Fuzzy sets and systems

Vol. 360, p. 33-48

Date of publication: 2019

Fuzzy sets and systems

DOI: 10.1016/j.fss.2018.09.002

Date of publication: 2018-09-06

Abstract:

Fuzzy subgroups are revisited considering their close relationship with indistinguishability operators (fuzzy equivalences) invariant under translations. Different ways to obtain new fuzzy subgroups from a given one are provided and different ways to characterize normal fuzzy subgroups are obtained. The idea of double coset of two (crisp) subgroups allow us to relate them via their equivalence classes. This is generalized to the fuzzy framework. The conditions in which a fuzzy relation R on a group G can be considered a fuzzy subgroup of G × G are obtained.]]>

International journal of uncertainty fuzziness and knowledge-based systems

Vol. 26, num. 2, p. 217-232

DOI: 10.1142/S0218488518500113

Date of publication: 2018-04-01

Abstract:

Extensionality is explored form different points of view. Extensional fuzzy subsets from a fuzzy equivalence relation E are considered as observable subsets with respect to the granularity generated by E. Interestingly, they are characterized as the fuzzy subsets that can be obtained as combinations of the fuzzy equivalence classes of E. Extensional mappings are characterized topologically and the set of extensional mappings between two universes are algebraically determined. Specifying the results to fuzzy mappings from a universe X onto [0, 1] an interpretation of type-2 fuzzy subsets of X as fuzzification of its type-1 fuzzy subsets is provided.]]>

Information sciences

Vol. 426, p. 117-130

DOI: 10.1016/j.ins.2017.10.023

Date of publication: 2018-02-01

Abstract:

Similarity Relations may be constructed from a set of fuzzy attributes. Each fuzzy attribute generates a simple similarity, and these simple similarities are combined into a complex similarity afterwards. The Representation Theorem establishes one such way of combining similarities, while averaging them is a different and more realistic approach in applied domains. In this paper, given an averaged similarity by a family of attributes, we propose a method to find families of new attributes having fewer elements that generate the same similarity. More generally, the paper studies the structure of this important class of fuzzy relations.]]>

Advances in intelligent systems and computing

Vol. 581, p. 40

DOI: 10.1007/978-3-319-59306-7_5

Date of publication: 2018-01-01

Abstract:

Fuzzy subgroups and T-vague groups are interesting fuzzy algebraic structures that have been widely studied. While fuzzy subgroups fuzzify the concept of crisp subgroup, T-vague groups can be identified with quotient groups of a group by a normal fuzzy subgroup and there is a close relation between both structures and T-indistinguishability operators (fuzzy equivalence relations). In this paper the functions that aggregate fuzzy subgroups and T-vague groups will be studied. The functions aggregating T-indistinguishability operators have been characterized [9] and the main result of this paper is that the functions aggregating T-indistinguishability operators coincide with the ones that aggregate fuzzy subgroups and T-vague groups. In particular, quasi-arithmetic means and some OWA operators aggregate them if the t-norm is continuous Archimedean.]]>

Journal of applied logic

Vol. 23, p. 16-26

DOI: 10.1016/j.jal.2016.11.003

Date of publication: 2017-09-01

Abstract:

A T-indistinguishability operator (or fuzzy similarity relation) E is called unidimensional when it may be obtained from one single fuzzy subset (or fuzzy criterion). In this paper, we study when a T-indistinguishability operator that has been obtained as an average of many unidimensional ones is unidimensional too. In this case, the single fuzzy subset used to generate E is explicitly obtained as the quasi-arithmetic mean of all the fuzzy criteria primarily involved in the construction of E.]]>

Fuzzy sets and systems

p. 1-14

DOI: 10.1016/j.fss.2017.10.006

Date of publication: 2017-01-01

Abstract:

This paper generalizes (fuzzifies) actions of a monoid or group on a set to deal with situations where imprecision and uncertainty are present. Fuzzy actions can handle the granularity of a set or even create it by defining a fuzzy equivalence relation on it.]]>

Applied artificial intelligence

Vol. 29, num. 5, p. 514-530

DOI: 10.1080/08839514.2015.1026663

Date of publication: 2015-05-28

Abstract:

This article studies T-preorders that can be generated in a natural way by a single fuzzy subset. These T-preorders are called one-dimensional and are of great importance, because every T-preorder can be generated by combining one-dimensional T-preorders.; In this article, the relation between fuzzy subsets generating the same T-preorder is given, and one-dimensional T-preorders are characterized in two different ways: They generate linear crisp orderings on X and they satisfy a Sincov-like functional equation. This last characterization is used to approximate a given T-preorder by a one-dimensional one by relating the issue to Saaty matrices used in the Analytical Hierarchical Process. Finally, strong complete T-preorders, important in decision-making problems, are also characterized.]]>

Information sciences

Vol. 237, p. 261-270

DOI: 10.1016/j.ins.2013.03.017

Date of publication: 2013-07-10

Abstract:

