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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>
 +)]