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Soft Computing

, Volume 23, Issue 3, pp 735–745 | Cite as

Congruences and ideals in generalized pseudoeffect algebras revisited

  • S. PulmannováEmail author
Foundations
  • 73 Downloads

Abstract

The first part of the present paper is an enhancement of the paper (Foulis et al. in Order 33:311–332, 2016). A new type of congruences on generalized pseudoeffect algebras (GPEAs), called R1-congruences, is introduced, which is in one-to-one correspondence with normal R1-ideals. The notion of Riesz congruences is reconsidered, and they are defined as congruences which are in one-to-one correspondence with normal Riesz ideals. In upward directed GPEAs, in particular in pseudoeffect algebras, both these types of congruences as well as ideals coincide. Conditions under which congruences and ideals in a GPEA P may be extended to a \(\gamma \)-unitization of U of P are clarified. In the last part of the paper, subcentral and central ideals in GPEAs and their relations to subdirect and direct decompositions are studied.

Keywords

Effect algebra Pseudoeffect algebra Generalized pseudoeffect algebra R1-congruence R1-ideal Riesz congruence Riesz ideal Unitization Subcentral ideal 

Notes

Acknowledgements

The author was supported by Research and Development Support Agency under the Contract APVV-16-0073 and Grant VEGA 2/0069/16.

Compliance with ethical standards

Conflict of interest

The author declares that she has no conflict of interests.

Ethical approval

This article does not contain any studies with human participants or animals performed by the author.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia

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