=Paper=
{{Paper
|id=Vol-3308/paper2
|storemode=property
|title=Graded concept lattices in fuzzy rough set theory
|pdfUrl=https://ceur-ws.org/Vol-3308/Paper02.pdf
|volume=Vol-3308
|authors=Alexander Šostak,Māris Krastiņš,Ingrīda Uļjane
|dblpUrl=https://dblp.org/rec/conf/cla/SostakKU22
}}
==Graded concept lattices in fuzzy rough set theory==
https://ceur-ws.org/Vol-3308/Paper02.pdf
Graded concept lattices in fuzzy rough set theory
Alexander Šostak1,2,∗ , Māris Krastiņš2 and Ingrīda Uļjane1,2
1
Institute of Mathematics and CS University of Latvia, Riga, LV-1459, Latvia
2
Department of Mathematics, University of Latvia, Riga, LV-1004, Latvia
Abstract
Noting certain limitations of fuzzy concept lattices in rough set theory in terms of their possible appli-
cations, we present here a more flexible notion of a graded fuzzy concept lattice in rough set theory.
We establish initial facts about the object-oriented version of graded fuzzy concept lattice and illustrate
them with some examples, of both theoretical and practical nature.
Keywords
Fuzzy relations, fuzzy rough sets, measure of inclusion, object-oriented fuzzy concepts, graded fuzzy
concept lattices in rough set theory
1. Introduction
Formal concept analysis was initiated by R. Wille and B. Ganter [28], [8] in 80-ties of the previous
century. The principal subject of the study and at the same time the main tool of research in
formal concept analysis are formal concepts and concept lattices. At present concept analysis
and corresponding theory of concept lattices is a well developed field of mathematics with
many practical applications. The basics of the fuzzy concept analysis and the corresponding
theory of fuzzy concept lattices were mainly developed by R. Bĕlohlávek [1], see also [2], etc.
I. Düntch and G. Gediga [7] who worked in the field of model logic introduced approximation-
type operators by means of a binary relation. In addition, they used these operators for an
alternative notion of a concept lattice, now known as a property-oriented concept lattice, see,
e.g. [29], [16] et. al. Afterwards, Y.Y. Yao [29] defined an object-oriented concept lattice, in a
certain sense dual to the property-oriented concept lattice. Besides, Y.Y. Yao has highlighted the
important feature of these lattices, namely, that they can be realized as the analogue of “classic”
Wille-Ganter-Bĕlohlávek concept lattices in rough set theory. The fuzzy version of object and
property-oriented concept lattices was considered in [6]. In addition, this paper contains deep
analysis of relations between the three kinds of (fuzzy) concept lattices, i.e. between Wille-
Ganter-Bĕlohlávek (or WGB for short) (fuzzy) concept lattices, and two (fuzzy) concept lattices
in rough set theory, namely the property-oriented and the object-oriented (fuzzy) concept
lattices.
Published in Pablo Cordero, Ondrej Kridlo (Eds.): The 16𝑡ℎ International Conference on Concept Lattices and Their
Applications, CLA 2022, Tallinn, Estonia, June 20–22, 2022, Proceedings, pp. 19–33.
∗
Corresponding author.
Envelope-Open aleksandrs.sostaks@lu.lv (A. Šostak); mk18032@lu.lv (M. Krastiņš); ingrida.uljane@lu.lv (I. Uļjane)
Orcid 0000-0002-0252-8306 (I. Uļjane)
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
Concept lattices, both “classic” and concept lattices in rough set theory, have found many
important applications outside pure mathematics. Specifically, the technique based on concept
lattices is used in medicine, biology, healthcare, sociology, etc. On the other hand, we know
examples of only fragmentary applications of fuzzy concept lattices (of both types) in practice.
The reason for this is that the precise matching between extents and intents in fuzzy environment
is nearly impossible, see some comments in this concern in Subsection 4.3. To overcome this
problem we suggest to replace the notion of a fuzzy concept by a more general notion of a
graded fuzzy concept and to develop a more flexible theory of graded fuzzy concept lattices. To
realize this approach we start with the notion of a preconcept, viewing it as a certain “potential
concept”, and then estimate “conceptuality” of a fuzzy preconcept by means of fuzzy logic tools.
In the result we obtain a graded fuzzy concept lattice. We study basic properties of graded fuzzy
concept lattices and illustrate them by some specific examples.
