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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
(x⊕y)⊕z=x⊕(y⊕z)
x⊕0=x
x⊕1=1
(x−⊕y−)∼=(x∼⊕y∼)−
(x⊕y∼)−⊕x=y⊕(x−⊕y)∼
x⊕(y−⊕x)∼=y⊕(x−⊕y)∼
x−∼=x
0−=1
Morphisms
Let A and B be pseudo MV-algebras. A morphism from A to B is a function h:A→B that is a homomorphism:
h(x⊕y)=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
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 |