Fuzzy transitivity is a key property for many fuzzy relational structures, such as fuzzy preorders and equivalences. Theoretical models for practical problems which are based on fuzzy relations make use of continuous scales, mostly the unit interval [0, 1]. Practical implementation of these models though, involves their discretization into finite scales, which generally results in some loss of transitivity. In this paper we study if there are any transitivity preserving discretization strategies. Also, we evaluate the loss of transitivity in some commonly used discretization approaches.]]>

International journal of uncertainty fuzziness and knowledge-based systems

Vol. 20, num. 2, p. 167-183

DOI: 10.1142/S0218488512500080

Date of publication: 2012-04

Abstract:

An isomorphism f between two continuous Archimedean t-norms T and T' transforms a T-indistinguishability operator E into a T'-indistinguishability operator f ° E and many interesting properties of E are transfered to f ° E by f. This paper generalizes this result in order to relate indistinguishability operators with respect to two non isomorphic continuous Archimedean t-norms. This will allow us to transfer definitions and properties from strict to non-strict Archimedean t-norms and vice versa.

An isomorphism f between two continuous Archimedean t-norms T and T′ transforms a T-indistinguishability operator E into a T′-indistinguishability operator f ∘ E and many interesting properties of E are transfered to f ∘ E by f. This paper generalizes this result in order to relate indistinguishability operators with respect to two non isomorphic continuous Archimedean t-norms. This will allow us to transfer definitions and properties from strict to non-strict Archimedean t-norms and vice versa.]]>

International journal of uncertainty fuzziness and knowledge-based systems

Vol. 16, num. 2, p. 129-145

DOI: 10.1142/S0218488508005108

Date of publication: 2008-04

Abstract:

A new map (¿E) between fuzzy subsets of a universe X endowed with a T-indistinguishability operator E is introduced. The main feature of ¿E is that it has the columns of E as fixed points, and thus it provides us with a new criterion to decide whether a generator is a column. Two well known maps (fE and ¿E) are also reviewed, in order to compare them with ¿E. Interesting properties of the fixed points of ¿E and ¿2 E are studied. Among others, the fixed points of ¿E (Fix(¿E)) are proved to be the maximal fuzzy points of (X,E) and the fixed points of ¿2 E coincide with the Image of ¿E. An isometric embedding of X into Fix(¿E) is established and studied.

A new map (ΛE) between fuzzy subsets of a universe X endowed with a T-indistinguishability operator E is introduced. The main feature of ΛE is that it has the columns of E as fixed points, and thus it provides us with a new criterion to decide whether a generator is a column. Two well known maps (φE and ψE) are also reviewed, in order to compare them with ΛE. Interesting properties of the fixed points of ΛE and Λ2 E are studied. Among others, the fixed points of ΛE (Fix(ΛE)) are proved to be the maximal fuzzy points of (X,E) and the fixed points of Λ2 E coincide with the Image of ΛE. An isometric embedding of X into Fix(ΛE) is established and studied.]]>

Fuzzy sets and systems

Vol. 120, num. 3, p. 415-422

DOI: 10.1016/S0165-0114(99)00133-5

Date of publication: 2001-06

Abstract:

Indistinguishability operators fuzzify the concept of equivalence relation and have been proved a useful tool in theoretical studies as well as in di0erent applications such as fuzzy control or approximate reasoning. One interesting problem is their construction. There are di0erent ways depending on how the data are given and on their future use. In this paper, the length of an indistinguishability operator is de2ned and it is used to relate its generation via max-T product and via the representation theorem when T is an Archimedean t-norm. The link is obtained taking into account that indistinguishability operators generate betweenness relations. The study is also extended to decomposable operators.]]>

International journal of general systems

Vol. 29, num. 4, p. 554-568

DOI: 10.1080/03081070008960961

Date of publication: 2000-09

Abstract:

The upper and lower approximations of a fuzzy subset with respect to an indistinguish-ability operator are studied. Their relations with fuzzy rough sets are also investigated]]>

International journal of uncertainty fuzziness and knowledge-based systems

Vol. 7, num. 5, p. 475-482

DOI: 10.1142/S0218488599000428

Date of publication: 1999-10

Abstract:

Defuzzification is an essential problem in fuzzy systems that it is always solved in a heuristic way. The aim of this work is to give a semantic interpretation to this process with the help of indistinguishability operators]]>

International journal of uncertainty fuzziness and knowledge-based systems

Vol. 7, num. 3, p. 203-212

DOI: 10.1142/S0218488599000143

Date of publication: 1999-06

Abstract:

The most common ways used to generate indistinguishability operators, namelyas transitive closure of reexive and symmetric fuzzy relation, via the Representation Theorem and as decomposable relations, is related for archimedean t-norms introducing the notion of length of indistinguishability operators.

The most common ways used to generate indistinguishability operators, namely as transitive closure of reexive and symmetric fuzzy relation, via the Representation Theorem and as decomposable relations, is related for archimedean t-norms introducing the notion of length of indistinguishability operators.]]>