## Quasi-MV-algebras

Abbreviation: qMV

### Definition

A quasi-MV-algebra1) is a structure $\mathbf{A}=\langle A, \oplus, ', 0, 1\rangle$ such that

$(x\oplus y)\oplus z = x\oplus(y\oplus z)$

$x''=x$

$x \oplus 1 = 1$

$(x'\oplus y)'\oplus y = (y'\oplus x)'\oplus x$

$(x\oplus 0)' = x'\oplus 0$

$(x\oplus 0)\oplus 0 = x\oplus 0$

$0' = 1$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be MV-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:

$h(x\oplus y)=h(x)\oplus h(y)$, $h(x')=h(x)'$, $h(0)=0$

### Examples

The standard qMV-algebra is $\mathbf S=\langle [0,1]^2,\oplus, ', \mathbf 0, \mathbf 1\rangle$ where $\langle a,b\rangle\oplus \langle c,d\rangle=\langle \min(1,a+c), \frac12\rangle$, $\langle a,b\rangle'=\langle 1-a,1-b\rangle$, $\mathbf 0=\langle 0,\frac12\rangle$ and $\mathbf 1=\langle 1,\frac12\rangle$.

### Basic results

The variety of qMV-algebras is generated by the standard qMV-algebra.

The operation $\oplus$ is commutative: $x\oplus y = y\oplus x$.

Every qMV-algebra that satisfies $x\oplus 0 = x$ is an MV-algebra.

### Properties

Classtype variety decidable decidable no unbounded no no no no no yes no yes

### Finite members

 $n$ # of algs # of si's 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 6 7 14 15 31 32 65 68 0 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

### References

1) A. Ledda, M. Konig, F. Paoli and R. Giuntini, MV algebras and quantum computation, Studia Logica, 82(2), 2006, 245–270