In the paper [26] we used similar ideas to develop a graded approach to WGB-fuzzy concept
lattices. However, as different from graded WGB-fuzzy concept lattices, here we base on forward
and backward powerset operators induced by fuzzy relations. As the result the structure of
graded fuzzy concept lattices in rough set theory relays on isotone Galois connection (or
adjunction) as different from WGB-fuzzy concept lattices constructed in accordance with
antitone Galois connection.
The paper is structured as follows. In the second, preliminary, section, we recall and refine
definitions used in the paper and clarify the framework in which our research is carried out. In
the third section forward and backward operators induced by fuzzy relations are defined and
used for studying derivation operators, which in turn form the basis of fuzzy concept lattices
in fuzzy rough set theory. In section 4, which is central to the article, we develop a gradation
approach to concept lattices in fuzzy rough set theory. The fifth section is devoted to examples
illustrating our approach in particular cases determined by specific choice of a 𝑡-norm involved
in the definition of a fuzzy relation and the choice of fuzzy sets of potential objects and potential
properties. In the sixth section we consider some examples of practical nature where graded
fuzzy concepts could be useful. In the last, conclusion section some prospects are sketched for
the further studies on the basis of this paper.
2. Preliminaries
Lattices, quantales and residuated lattices. We recall here some well known concepts
from the theory of lattices see, e.g. [3], [13], [17], in order to clarify the terms used in the paper.
In our paper 𝐿 = (𝐿, ≤, ∧, ∨) denotes a complete lattice, that is a lattice in which joins ⋁ 𝑀
and meets ⋀ 𝑀 of all subsets 𝑀 ⊆ 𝐿 exist. In particular 0 ∈ 𝐿 and 1 ∈ 𝐿 are the bottom and the
top elements of 𝐿 respectively. A complete lattice 𝐿 is called join-distributive if 𝑎 ∧ (⋁𝑖∈𝐼 𝑏𝑖 ) =
⋁𝑖∈𝐼 (𝑎 ∧ 𝑏𝑖 ) for every 𝑎 ∈ 𝐿 and every {𝑏𝑖 ∣ 𝑖 ∈ 𝐼 } ⊆ 𝐿. Dually, a complete lattice 𝐿 is called
meet-distributive if 𝑎 ∨ (⋀𝑖∈𝐼 𝑏𝑖 ) = ⋀𝑖∈𝐼 (𝑎 ∨ 𝑏𝑖 ). A complete lattice is called bi-distributive if it is
join- and meet-distributive.
Let 𝐿 be a complete lattice and ∗ ∶ 𝐿 × 𝐿 → 𝐿 be a binary associative monotone operation.
The tuple (𝐿, ≤, ∧, ∨, ∗) is called a quantale [21] if ∗ distributes over arbitrary joins:
𝑎 ∗ (⋁𝑖∈𝐼 𝑏𝑖 ) = ⋁𝑖∈𝐼 (𝑎 ∗ 𝑏𝑖 ), (⋁𝑖∈𝐼 𝑏𝑖 ) ∗ 𝑎 = ⋁𝑖∈𝐼 (𝑏𝑖 ∗ 𝑎) ∀𝑎 ∈ 𝐿, {𝑏𝑖 |𝑖 ∈ 𝐼 } ⊆ 𝐿.
A quantale is integral if the top element acts as the unit, i.e. 1 ∗ 𝑎 = 𝑎. A quantale is commutative,
if 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎 for all 𝑎, 𝑏 ∈ 𝐿. In what follows by a quantale we mean a commutative integral
quantale. A typical example of a quantale is the unit interval endowed with a lower semi-
continuous 𝑡-norm, see, e.g. [15].
In a quantale a further binary operation ↦∶ 𝐿 × 𝐿 → 𝐿, the residuum, can be introduced
as associated with operation ∗ of the quantale (𝐿, ≤, ∧, ∨, ∗) via the Galois connection, that is
𝑎 ∗ 𝑏 ≤ 𝑐 ⟺ 𝑎 ≤ 𝑏 ↦ 𝑐 for all 𝑎, 𝑏, 𝑐 ∈ 𝐿.
A quantale (𝐿, ≤, ∧, ∨, ∗) provided with the derived operation ↦, that is the tuple (𝐿, ≤, ∧, ∨, ∗, ↦),
is known also as a (complete) residuated lattice [17].
In the next proposition we list basic properties of the residuum that can be found in the
works of different authors, see, e.g. [12].
