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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
(x⊕y)⊕z=x⊕(y⊕z)
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 |