The Hahn Embedding Theorem for a Class of Residuated Semigroups

Jenei, Sándor

doi:10.1007/s11225-019-09893-y

Cited by 8 publications

(41 citation statements)

References 19 publications

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“…The main result of [26] (hereinafter the original statement) falsely states that every algebra in O can be constructed by applying finitely many times the partial lex product construction using totally ordered abelian groups. However, it holds true that (hereinafter the correct statement) every algebra in O can be constructed by applying finitely many times the partial sublex product construction using totally ordered abelian groups.…”

Section: The Errormentioning

confidence: 99%

“…However, it holds true that (hereinafter the correct statement) every algebra in O can be constructed by applying finitely many times the partial sublex product construction using totally ordered abelian groups. In the sequel we explain why the original statement is false, we introduce the partial sublex product construction, and show what modifications are needed in [26] to obtain the correct statement. Knowledge of notions in [26] will be assumed.…”

Section: The Errormentioning

confidence: 99%

“…In the sequel we explain why the original statement is false, we introduce the partial sublex product construction, and show what modifications are needed in [26] to obtain the correct statement. Knowledge of notions in [26] will be assumed.…”

Section: The Errormentioning

confidence: 99%

“…We present the necessary changes to be made in [26] to obtain the correct statement. The main contribution (the novelty) of the present corrigendum is the fixing of Lemma 6.2, which will be done in Lemma B.4.…”

Section: How To Fix the Errormentioning

confidence: 99%

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Correction to: The Hahn Embedding Theorem for a Class of Residuated Semigroups

Jenei

2021

Stud Logica

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Let O be the class of odd involutive FLe-chains which admit finitely many idempotent elements. The price for describing the structure of such a rich class as O in [26] by such simple mathematical objects as linearly ordered abelian groups was that the related construction method, called partial lex product was quite complex. But in order to describe the structure of O, even the notion of partial lex products is not sufficiently general. One more tweak is needed, a slightly even more complex construction, called partial sublex product, introduced here.

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Section: The Errormentioning

confidence: 99%

Section: The Errormentioning

confidence: 99%

Section: The Errormentioning

confidence: 99%

Section: How To Fix the Errormentioning

confidence: 99%

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Correction to: The Hahn Embedding Theorem for a Class of Residuated Semigroups

Jenei

2021

Stud Logica

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“…In order to narrow the gap between these two extremal classes, in [26,27] a deep knowledge has been gained about the structure of odd involutive FL e -chains, including a Hahn-type embedding theorem and a representation theorem by means of linearly ordered abelian groups and there-introduced constructions, called partial lex products [27] and the more general partial sublex products [26]: all odd involutive FL e -chains which have finitely many idempotent elements have a partial sublex product group-representation. Square odd involutive FL e -algebras are those which admit a partial lex product group-representation.…”

Section: Introductionmentioning

confidence: 99%

Group-Like Uninorms

Jenei¹

2020

IJCIS

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Uninorms play a prominent role both in the theory and the applications of aggregations and fuzzy logic. In this paper a class of uninorms, called group-like uninorms will be introduced and a complete structural description will be given for a large subclass of them. First, the four versions of a general construction-called partial lex product-will be recalled. Then two particular variants of them will be specified: the first variant constructs, starting from ℝ (the additive group of the reals) and modifying it in some way by ℤ's (the additive group of the integers) what we will coin basic group-like uninorms, whereas the second variant can enlarge any group-like uninorm by a basic group-like uninorm resulting in another group-like uninorm. All grouplike uninorms obtained this way are "square" and have finitely many idempotent elements. On the other hand, we prove that any square group-like uninorm which has finitely many idempotent elements can be constructed by consecutive applications of the second variant (finitely many times) using only basic group-like uninorms as building blocks. Any basic group-like uninorm can be built by the first variant using only ℝ and ℤ, and any square group-like uninorm which has finitely many idempotent elements can be built using the second variant using only basic group-like uninorms: ultimately, all such uninorms can be built from ℝ and ℤ. In this way a complete characterization for square group-like uninorms which possess finitely many idempotent elements is given. The characterization provides, for potential applications in several fields of fuzzy theory or aggregation theory, the whole spectrum of choice of those square group-like uninorms which possess finitely many idempotent elements.

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