=====Quasi-MV-algebras=====

Abbreviation: **qMV**


====Definition====
A \emph{quasi-MV-algebra}[(LKPG2006)] 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 |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  |decidable |
^[[First-order theory]]  | |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |no |
^[[Congruence modular]]  |no |
^[[Congruence n-permutable]]  |no |
^[[Congruence e-regular]]  |no |
^[[Congruence uniform]]  |no |
^[[Congruence extension property]]  |yes |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  |no |
^[[Amalgamation property]]  |yes  |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |


====Finite members====

^$n$       | 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 |
^# of algs | 1 | 2 | 3 | 6 | 7 | 14| 15| 31| 32| 65 | 68 |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
^# of si's | 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  |

====Subclasses====
[[MV-algebras]] 


====Superclasses====

====References====

[(LKPG2006>
A. Ledda, M. Konig, F. Paoli and R. Giuntini, 
\emph{MV algebras and quantum computation}, 
Studia Logica, \textbf{82}(2), 2006, 245--270)]