Abbreviation: psMV
A \emph{pseudo MV-algebra}1) (or \emph{psMV-algebra} for short) is a structure $\mathbf{A}=\langle A, \oplus, ^-, ^\sim, 0, 1\rangle$ such that
$(x\oplus y)\oplus z = x\oplus(y\oplus z)$
$x\oplus 0 = x$
$x\oplus 1 = 1$
$(x^-\oplus y^-)^\sim = (x^\sim\oplus y^\sim)^-$
$(x\oplus y^\sim)^-\oplus x = y\oplus (x^-\oplus y)^\sim$
$x\oplus (y^-\oplus x)^\sim = y\oplus (x^-\oplus y)^\sim$
$x^{-\sim}=x$
$0^- = 1$
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).
$0+x=x$, $1+x=1$, $x^{\sim-}=x$, $0^\sim=1$ 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.
Classtype | variety |
---|---|
Equational theory | decidable |
Quasiequational theory | undecidable |
First-order theory | undecidable |
Locally finite | no |
Residual size | unbounded |
Congruence distributive | yes |
Congruence modular | yes |
Congruence n-permutable | yes |
Congruence e-regular | yes |
Congruence uniform | yes |
Congruence extension property | yes |
Definable principal congruences | |
Equationally def. pr. cong. | |
Amalgamation property | |
Strong amalgamation property | |
Epimorphisms are surjective |
$n$ | 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 |