Geodesic
States on unital partially-ordered groups
Kybernetika, Tome 38 (2002) no. 3, pp. 297-318 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We study states on unital po-groups which are not necessarily commutative as normalized positive real-valued group homomorphisms. We show that in contrast to the commutative case, there are examples of unital po-groups having no state. We introduce the state interpolation property holding in any Abelian unital po-group, and we show that it holds in any normal-valued unital $\ell $-group. We present a connection among states and ideals of po-groups, and we describe extremal states on the state space of unital po-groups.
We study states on unital po-groups which are not necessarily commutative as normalized positive real-valued group homomorphisms. We show that in contrast to the commutative case, there are examples of unital po-groups having no state. We introduce the state interpolation property holding in any Abelian unital po-group, and we show that it holds in any normal-valued unital $\ell $-group. We present a connection among states and ideals of po-groups, and we describe extremal states on the state space of unital po-groups.
Classification : 06B10, 06F15
Keywords: non-commutative group; partially ordered groups 
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     title = {States on unital partially-ordered groups},
     journal = {Kybernetika},
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     volume = {38},
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     zbl = {1265.06052},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2002_38_3_a6/}
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Dvurečenskij, Anatolij. States on unital partially-ordered groups. Kybernetika, Tome 38 (2002) no. 3, pp. 297-318. http://geodesic.mathdoc.fr/item/KYB_2002_38_3_a6/

[1] Bigard A., Keimel K., Wolfenstein S.: Groupes et Anneax Réticulés. Springer–Verlag, Berlin – Heidelberg – New York 1981

[2] Birkhoff G.: Lattice theory. Amer. Math. Soc. Colloq. Publ. 25 (1967) | MR | Zbl

[3] Chovanec F.: States and observables on MV-algebras. Tatra Mountains Math. Publ. 3 (1993), 55–65 | MR | Zbl

[4] Nola A. Di, Georgescu G., Iorgulescu A.: Pseudo-BL-algebras, I, II. Multi. Val. Logic, to appear

[5] Dvurečenskij A.: Pseudo MV-algebras are intervals in $\ell $-groups. J. Austral. Math. Soc. 72 (2002), to appear | DOI | MR | Zbl

[6] Dvurečenskij A.: States on pseudo MV-algebras. Studia Logica 68 (2001), 301–327 | DOI | MR | Zbl

[7] Dvurečenskij A.: States and idempotents of pseudo MV-algebras. Tatra Mountains. Math. Publ. 22 (2001), 79–89 | MR | Zbl

[8] Dvurečenskij A., Kalmbach G.: States on pseudo MV-algebras and the hull-kernel topology. Atti Sem. Mat. Fis. Univ. Modena 50 (2002), 131–146 | MR | Zbl

[9] Dvurečenskij A., Vetterlein T.: Pseudoeffect algebras. I. Basic properties. Internat. J. Theoret. Phys. 40 (2001), 685–701 | DOI | MR | Zbl

[10] Dvurečenskij A., Vetterlein T.: Pseudoeffect algebras. II. Group representations. Interat. J. Theoret. Phys. 40 (2001), 703–726 | DOI | MR | Zbl

[11] Dvurečenskij A., Vetterlein T.: Congruences and states on pseudo-effect algebras. Found. Phys. Lett. 14 (2001), 425–446 | DOI | MR

[12] Fuchs L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford – London – New York – Paris 1963 | MR | Zbl

[13] Georgescu G.: Bosbach states on pseudo-BL algebras. Soft Computing, to appear

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

[15] Goodearl K. R.: Partially Ordered Abelian Groups with Interpolation. (Math. Surveys Monographs 20), Amer. Math. Soc., Providence, Rhode Island 1986 | MR | Zbl

[16] Mundici D.: Averaging the truth-value in Łukasiewicz logic. Studia Logica 55 (1995), 113–127 | DOI | MR | Zbl

[17] Rachůnek J.: A non-commutative generalization of MV-algebras. Czechoslovak Math. J. 52 (2002), 255–273 | DOI | Zbl