Equational Characterization of All Varieties of MV-Algebras

Nola, Antonio Di; Lettieri, A.

doi:10.1006/jabr.1999.7900

Cited by 57 publications

(43 citation statements)

References 5 publications

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“…There are more general MV-algebras which combine infinitesimals and non-infinitesimals. Their complete characterization follows from [14,5], see also [4]. The semantics based on such MV-algebras could describe two types of changes of truth values-"big" ones which may model the Sorites Paradox and infinitesimal ones for which this paradox applies (as in Classical Logic).…”

Section: What Can and What Cannot Be Expressed In Rational Or Perfectmentioning

confidence: 99%

Infinitesimals and Pavelka logic

Turunen¹,

Navara²

2015

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

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Section: What Can and What Cannot Be Expressed In Rational Or Perfectmentioning

confidence: 99%

Infinitesimals and Pavelka logic

Turunen¹,

Navara²

2015

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

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“…An equational basis relative to M V for every subvariety of MV-algebras was presented by Di Nola and Lettieri (1999).…”

Section: The Lattice Of Subvarieties Of Pseudo Mv-algebrasmentioning

confidence: 99%

How Do $$\ell $$-Groups and Po-Groups Appear in Algebraic and Quantum Structures?

Dvurečenskij

2014

Petr Hájek on Mathematical Fuzzy Logic

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“…For (i) we have (8) and (13) = (((xy) • y)y)(xy) by (H) = ((xy) • y)(y • (xy)) by (8) = 0 by (3) and (13).…”

Section: Lemma 1 If a Bck-algebra A With The Operation (S) Satisfiesmentioning

confidence: 99%

“…A general approach to the theory of MV-algebras can be found in the book [11] and the paper [16]. The lattice of all subvarieties of MV-algebras was studied in [20], and a complete equational presentation of these subvarieties can be found in [13]. …”

mentioning

confidence: 99%

The variety of all commutative BCK-algebras is generated by its finite members as a quasivariety

Palasinski

2012

Demonstratio Mathematica

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Abstract. We prove the result announced by the title as well as some of its consequences.It is well known that the variety of Łukasiewicz algebras is generated by its finite members (see e.g. [20]). W. Blok and I. Ferreirim [3] proved that the variety of all Łukasiewicz algebras is generated by its finite members as a quasi-variety. In this paper we show that this is also the case for the variety of all commutative BCK-algebras as well as some of its subvarieties. It is worth to note that unlike the case of subvarieties of the variety of all Łukasiewicz algebras, there are subvarieties of the variety of all commutative BCK-algebras which are not generated by their finite members [29,38].The main result of the paper was obtained in early '90s. Since then there were many papers on BCK-algebras, BCK-algebras with condition (S) (pocrims) and bounded commutative BCK-algebras (Wajsberg algebras, MV-algebras). Let me list some important papers suggested by an anonymous referee.• A good study of the varieties of BCK-algebras (including some aspects of commutative BCK-algebras) was done by W. Blok and J. Raftery in [5]. The references given in this paper provide a fairly comprehensive vision of the status of BCK-algebras until 1995. The implicative presentation of BCK-algebras tends to be used today, because they are the implicational subreduct of commutative integral residuated lattices.• BCK-algebras with condition S are currently known as pocrims: partially ordered, commutative, residuated, integral monoids. The varieties of were also studied by W. Blok and J. Raftery in [6]. This paper contains a complete bibliography on pocrims and BCK-algebras.• Partially naturally ordered commutative monoids with residuation are a special case of pocrims known as hoops, and they are studied in [4,2,3].• Bounded Commutative BCK-algebras, in their implicational presentation, are also known as CN-algebras [20] and Wajsberg algebras [15]. In fact, they are term-wise equivalent to Chang's MV-algebras introduced in [9] (see also [20, 15, 23] and [11]). MV-algebras can be seen as the unit segment of commutative lattice ordered groups with a strong unit (see [10] and [20] for and [24] for the general case).• The literature on MV-algebras is very large. Since Mundici's work [23], in which he studies the relationship between the many valued Łukasiewicz logic and AF C*-algebras, using the categorical equivalence between MValgebras and commutative ℓ-groups with strong unit, the theory of the MV-algebras has developed a great deal in different directions. A general approach to the theory of MV-algebras can be found in the book [11] and the paper [16]. The lattice of all subvarieties of MV-algebras was studied in [20], and a complete equational presentation of these subvarieties can be found in [13].

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Equational Characterization of All Varieties of MV-Algebras

Cited by 57 publications

References 5 publications

Infinitesimals and Pavelka logic

Infinitesimals and Pavelka logic

How Do $$\ell $$-Groups and Po-Groups Appear in Algebraic and Quantum Structures?

The variety of all commutative BCK-algebras is generated by its finite members as a quasivariety

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