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Pseudo MV-algebras

Abbreviation: psMV

Definition

A \emph{pseudo MV-algebra}1) (or \emph{psMV-algebra} for short) is a structure A=A,,,,0,1 such that

(xy)z=x(yz)

x0=x

x1=1

(xy)=(xy)

(xy)x=y(xy)

x(yx)=y(xy)

x=x

0=1

Morphisms

Let A and B be pseudo MV-algebras. A morphism from A to B is a function h:AB that is a homomorphism:

h(xy)=h(x)h(y), h(x)=h(x), h(0)=0 (h(x)=h(x) and h(1)=1 follow from these).

Examples

Basic results

0+x=x, 1+x=1, x=x, 0=1 and axiom A7 in2) follow from the above axioms.

Pseudo MV-algebras are term-equivalent to divisible involutive residuated lattices.

Every psMV-algebra is obtained from an interval in a lattice-ordered group3).

Every finite psMV-algebra is commutative.

Every commutative psMV-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 1 2 1 2 1 3 2 2 1 4 1 2 2 5 1 4 1 4 2 2 1 7 2
# of si's 0 1 1 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), 2) S. Georgescu and A. Iorgulescu, \emph{Pseudo-MV algebras}, Multiple Valued Logic, \textbf{6}, 2001, 95–135
3) A. Dvurecenskij, \emph{Pseudo MV-algebras are intervals in -groups}, Journal of the Australian Mathematical Soc. Ser. 72, (2002), 427-–445

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