generalized_mv-algebras - MathStructures

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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>
)]

Trace: • boolean_spaces • partial_groupoids • abelian_partially_ordered_groups • finite_semilattices_with_identity • abelian_ordered_groups • directed_partial_orders • groups • solvable_groups • commutative_integral_ordered_monoids • idempotent_semirings_with_identity_and_zero