Proposition 1. Basic properties of the residium:
(1) (⋁𝑖 𝑎𝑖 ) ↦ 𝑏 = ⋀𝑖 (𝑎𝑖 ↦ 𝑏) for all {𝑎𝑖 ∣ 𝑖 ∈ 𝐼 } ⊆ 𝐿, for all 𝑏 ∈ 𝐿;
(2) 𝑎 ↦ (⋀𝑖 𝑏𝑖 ) = ⋀𝑖 (𝑎 ↦ 𝑏𝑖 ) for all 𝑎 ∈ 𝐿, for all {𝑏𝑖 ∣ 𝑖 ∈ 𝐼 } ⊆ 𝐿;
(3) 1𝐿 ↦ 𝑎 = 𝑎 for all 𝑎 ∈ 𝐿;
(4) 𝑎 ↦ 𝑏 = 1𝐿 whenever 𝑎 ≤ 𝑏;
(5) 𝑎 ∗ (𝑎 ↦ 𝑏) ≤ 𝑏 for all 𝑎, 𝑏 ∈ 𝐿;
(6) (𝑎 ↦ 𝑏) ∗ (𝑏 ↦ 𝑐) ≤ 𝑎 ↦ 𝑐 for all 𝑎, 𝑏, 𝑐 ∈ 𝐿;
(7) 𝑎 ↦ 𝑏 ≤ (𝑎 ∗ 𝑐 ↦ 𝑏 ∗ 𝑐) for all 𝑎, 𝑏, 𝑐 ∈ 𝐿;
(8) 𝑎 ∗ 𝑏 ≤ 𝑎 ∧ 𝑏 for any 𝑎, 𝑏 ∈ 𝐿;
(9) (𝑎 ∗ 𝑏) ↦ 𝑐 = 𝑎 ↦ (𝑏 ↦ 𝑐) for any 𝑎, 𝑏, 𝑐 ∈ 𝐿.
Fuzzy sets and fuzzy relations. The concept of a fuzzy set was introduced by L.A. Zadeh
[31] and then extended to a more general concept of an 𝐿-fuzzy set by J.A. Goguen [11] where
𝐿 is a complete lattice, in particular a quantale. Given a set 𝑋 its 𝐿-fuzzy subset is a mapping
𝐴 ∶ 𝑋 → 𝐿. The lattice and the quantale structure of 𝐿 is extended point-wise to the 𝐿-exponent
of 𝑋, that is to the set 𝐿𝑋 of all 𝐿-fuzzy subsets of 𝑋. An 𝐿-fuzzy relation between two sets 𝑋 and
𝑌 is an 𝐿-fuzzy subset of the product 𝑋 × 𝑌, that is a mapping 𝑅 ∶ 𝑋 × 𝑌 → 𝐿, see, e.g. [27], [32].
An 𝐿-fuzzy relation 𝑅 is called left connected if ⋀𝑦∈𝑌 ⋁𝑥∈𝑋 𝑅(𝑥, 𝑦) = 1. If for every 𝑦 ∈ 𝑌 there
exists 𝑥 ∈ 𝑋 such that 𝑅(𝑥, 𝑦) = 1, then 𝑅 is called strongly left connected. An 𝐿-fuzzy relation
𝑅 is called right connected if ⋀𝑥∈𝑋 ⋁𝑦∈𝑌 𝑅(𝑥, 𝑦) = 1. If for every 𝑥 ∈ 𝑋 there exists 𝑦 ∈ 𝑌 such
that 𝑅(𝑥, 𝑦) = 1, then 𝑅 is called strongly right connected. An 𝐿-fuzzy relation 𝑅 ∶ 𝑋 × 𝑌 → 𝐿 is
called connected (strongly connected) if it is both left and right connected (respectively strongly
left and strongly right connected). Since in the paper 𝐿 is an arbitrary but a fixed lattice, we
shall omit the prefix 𝐿 and speak just of fuzzy sets and fuzzy relations.
Measure of inclusion of 𝐿-fuzzy sets. The gradation of a preconcept lattice presented
below is based on the fuzzy inclusion between fuzzy sets. We give here a brief introduction
into this field.
