Sqrt-quasi-MV-algebras

Abbreviation: sqMV

Definition

A $\sqrt{'}$quasi-MV-algebra1) 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 MV-algebras. A morphism from $\mathbf{A} $ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:

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

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

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

Superclasses

References


1) R. Giuntini, A. Ledda, F. Paoli, Expanding quasi-MV algebras by a quantum operator, Studia Logica, 87, 2007, 99–128