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Quasi-MV-algebras

Abbreviation: qMV

Definition

A \emph{quasi-MV-algebra}1) is a structure A=A,,,0,1 such that

(xy)z=x(yz)

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

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

Superclasses

References


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

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