Adding structure to MV-algebras

Montagna, Franco; Panti, Giovanni

doi:10.1016/s0022-4049(00)00169-9

Cited by 36 publications

(16 citation statements)

References 22 publications

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“…Basically, the idea we follow to achieve the characterisation we are looking for is an extension of the technique used in [20], where, among others, the following result is proved.…”

Section: Definition 38mentioning

confidence: 99%

Advances in the theory of L algebras

Marchioni

Spada

2010

Logic Journal of IGPL

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Recently an expansion of LΠ 1 2 logic with fixed points has been considered [23]. In the present work we study the algebraic semantics of this logic, namely µ LΠ algebras, from algebraic, model theoretic and computational standpoints.We provide a characterisation of free µ LΠ algebras as a family of particular functions from [0, 1] n to [0,1]. We show that the first-order theory of linearly ordered µ LΠ algebras enjoys quantifier elimination, being, more precisely, the model completion of the theory of linearly ordered LΠ 1 2 algebras. Furthermore, we give a functional representation of any LΠ 1 2 algebra in the style of Di Nola Theorem for MV-algebras and finally we prove that the equational theory of µ LΠ algebras is in PSPACE.

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“…Basically, the idea we follow to achieve the characterisation we are looking for is an extension of the technique used in [20], where, among others, the following result is proved.…”

Section: Definition 38mentioning

confidence: 99%

Advances in the theory of L algebras

Marchioni

Spada

2010

Logic Journal of IGPL

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“…These algebras are known as PMValgebras or PŁ-algebras (see [7] and [6]). Then we combine these two extensions and get algebras known as PMV 4 -algebras or PŁ 0 4 -algebras (see [8] and [6])…”

Section: Preliminariesmentioning

confidence: 99%

A note to the definition of the ŁΠ-algebras

Cintula

2005

Soft Comput

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The ŁP-algebras are interesting algebraic structures. They were introduced by Esteva, Godo, and Montagna. These algebras are closely related to the wellknown MV-algebras and P-algebras. They have two sets of operations and reduct to one set is an MV-algebra and to the other one is a P-algebra.Other important reducts of ŁP-algebras are so-called PŁ-algebras (an MV-algebras enriched with product connective) and P $ -algebras (P-algebras with additional involutive negation).The definition of ŁP-algebras is based on its MValgabraic reduct and the additional identities. In this paper we present how to base this definition on the other reducts we mention above. We present minimal sets of conditions which has to be added to the defining conditions of those reducts in order to obtain ŁP-algebras.Definition 2. A MV-algebra is a structure L = ðL; È; À Ł ; 0 L Þ such that: Soft Comput (2005) 9: 575-578

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“…First of all, Mundici's equivalence between MV-algebras and lattice-ordered abelian groups with strong unit [Mu86] may be extended to a class of algebras which at the same time constitutes the algebraic semantics for an extension of Lukasiewicz Logic, and forms a category equivalent to a suitable category of lattice-ordered rings. This idea has been developed in [DND01], [M00], [EGM01], [M01] and [MP01]. The paper [DND01] treats products in a general setting, whereas the other papers describe a product which is closely related to the product in the reals.…”

Section: Introductionmentioning

confidence: 99%

“…Then ⊕, , · and / are the truncations to [0, 1] of the four basic operations, sum, subtraction, product and division. The papers [M00], [EGM01], [M01] and [MP01] contain a detailed treatment of LΠ-algebras. In particular, it is shown that these algebras constitute a discriminator variety, hence in every subdirectly irreducible LΠ-algebra universal properties can be expressed by means of identities.…”

Section: Introductionmentioning

confidence: 99%

“…This may cause problems when using LΠ in the treatment of approximate data, because a small error in the data may cause relevant errors in the output. Moreover, while free LΠ-algebras have been fully described in [MP01], the study of free M V -algebras with product and possibly with other continuous operators seems at the same time more difficult and more attractive. For instance, it seems to be connected with the Birkhoff-Pierce conjecture, see [K95] and [MP01].…”

Section: Introductionmentioning

confidence: 99%

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Subreducts of MV-algebras with product and product residuation

Montagna

2005

Algebra univers.

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Recently, MV -algebras with product have been investigated from different points of view. In particular, in [EGM01], a variety resulting from the combination of MV -algebras and product algebras (see [H98]) has been introduced. The elements of this variety are called LΠ-algebras. In this paper we treat subreducts of LΠ-algebras, with emphasis on quasivarieties of subreducts whose basic operations are continuous in the order topology. We give axiomatizations of the most interesting classes of subreducts, and we connect them with other algebraic classes of algebras, like f -rings and Wajsberg hoops, as well as to structures of co-infinitesimals of LΠ-algebras. In some cases, connections are given by means of equivalences of categories.

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Adding structure to MV-algebras

Cited by 36 publications

References 22 publications

Advances in the theory of L algebras

Advances in the theory of L algebras

A note to the definition of the ŁΠ-algebras

Subreducts of MV-algebras with product and product residuation

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