales: Algebras and Categories in the Semantics of Linear Logic}, Proofs and Diagrams, CSLI Publications, Stanford, 123-142, 2005, [[https://research.nuance.com/wp-content/uploads/2014/10/Lineales-algebras-and-categories-in-the-semantics-of-Linear-Logic.pdf]])]
-)]+

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