## 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:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$

Example 1:

### Properties

Classtype variety decidable [http://www.chapman.edu/~jipsen/lgroups/GMVDecisionProc.html implementation] no unbounded yes yes yes, $n=2$ no yes no

### Finite members

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= & f(4)= & f(5)= & f(6)= & \end{array}$