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commutative_residuated_partially_ordered_monoids [2010/07/29 15:46] external edit
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 $x\le y\Longrightarrow h(x)\le h(y)$.
-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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====Subclasses==== ====Subclasses====
-  [[Commutative residuated lattices]] expansion+[[Commutative residuated lattices]] expansion
-  [[Pocrims]] same type+[[Pocrims]] same type
====Superclasses==== ====Superclasses====
-  [[Residuated partially ordered monoids]] supervariety+[[Residuated partially ordered monoids]] supervariety
-  [[Commutative partially ordered monoids]] subreduct+[[Commutative partially ordered monoids]] subreduct
====References==== ====References====
-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]])]