• PDF / 321,350 Bytes
• 11 Pages / 595.276 x 790.866 pts Page_size
• 61 Downloads / 206 Views

DOWNLOAD

REPORT

---

FOUNDATIONS

EBL-algebras Hongxing Liu1 Published online: 10 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract In this paper, we define the notion of EBL-algebras, which are generalizations of BL-algebras and EMV-algebras. The notions of ideals, congruences and filters in EBL-algebras are introduced, and their mutual relationships are investigated. There is a one-to-one correspondence between the set of all ideals in an EBL-algebra and the set of all congruences on an EBL-algebra. Moreover, we give a representation theorem on EBL-algebras. Every proper EBL-algebras under some condition can be embedded into an EBL-algebras with a top element as an ideal. Keywords EBL-algebra · EMV-algebra · BL-algebra · MV-algebra · Ideal

1 Introduction The notion of MV-algebras was introduced by Chang (1958) as an algebraic counterpart of the Lukasiewicz system of many-valued logic. Mundici (1986) proved that the category of MV-algebras and the category of unital Abelian l-groups are categorical equivalent. MV-algebra theory has been deeply developed, and MV-algebras have been applied to many other parts of mathematics. BL-algebras were defined by Hájek (1998) as an algebraic counterpart of Hájek Basic Logic. MV-algebras are particular case of BL-algebras. In fact, an MV-algebra M is a BL-algebra such that (x − )− = x, for all x ∈ M. The two algebras are closely related. Many results on MV-algebras also valid in the settings of BL-algebras. However, their proofs are usually quite different. Lele and Nganou introduced the notion of ideals in BL-algebras. Ideals and filters behave quite differently in BL-algebras. This is not similar to the case of MV-algebras. Dvureˇcenskij and Zahiri (2019a) defined EMV-algebras, which extend the notion of MV-algebras. In fact, EMValgebras do not have a top element. The authors gave some basic properties of EMV-algebras. Also, the notions of ideals, congruences and filters are introduced and their relationships

are investigated. Also, semisimple EMV-algebras are characterized. They proved that every proper EMV-algebra can be embedded into an EMV-algebra with a top element as a maximal ideal. Recently, there are many works on EMV-algebras (Dvureˇcenskij and Zahiri 2019b). EMV-algebras are unbounded generalization of MValgebras. Inspired by the paper (Dvureˇcenskij and Zahiri 2019a), we give the notion of EBL-algebras and study the properties of these algebras. The paper is constructed as follows. In Sect. 2, we shall give some notions and results which are useful in the paper. In Sect. 3, we introduce EBL-algebras and give some properties of EBL-algebras. In Sect. 4, we define ideals and congruences in EBL-algebras. We present some properties of ideals and study the relationship between ideals and congruences. In Sect. 5, we define filters of EBLalgebras and study the relationship between filters and ideals. Also, we study the properties of prime filters. Moreover, we give a representation theorem of EBL-algebras. That is, every proper EBL-algebra can be embedded int