In order to fuzzify the inclusion relation 𝐴 ⊆ 𝐵 “a fuzzy set 𝐴 is a subset of a fuzzy set 𝐵”,
we have to interpret it as a certain fuzzy inclusion ↪ based on ”if - then” rule, that is on some
implication ⇒ on the lattice 𝐿. In the result we come to the formula 𝐴 ↪ 𝐵 = inf𝑥∈𝑋 (𝐴(𝑥) ⇒
𝐵(𝑥)). As far as we know, this approach was applied for the first time in [23], see also [24],
where it was based on the Kleene-Dienes implication ⇒. Later the fuzzified relation of inclusion
between fuzzy sets was studied and used by many authors, see, e.g. [14], [22], [4], [5], et al. In
most of the papers the implication ⇒ was defined by means of residuum ↦ of the underlying
quantale (𝐿, ≤, ∧, ∨, ∗).
Definition 1. By setting 𝐴 ↪ 𝐵 = ⋀𝑥∈𝑋 (𝐴(𝑥) ↦ 𝐵(𝑥)) for all 𝐴, 𝐵 ∈ 𝐿𝑋 , we obtain a mapping
↪∶ 𝐿𝑋 × 𝐿𝑋 → 𝐿. We call 𝐴 ↪ 𝐵 by the measure of inclusion of a fuzzy set 𝐴 into the fuzzy set 𝐵.
Let 𝐴 ↩ 𝐵 =𝑑𝑒𝑓 𝐵 ↪ 𝐴. We denote 𝐴 ≅ 𝐵 =𝑑𝑒𝑓 (𝐴 ↪ 𝐵) ∧ (𝐵 ↪ 𝐴) and view it as the degree of
equality of fuzzy sets 𝐴 and 𝐵.
Proposition 2. [10]. Properties of the mapping ↪∶ 𝐿𝑋 × 𝐿𝑋 → 𝐿:
(1) (⋁𝑖 𝐴𝑖 ) ↪ 𝐵 = ⋀𝑖 (𝐴𝑖 ↪ 𝐵) for all {𝐴𝑖 ∣ 𝑖 ∈ 𝐼 } ⊆ 𝐿𝑋 and for all 𝐵 ∈ 𝐿𝑋 ;
(2) 𝐴 ↪ (⋀𝑖 𝐵𝑖 ) = ⋀𝑖 (𝐴 ↪ 𝐵𝑖 ) for all 𝐴 ∈ 𝐿𝑋 , and for all {𝐵𝑖 ∣ 𝑖 ∈ 𝐼 } ⊆ 𝐿𝑋 ;
(3) 𝐴 ↪ 𝐵 = 1 whenever 𝐴 ≤ 𝐵;
(4) 1𝑋 ↪ 𝐴 = ⋀𝑥 𝐴(𝑥) for all 𝐴 ∈ 𝐿𝑋 where 1𝑋 ∶ 𝑋 → 𝐿 is a constant function with the value
1 ∈ 𝐿;
(5) (𝐴 ↪ 𝐵) ≤ (𝐴 ∗ 𝐶 ↪ 𝐵 ∗ 𝐶) for all 𝐴, 𝐵, 𝐶 ∈ 𝐿𝑋 ;
(6) (𝐴 ↪ 𝐵) ∗ (𝐵 ↪ 𝐶) ≤ (𝐴 ↪ 𝐶) for all 𝐴, 𝐵, 𝐶 ∈ 𝐿𝑋 ;
(7) (⋀𝑖 𝐴𝑖 ) ↪ (⋀𝑖 𝐵𝑖 ) ≥ ⋀𝑖 (𝐴𝑖 ↪ 𝐵𝑖 ) for all {𝐴𝑖 ∶ 𝑖 ∈ 𝐼 }, {𝐵𝑖 ∶ 𝑖 ∈ 𝐼 } ⊆ 𝐿𝑋 ;
(8) (⋁𝑖 𝐴𝑖 ) ↪ (⋁𝑖 𝐵𝑖 ) ≥ ⋀𝑖 (𝐴𝑖 ↪ 𝐵𝑖 ) for all {𝐴𝑖 ∶ 𝑖 ∈ 𝐼 }, {𝐵𝑖 ∶ 𝑖 ∈ 𝐼 } ⊆ 𝐿𝑋 .
3. Forward and backward operators induced by fuzzy relations
Let 𝑅 ∶ 𝑋 × 𝑌 → 𝐿 be a fuzzy relation. We refer to the set 𝑋 as the domain and to the set 𝑌 as
the codomain of the fuzzy relation 𝑅. The forward and backward lower and upper operators
induced by fuzzy relations can be found in the works of different authors and under various
names. Since our goal is to apply them in the framework of concept analysis we interpret them
here also as derivation operators in the definition of a fuzzy concept.
