Strongly algebraically closed orthomodular near semirings
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Strongly algebraically closed orthomodular near semirings Ali Molkhasi1
· K. P. Shum2
Received: 29 May 2019 / Accepted: 20 July 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract The notion of q -compactness on a modular near ring is introduced and considered. Our aim is to show that if an induced lattice with an antitone involution on an orthogonal near semiring is complete and q -compact, then the induced lattice is a strongly algebraically closed lattice. In particular, an open question proposed by A. Di-Nola, G. Georgescu and A. Iorgulescu about the connections of pseudo-BL algebras with other algebraic structures in Di Nola et al. (Mult Val Logic 8:717–750, 2002) is answered. Keywords Łukasiewicz semirings · Orthomodular near semirings · Strongly algebraically closed algebras · Pseudo-BL-algebras Mathematics Subject Classification 16Y60 · 06E05 · 13L05
1 Introduction MV-algebras are models of an equational theory in universal algebra. Recently, as a common generalization of MV-algebras and orthomodular lattices, the term “basic algebras” was mentioned by Chajda et al. [5]. It is also noticed by Bonzio et al. [2] that the “basic algebras” can be represented by Łukasiewicz near semirings and orthomodular near semirings. The aim of the paper is to study the connection among the notion of q -compactness, (strongly) algebraically closed algebras, orthomodular near semirings, and Łukasiewicz near semirings. In particular, the analysis is carried out considering some specific class of algebras, such as Łukasiewicz near semirings, orthomodular near semirings and pseudo-BL algebras: for each one of them, our goal is that q -compactness (of the lattice reduct) implies strongly algebraically closed (see [2,11]).
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Ali Molkhasi [email protected] K. P. Shum [email protected]
1
Department of Mathematics, Faculty of Mathematical Sciences, University of Farhangian, Tehran, Iran
2
Institute of Mathematics, Yunnan University, Kunming, People’s Republic of China
123
A. Molkhasi, K. P. Shum
The paper is organized as follows. In Sect. 2, we recall some definitions and results which will be used in the following. In Sect. 3, we introduce the q -compactness for orthomodular near semirings and we prove that if the induced lattice with an antitone involution of an orthomodular near semiring is a complete lattice and q -compact, then it is a strongly algebraically closed lattice. In Sect. 4, we investigate the connections of pseudo-BL algebras with strongly algebraically closed algebras and we answer to a problem, which was proposed by Di Nola et al. [11] about the connections of pseudo-BL algebras with other algebraic structures. Finally, in the last section of the paper we study the connection of Łukasiewicz semirings with other algebraic structures, for example, Wajsberg algebras, BE-algebras, residuated–bounded lattices, and RS–BL-algebras.
2 Preliminaries Groupoids were introduced by Brandt in his 1926 paper [3] and semilattices can be equivalently presented as ordered sets as w
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