# Differences

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generalized_mv-algebras [2010/07/29 15:46] (current)
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+=====Generalized MV-algebras=====
+Abbreviation: **GMV**
+
+====Definition====
+A \emph{generalized MV-algebra} is a [[residuated lattices]]
+$\mathbf{L}=\langle L,\vee, \wedge, \cdot, e, \backslash, /\rangle$ such that
+
+$x\vee y=x/(y\backslash x\wedge e)$, $x\vee y=(x/y\wedge e)\backslash y$
+
+==Morphisms==
+Let $\mathbf{L}$ and $\mathbf{M}$ be generalized MV-algebras. A
+morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$
+that is a homomorphism:
+
+$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$,
+$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash +y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+^[[Classtype]]  |variety |
+^[[Equational theory]]  |decidable [http://www.chapman.edu/~jipsen/lgroups/GMVDecisionProc.html implementation] |
+^[[Quasiequational theory]]  | |
+^[[First-order theory]]  | |
+^[[Locally finite]]  |no |
+^[[Residual size]]  |unbounded |
+^[[Congruence distributive]]  |yes |
+^[[Congruence modular]]  |yes |
+^[[Congruence n-permutable]]  |yes, $n=2$ |
+^[[Congruence regular]]  |no |
+^[[Congruence e-regular]]  |yes |
+^[[Congruence uniform]]  |no |
+^[[Congruence extension property]]  | |
+^[[Definable principal congruences]]  | |
+^[[Equationally def. pr. cong.]]  | |
+^[[Amalgamation property]]  | |
+^[[Strong amalgamation property]]  | |
+^[[Epimorphisms are surjective]]  | |
+
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &\\ +f(4)= &\\ +f(5)= &\\ +f(6)= &\\ +\end{array}$
+
+====Subclasses====
+
+[[Commutative generalized MV-algebras]]
+
+[[Integral generalized MV-algebras]]
+
+[[MV-algebras]]
+
+
+====Superclasses====
+[[Generalized BL-algebras]]
+
+
+====References====
+
+[(Ln19xx>
+)]

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