Definition 2.
(□ ) - 𝑅□ denotes the lower forward operator 𝑅⇒ ∶ 𝐿𝑋 → 𝐿𝑌 . That is, given 𝐴 ∈ 𝐿𝑋 and 𝑦 ∈ 𝑌, let
𝑅□ (𝐴)(𝑦) =𝑑𝑒𝑓 𝐴□ (𝑦) = ⋀𝑥∈𝑋 (𝑅(𝑥, 𝑦) ↦ 𝐴(𝑥)).
(♦ ) - 𝑅♦ denotes the upper forward operator 𝑅→ ∶ 𝐿𝑋 → 𝐿𝑌 . That is, given 𝐴 ∈ 𝐿𝑋 and 𝑦 ∈ 𝑌, let
𝑅♦ (𝐴)(𝑦) =𝑑𝑒𝑓 𝐴♦ (𝑦) = ⋁𝑥∈𝑋 (𝑅(𝑥, 𝑦) ∗ 𝐴(𝑥)).
(■ ) - 𝑅■ denotes the lower backward operator 𝑅⇐ ∶ 𝐿𝑌 → 𝐿𝑋 . That is, given 𝐵 ∈ 𝐿𝑌 and 𝑥 ∈ 𝑋, let
𝑅■ (𝐵)(𝑥) =𝑑𝑒𝑓 𝐵■ (𝑥) = ⋀𝑦∈𝑌 (𝑅(𝑥, 𝑦) ↦ 𝐵(𝑦)).
(♦ ) - 𝑅♦ denotes the upper backward operator 𝑅← ∶ 𝐿𝑌 → 𝐿𝑋 . That is, given 𝐵 ∈ 𝐿𝑌 and 𝑥 ∈ 𝑋, let
𝑅♦ (𝐵)(𝑥) =𝑑𝑒𝑓 𝐵♦ (𝑥) = ⋁𝑦∈𝑌 (𝑅(𝑥, 𝑦) ∗ 𝐵(𝑦)).
The proof of the following three propositions is easy and can be found in [25]:
Proposition 3. Let 𝑋 , 𝑌 be sets and 𝑅 ∶ 𝑋 × 𝑌 → 𝐿 a fuzzy relation. Then
(1) 𝐴1 , 𝐴2 ∈ 𝐿𝑋 , 𝐴1 ≤ 𝐴2 ⟹ 𝐴□ □ ♦
1 ≤ 𝐴2 ; 𝐴1 ≤ 𝐴2 ;
♦
(2) 𝐵1 , 𝐵2 ∈ 𝐿𝑌 , 𝐵1 ≤ 𝐵2 ⟹ 𝐵1■ ≤ 𝐵2■ ; 𝐵1♦ ≤ 𝐵2♦ .
Proposition 4. If a fuzzy relation is strongly left connected, then 𝑅□ ≤ 𝑅♦ . If 𝑅 ∶ 𝑋 × 𝑌 → 𝐿 is
strongly right connected, then 𝑅■ ≤ 𝑅♦ .
Proposition 5.
(1) 𝑅□ (1𝑋 ) = 1𝑌 and 𝑅♦ (0𝑋 ) = 0𝑌 . If 𝑅 is left connected, then 𝑅□ (𝑎𝑋 ) = 𝑅♦ (𝑎𝑋 ) = 𝑎𝑌 for every
𝑎 ∈ 𝐿.
(2) 𝑅■ (1𝑌 ) = 1𝑋 and 𝑅♦ (0𝑌 ) = 0𝑋 . If 𝑅 is right connected, then 𝑅■ (𝑏𝑌 ) = 𝑅♦ (𝑏𝑌 ) = 𝑏𝑋 for every
𝑏 ∈ 𝐿.
Proposition 6. Let {𝐴𝑖 ∣ 𝑖 ∈ 𝐼 } ⊆ 𝐿𝑋 and {𝐵𝑖 ∣ 𝑖 ∈ 𝐼 } ⊆ 𝐿𝑌 . Then:
□ ■
(1) (⋀𝑖∈𝐼 𝐴𝑖 ) = ⋀𝑖∈𝐼 𝐴□
𝑖 ; (3) (⋀𝑖∈𝐼 𝐵𝑖 ) = ⋀𝑖∈𝐼 𝐵𝑖■ ;
♦ ♦
(2) (⋁𝑖∈𝐼 𝐴𝑖 ) = ⋁𝑖∈𝐼 𝐴♦
𝑖 ; (4) (⋁𝑖∈𝐼 𝐵𝑖 ) = ⋁𝑖∈𝐼 𝐵𝑖♦ .
