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Licensed Unlicensed Requires Authentication Published by De Gruyter July 19, 2019

On EMV-Semirings

R. A. Borzooei, M. Shenavaei, A. Di Nola and O. Zahiri
From the journal Mathematica Slovaca

Abstract

The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that any EMV-semiring is a distributive lattice. Moreover, we define an EMV-semiring homomorphism and show that the categories of EMV-semirings and the category of EMV-algebras are isomorphic. We also define the concepts of GI-simple and DLO-semiring and prove that every EMV-semiring is a GI-simple and a DLO-semiring. Finally, we propose a representation for EMV-semirings, which proves that any EMV-semiring is either an MV-semiring or can be embedded into an MV-semiring as a maximal ideal.

MSC 2010: Primary 06C15; 06D35

  1. (Communicated by Anatolij Dvurečenskij )

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Received: 2018-06-16
Accepted: 2018-11-17
Published Online: 2019-07-19
Published in Print: 2019-08-27

© 2019 Mathematical Institute Slovak Academy of Sciences