# Differences

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involutive_residuated_lattices [2010/07/29 15:46]
127.0.0.1 external edit
involutive_residuated_lattices [2012/07/18 23:24] (current)
jipsen
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====Definition==== ====Definition====
-An \emph{involutive residuated lattice} is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \neg\rangle$ of type $\langle 2, 2, 2, 0, 1\rangle$ such that+An \emph{involutive residuated lattice} is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \sim, -\rangle$ of type $\langle 2, 2, 2, 0, 1, 1\rangle$ such that
$\langle A, \vee, \wedge, \neg\rangle$ is an [[involutive lattice]] $\langle A, \vee, \wedge, \neg\rangle$ is an [[involutive lattice]]
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$xy\le z\iff x\le \neg(y(\neg z))\iff y\le \neg((\neg z)x)$ $xy\le z\iff x\le \neg(y(\neg z))\iff y\le \neg((\neg z)x)$
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-Remark: This is a template.
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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== ==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 involutive residuated lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
-$h(x ... y)=h(x) ... h(y)$+$h(x \vee y)=h(x) \vee h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h({\sim}x)={\sim}h(x)$ and $h(1)=1$.
====Definition==== ====Definition====
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====Subclasses==== ====Subclasses====
-  [[...]] subvariety+[[...]] subvariety
-  [[...]] expansion+[[...]] expansion
====Superclasses==== ====Superclasses====
-  [[...]] supervariety+[[...]] supervariety
-  [[...]] subreduct+[[...]] subreduct