Geodesic
Holland’s theorem for pseudo-effect algebras
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 47-59 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We give two variations of the Holland representation theorem for $\ell $-groups and of its generalization of Glass for directed interpolation po-groups as groups of automorphisms of a linearly ordered set or of an antilattice, respectively. We show that every pseudo-effect algebra with some kind of the Riesz decomposition property as well as any pseudo $MV$-algebra can be represented as a pseudo-effect algebra or as a pseudo $MV$-algebra of automorphisms of some antilattice or of some linearly ordered set.
We give two variations of the Holland representation theorem for $\ell $-groups and of its generalization of Glass for directed interpolation po-groups as groups of automorphisms of a linearly ordered set or of an antilattice, respectively. We show that every pseudo-effect algebra with some kind of the Riesz decomposition property as well as any pseudo $MV$-algebra can be represented as a pseudo-effect algebra or as a pseudo $MV$-algebra of automorphisms of some antilattice or of some linearly ordered set.
Classification : 03B50, 03G12, 06F20
Keywords: pseudo-effect algebra; pseudo $MV$-algebra; antilattice; prime ideal; automorphism; unital po-group; unital $\ell $-group 
@article{CMJ_2006_56_1_a4,
     author = {Dvure\v{c}enskij, Anatolij},
     title = {Holland{\textquoteright}s theorem for pseudo-effect algebras},
     journal = {Czechoslovak Mathematical Journal},
     pages = {47--59},
     year = {2006},
     volume = {56},
     number = {1},
     mrnumber = {2206286},
     zbl = {1164.06329},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a4/}
}
TY  - JOUR
AU  - Dvurečenskij, Anatolij
TI  - Holland’s theorem for pseudo-effect algebras
JO  - Czechoslovak Mathematical Journal
PY  - 2006
SP  - 47
EP  - 59
VL  - 56
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a4/
LA  - en
ID  - CMJ_2006_56_1_a4
ER  - 
%0 Journal Article
%A Dvurečenskij, Anatolij
%T Holland’s theorem for pseudo-effect algebras
%J Czechoslovak Mathematical Journal
%D 2006
%P 47-59
%V 56
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a4/
%G en
%F CMJ_2006_56_1_a4
Dvurečenskij, Anatolij. Holland’s theorem for pseudo-effect algebras. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 47-59. http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a4/

[1] A.  Dvurečenskij: Pseudo $MV$-algebras are intervals in $\ell $-groups. J.  Austral. Math. Soc. 72 (2002), 427–445. | DOI | MR

[2] A.  Dvurečenskij: Ideals of pseudo-effect algebras and their applications. Tatra Mt. Math. Publ. 27 (2003), 45–65. | MR

[3] A.  Dvurečenskij, S. Pulmannová: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht, Ister Science, Bratislava, 2000. | MR

[4] A.  Dvurečenskij, T. Vetterlein: Pseudoeffect algebras. I.  Basic properties. Inter. J.  Theor. Phys. 40 (2001), 685–701. | MR

[5] A.  Dvurečenskij, T. Vetterlein: Pseudoeffect algebras. II.  Group representations. Inter. J.  Theor. Phys. 40 (2001), 703–726. | MR

[6] G.  Georgescu, A.  Iorgulescu: Pseudo-$MV$ algebras. Multi. Val. Logic 6 (2001), 95–135. | MR

[7] A. M. W.  Glass: Polars and their applications in directed interpolation groups. Trans. Amer. Math. Soc. 166 (1972), 1–25. | DOI | MR | Zbl

[8] P.  Hájek: Observations on non-commutative fuzzy logic. Soft Computing 8 (2003), 38–43. | DOI

[9] C.  Holland: The lattice-ordered group of automorphism of an ordered set. Michigan Math.  J. 10 (1963), 399–408. | DOI | MR | Zbl