### Table of Contents

## Pseudo MV-algebras

Abbreviation: **psMV**

### Definition

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

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

$x\oplus 0 = x$

$x\oplus 1 = 1$

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

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

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

$x^{-\sim}=x$

$0^- = 1$

##### 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 in^{2)} 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 group^{3)}.

Every finite psMV-algebra is commutative.

Every commutative psMV-algebra is an MV-algebra.

### Properties

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 |

### 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 $\ell$-groups}, Journal of the Australian Mathematical Soc. Ser. 72, (2002), 427-–445