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Abelian extensions of partially ordered partial monoids

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Abstract

In this paper we introduce the notion of a partially ordered partial monoid (po PM), and we study extensions of cancellative po PMs by abelian groups. We concentrate on the so-called central extensions, and prove that every such extension is an F-product of a po PM by an abelian group defined by a cocycle \(f.\) Like in partially ordered groups, the extension can be ordered by means of special sets. We also compare extensions of po PMs with the extensions of their universal groups.

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References

  • Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer Acad. Publ./Ister Science, Dordrecht/Bratislava

  • Dvurečenskij A, Vetterlein T (2001a) Pseudoeffect algebras. I. Basic properties. Int J Theor Phys 40:685–701

    Article  MATH  Google Scholar 

  • Dvurečenskij A, Vetterlein T (2001b) Pseudoeffect algebras. II. Group representation. Int J Theor Phys 40:703–726

    Article  MATH  Google Scholar 

  • Dvurečenskij A, Vetterlein T (2001c) Generalized pseudo-effect algebras. In: DiNole A, Gerla G (eds) Lectures on soft computing and fuzzy logic. Physica-Verlag, Heidelberg

  • Everett Jr CJ (1942) An extension theory for rings. Am J Math 64:363–370

    Article  MathSciNet  MATH  Google Scholar 

  • Evseev AE (1989a) A survey of partial grupoids. In: Lyapin ES (ed) Properties of semigroups. Gos. Ped. Inst. Leningrad, 1984, pp 39–76 (Russian) (English translation: Am Math Soc Transl 139:139–163)

  • Feldman DW, Wilce A (1998) Abelian extensions of quantum logics. Int J Theor Phys 47:39–43

    Article  MathSciNet  Google Scholar 

  • Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1325–1346

    Article  MathSciNet  Google Scholar 

  • Fuchs L (1965) Riesz groups. Ann Scuola Norm Sup Pisa 19:1–34

    MathSciNet  MATH  Google Scholar 

  • Fuchs L (1950) The extension of partially ordered groups. Acta Math Acad Sci Hungar 1:118–124

    Article  MathSciNet  MATH  Google Scholar 

  • Goodearl KR (1986) Partially ordered Abelian groups with interpolation. Am Math Soc Providence, Rhode Island

  • Gudder SP, Pulmannová S (1997) Quotients of partial abelian monoids. Algebra Universalis 38:395–421

    Article  MathSciNet  MATH  Google Scholar 

  • Hall Jr M (1959) The theory of groups. Macmillan, New York

  • Jenča G, Pulmannová S (2002) Quotients of partial abelian monoids and the Riesz decomposition property. Algebra Universalis 47:443–477

    Article  MathSciNet  MATH  Google Scholar 

  • Kôpka F, Chovanec F (1993) D-posets. Math Slovaca 43:23–34

    Google Scholar 

  • Lyapin ES, Evseev AE (1991) Partial Grupoids. Ross Gos Ped Inst, St.-Petersburg (Russian)

  • Murphy GJ (1990) C*-algebras and operator theory. Academic Press, INC, Boston

  • Nánásiová O (1995) D-sets and groups. Int J Theor Phys 34:1637–1642

    Article  MATH  Google Scholar 

  • Nánásiová O, Pulmannová S (2001) Abelian extensions of difference sets. Tatra Mt Math Publ 22:1–18

    MathSciNet  Google Scholar 

  • Pulmannová S (2006) Extensions of partially ordered partial abelian monoids. Czech Math J 56:155–178

    Article  MATH  Google Scholar 

  • Teller R (1964) On the extensions of lattice ordered groups. Pacif J Math 14:709–718

    MathSciNet  MATH  Google Scholar 

  • Schreier O (1926a) Über die Erweiterung von Gruppen, I. Monatshefte Math Phys 34:165–180

    Article  MathSciNet  MATH  Google Scholar 

  • Schreier O (1926b) Über die Erweiterung von Gruppen. II. Hamburger Abh 4:321–346

    Article  Google Scholar 

Download references

Acknowledgments

The authors were supported by ERDF OP R&D Projects CE, QUTE ITMS 26240120009, meta-QUTE ITMS 26240120022; the grant VEGA No. 2/0059/12 SAV and LPP-0199-07.

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Correspondence to Elena Vinceková.

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Pulmannová, S., Vinceková, E. Abelian extensions of partially ordered partial monoids. Soft Comput 16, 1339–1346 (2012). https://doi.org/10.1007/s00500-012-0814-8

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