EMV-pairs

ABSTRACT

The class of EMV-algebras and the class of lattice generalized effect algebras with RDP which have enough idempotents (EMV-effect algebra for short) are termwise equivalent. We use it to generalize connections between MV-algebras and MV-pairs for the case of EMV-algebras. First, we show that each EMV-algebra M is a homomorphic image of $\mathrm{Br}\left(M\right)$, the generalized Boolean algebra R-generated by M. Then we introduce a concept of an EMV-pair and construct an EMV-algebra using the EMV-pair. We also show that each EMV-algebra can be induced in this way. Then, the concept of an EMV-pair as a generalization of an MV-pair is introduced and some related results are obtained. Finally, we study G-invariant ideals of an EMV-pair $(B,G)$.

KEYWORDS:

• EMV-algebra
• EMV-pair
• generalized Boolean algebra
• generalized effect algebra

Acknowledgments

The authors are very indebted to anonymous referees for their careful reading and suggestions which helped us to improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Omid Zahiri http://orcid.org/0000-0002-2178-9421

Additional information

Funding

The paper has been supported by the grant of the Slovak Research and Development Agency under contract APVV-16-0073 and the grant VEGA No. 2/0069/16 SAV.

Notes on contributors

Anatolij Dvurečenskij

Anatolij Dvurečenskij, DrSc, graduated in mathematics in 1972 at the Comenius University, ratislava. In 1977, he obtained his Ph.D. at the Institute of Mathematics of the Slovak Academy of Sciences, Bratislava, and in 1989 his degree of Doctor of Sciences, DrSc. In 1998 he was appointed by the state president as a full professor at the Comenius University. During 1972–1987 he worked at the Institute of Measurement of the Slovak Academy of Sciences, and since 1987 he has been at the Institute of Mathematics SAS, Bratislava. During 1979–1987 he was at the Laboratory of Computing Techniques and Automatization of the Joint Institute of Nuclear Research, Dubna, Russia, and during 1991–1992 he was an A. von Humboldt fellow at the Institute of Theoretical Physics, University of Cologne, Germany. His main scientific interests are quantum structures, algebraic structures, and queueing theory, and he is the author/coauthor of more than 280 scientific papers as well as of two monographs: Gleason's Theorem and its Applications, Kluwer Academic Publications 1993, and (S. Pulmannova, coauthor) New Trends in Quantum Structures, Kluwer Academic Publications 2000. Homepage: http://www.mat.savba.sk/∼dvurecenskij/.

Omid Zahiri

Omid Zahiri, obtained his Ph.D. in mathematics at Shahid Beheshti University, Tehran, Iran in 2013. During 2014–2018 he worked at University of Applied Science and Technology, Tehran, Iran. His main scientific interests are ordered algebras, lattice theory, and ordered groups and he is the author/coauthor of more than 30 scientific papers. Homepage: https://scholar.google.com/citations?user=PQEyfJgAAAAJ&hl=en.