Boolean Products of MV-Algebras: Hypernormal MV-Algebras

“…It is straightforward to see (as in the cases of PL-algebras and MV-algebras, see [12] and [11]) that A a is a BL-algebra, and that the correspondence…”

Section: Corollary 24 For Each Bl-algebramentioning

confidence: 99%

“…We are going to give an explicit description of this representation. Firstly, note that the following lemma can be obtained in a standard way (see, for instance, [11]): …”

Section: Theorem 25 a Nontrivial Bl-algebra A Is Directly Indecompomentioning

confidence: 99%

Free algebras in varieties of BL-algebras with a Boolean retract.

Cignoli¹,

Torrens²

2002

Algebra Universalis

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The aim of this paper is to give a description of the free algebras in varieties of BL-algebras having the Boolean retraction property. This description is given in terms of weak Boolean products over Cantor spaces. The stalks are obtained in a constructive way from free radical algebras. Radical algebras are obtained endowing the maximal radicals of BL-algebras with a unary operation corresponding to double negation. The radical algebras obtained from a variety of BL-algebras form themselves a variety, that in the cases of PLalgebras and bipartite MV-algebras can be identified with the class of cancellative hoops.

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“…MV-algebras Boolean product of (n + 1)-homogeneous (Post), [79] [55] subalgebras of Z 1 n ∩ [0, 1] (n + 1)-bounded, [25] Z 1 kι ∩ [0, 1], 1 ≤ kι ≤ n Liminary, [28] finite chains, [22] Locally finite, [26,27] subalgebras of Q ∩ [0, 1] Table 1. Classes of locally finite MV-algebras as boolean products of MV-chains.…”

Section: Locally Finite Mv-algebras and Their Relativesmentioning

confidence: 99%

MV-algebras: a variety for magnitudes with archimedean units

Gispert

Mundici

2005

Algebra univers.

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Chang's MV-algebras, on the one hand, are the algebras of the infinite-valued Lukasiewicz calculus and, on the other hand, are categorically equivalent to abelian latticeordered groups with a distinguished strong unit, for short, unital -groups. The latter are a modern mathematization of the time-honored euclidean magnitudes with an archimedean unit. While for magnitudes the unit is no less important than the zero element, its archimedean property is not even definable in first-order logic. This gives added interest to the equivalent representation of unital -groups via the equational class of MV-algebras. In this paper we survey several applications of this equivalence, and various properties of the variety of MV-algebras. PrologueIn the last twenty years the number of papers devoted to Chang's MV-algebras [14] has been increasing so rapidly that, since the year 2000 the AMS Classification Index contains the special item 06D35 for MV-algebras. To quote just a handful of books, the monograph [25] by Cignoli et al., is entirely devoted to MValgebras, Hajek's monograph [56] and Gottwald's book [54] devote ample space to these algebras. As shown in [37] and [94], MV-algebras also provide an interesting example of "quantum structures". The Handbook of Measure Theory [89] includes several chapters on MV-algebraic measure theory. As the Lindenbaum algebras of Lukasiewicz infinite-valued logic, MV-algebras are also considered in many surveys, e.g., [21,68,87]. In this paper we will present several main topics in the theory of MV-algebras which are of greater potential interest for the universal algebraist. We apologize to authors and readers for all omissions, mainly due to lack of space.

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Boolean Products of MV-Algebras: Hypernormal MV-Algebras

Cited by 22 publications

References 6 publications

Boolean Products of BL-Algebras

Boolean Products of BL-Algebras

Free algebras in varieties of BL-algebras with a Boolean retract.

MV-algebras: a variety for magnitudes with archimedean units

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