Pseudo MV-algebras

Abbreviation: psMV

Definition

A pseudo MV-algebra1) (or psMV-algebra for short) is a structure $\mathbf{A}=\langle A, \oplus, ^-, ^\sim, 0, 1\rangle$ such that

$(x\oplus y)\oplus z = x\oplus(y\oplus z)$

$x\oplus 0 = x$

$x\oplus 1 = 1$

$(x^-\oplus y^-)^\sim = (x^\sim\oplus y^\sim)^-$

$(x\oplus y^\sim)^-\oplus x = y\oplus (x^-\oplus y)^\sim$

$x\oplus (y^-\oplus x)^\sim = y\oplus (x^-\oplus y)^\sim$

$x^{-\sim}=x$

$0^- = 1$

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be pseudo MV-algebras. A morphism from $\mathbf{A} $ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:

$h(x\oplus y)=h(x)\oplus h(y)$, $h(x^-)=h(x)^-$, $h(0)=0$ ($h(x^\sim)=h(x)^\sim$ and $h(1)=1$ follow from these).

Examples

Basic results

$0+x=x$, $1+x=1$, $x^{\sim-}=x$, $0^\sim=1$ 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, Pseudo-MV algebras, Multiple Valued Logic, 6, 2001, 95–135
3) A. Dvurecenskij, Pseudo MV-algebras are intervals in $\ell$-groups, Journal of the Australian Mathematical Soc. Ser. 72, (2002), 427-–445