=====Sqrt-quasi-MV-algebras=====

Abbreviation: **sqMV**


====Definition====
A $\sqrt{'}$\emph{quasi-MV-algebra}[(GLP2007)] is a
structure $\mathbf{A}=\langle A, \oplus, \sqrt{'}, ', 0, 1, k\rangle$ such that $\sqrt{'}$ is a unary operation,

$\mathbf{A}=\langle A, \oplus, ', 0, 1\rangle$ is a [[quasi-MV-algebra]],

$x'=\sqrt{'}\sqrt{'}x$,

$k'=k$, and

$\sqrt{'}(x\oplus 0)\oplus 0=k$.

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be $\sqrt{'}$qMV-algebras. A morphism from $\mathbf{A}
$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism: 

$h(x\oplus y)=h(x)\oplus h(y)$, $h(\sqrt{'}x)=\sqrt{'}h(x)$, $h(0)=0$, $h(k)=k$.


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


====Basic results====
The variety of $\sqrt{'}$qMV-algebras is generated by the standard $\sqrt{'}$qMV-algebra.

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

Only the trivial $\sqrt{'}$qMV-algebra 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 | 1 | 2 | 2 | 5 | 5 | 8 | 8 | 16| 16 | 24 | 24 |   |   |   |   |   |   |   |   |   |   |   |   |   |
^# of si's | 0 | 1 | 1 | 0 | 2 | 0 | 0 |  |  |   |   |   |   |   |   |   |   |   |   |  |   |   |   |   |   |

====Subclasses====
[[Strongly cartesian sqrt-quasi-MV-algebras]] 


====Superclasses====

====References====

[(GLP2007>
R. Giuntini, A. Ledda, F. Paoli, 
\emph{Expanding quasi-MV algebras by a quantum operator}, 
Studia Logica, \textbf{87}, 2007, 99--128)]