Abbreviation: sqMV

A $\sqrt{'}$\emph{quasi-MV-algebra}¹⁾ 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

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

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:

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

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 $\mathbf k=\langle \frac12,\frac12\rangle$.

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.

Strongly cartesian sqrt-quasi-MV-algebras