On the structure of rotation-invariant semigroups

Jenei, Sándor

doi:10.1007/s00153-002-0165-8

Cited by 64 publications

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“…P r o o f. In [15], it is proved that DR(D) is an involutive residuated lattice. Clearly, there is no negation fixpoint in this algebra.…”

Section: Theorem 43 Let D Be a Gg-algebra The Disconnected Rotationmentioning

confidence: 99%

“…To obtain a proof of Theorem 4.7 we will prove first the following result: 1) Our definition of connected rotation is analogous but not exactly the same as the one given in [15]. The algebra CR(D) that we define is, according to [15], the connected rotation of the semigroup obtained by adding a lower bound to the GG-algebra D.…”

Section: Theorem 47 If a Is An Nm-algebra Then P (A) Is A Gg-algebramentioning

confidence: 99%

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Free nilpotent minimum algebras

Busaniche¹

2006

Mathematical Logic Qtrly

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“…P r o o f. In [15], it is proved that DR(D) is an involutive residuated lattice. Clearly, there is no negation fixpoint in this algebra.…”

Section: Theorem 43 Let D Be a Gg-algebra The Disconnected Rotationmentioning

confidence: 99%

Section: Theorem 47 If a Is An Nm-algebra Then P (A) Is A Gg-algebramentioning

confidence: 99%

Free nilpotent minimum algebras

Busaniche¹

2006

Mathematical Logic Qtrly

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“…3 By generalizing the concepts of perfect, local and bipartite algebra defined formerly for MV-algebras we have characterized several classes of IMTL-algebras. In particular, we have given a new insight to Jenei's disconnected rotation presented in [20] showing that the algebras obtained by this method are exactly the class of perfect IMTL-algebras. Futhermore, we have studied the variety generated by those algebras and we have proved that it is BP 0 .…”

Section: Discussionmentioning

confidence: 99%

“…Now, in order to introduce some methods for constructing IMTL-algebras defined by Jenei (see [20]), we need first to define the notion of prelinear semihoop. Definition 2.8 ( [13]).…”

Section: Preliminariesmentioning

confidence: 99%

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Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops

2005

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IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it.

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On n ‐contractive fuzzy logics

Horčı́k¹,

Noguera

Petrík

2007

Mathematical Logic Qtrly

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It is well known that MTL satisfies the finite embeddability property. Thus MTL is complete w. r. t. the class of all finite MTL-chains. In order to reach a deeper understanding of the structure of this class, we consider the extensions of MTL by adding the generalized contraction since each finite MTL-chain satisfies a form of this generalized contraction. Simultaneously, we also consider extensions of MTL by the generalized excluded middle laws introduced in [9] and the axiom of weak cancellation defined in [31]. The algebraic counterpart of these logics is studied characterizing the subdirectly irreducible, the semisimple, and the simple algebras. Finally, some important algebraic and logical properties of the considered logics are discussed: local finiteness, finite embeddability property, finite model property, decidability, and standard completeness.

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On the structure of rotation-invariant semigroups

Cited by 64 publications

References 0 publications

Free nilpotent minimum algebras

Free nilpotent minimum algebras

Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops

On n ‐contractive fuzzy logics

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