Proof. Let {𝐴𝑖 ∣ 𝑖 ∈ 𝐼 } ⊆ 𝐿𝑋 and 𝑦 ∈ 𝑌. Then (⋀𝑖∈𝐼 𝐴□ □
𝑖 ) (𝑦) = ⋀𝑖∈𝐼 𝐴𝑖 (𝑦) =
⋀𝑖∈𝐼 (⋀𝑥∈𝑋 (𝑅(𝑥, 𝑦) ↦ 𝐴𝑖 (𝑥))) = ⋀𝑥∈𝑋 (⋀𝑖∈𝐼 (𝑅(𝑥, 𝑦) ↦ 𝐴𝑖 (𝑥))) =
□
⋀𝑥∈𝑋 (𝑅(𝑥, 𝑦) ↦ ⋀𝑖∈𝐼 𝐴𝑖 (𝑥)) = (⋀𝑖∈𝐼 𝐴𝑖 ) (𝑦).
(⋁𝑖∈𝐼 𝐴♦ ♦
𝑖 ) (𝑦) = ⋁𝑖∈𝐼 𝐴𝑖 (𝑦) = ⋁𝑖∈𝐼 (⋁𝑥∈𝑋 (𝑅(𝑥, 𝑦) ∗ 𝐴𝑖 (𝑥))) =
♦
⋁𝑥∈𝑋 (⋁𝑖∈𝐼 (𝑅(𝑥, 𝑦) ∗ 𝐴𝑖 (𝑥))) = ⋁𝑥∈𝑋 ((𝑅(𝑥, 𝑦) ∗ ⋁𝑖∈𝐼 𝐴𝑖 (𝑥)) = (⋁𝑖∈𝐼 𝐴𝑖 ) (𝑦).
Our next goal is to consider the successive composition of these operators. We denote
compositions 𝑅♦ ∘ 𝑅□ = 𝑅□♦ ; 𝑅♦ ∘ 𝑅■ = 𝑅■♦ , 𝑅□ ∘ 𝑅♦ = 𝑅♦□ 𝑅■ ∘ 𝑅♦ = 𝑅♦■ . Respectively,
given fuzzy sets 𝐴 ∈ 𝐿𝑋 and 𝐵 ∈ 𝐿𝑌 , denote 𝑅□♦ (𝐴) = 𝐴□♦ , 𝑅♦■ (𝐴) = 𝐴♦■ ; 𝑅■♦ (𝐵) = 𝐵■♦
and 𝑅♦□ (𝐵) = 𝐵♦□ .
Theorem 1. For every 𝐴 ∈ 𝐿𝑋 and every 𝐵 ∈ 𝐿𝑌 it holds:
(1) 𝐴□♦ ≤ 𝐴 ≤ 𝐴♦■ ;
(2) 𝐵■♦ ≤ 𝐵 ≤ 𝐵♦□ .
Proof. Let 𝑦 ∈ 𝑌. Then 𝐴□♦ (𝑥) = ⋁𝑦∈𝑌 (𝑅(𝑥, 𝑦) ∗ 𝐴□ (𝑦)) =
⋁𝑦∈𝑌 (𝑅(𝑥, 𝑦) ∗ (⋀𝑥 ′ ∈𝑋 (𝑅(𝑥 ′ , 𝑦) ↦ 𝐴(𝑥 ′ ))) ≤
⋁𝑦∈𝑌 (𝑅(𝑥, 𝑦) ∗ (𝑅(𝑥, 𝑦) ↦ 𝐴(𝑥))) ≤ 𝐴(𝑥);
on the other hand 𝐴♦■ (𝑥) = ⋀𝑦∈𝑌 (𝑅(𝑥, 𝑦) ↦ 𝐴♦ (𝑥)) =
⋀𝑦∈𝑌 (𝑅(𝑥, 𝑦) ↦ (⋁𝑥 ′ ∈𝑋 𝑅(𝑥 ′ , 𝑦) ∗ 𝐴(𝑥 ′ ))) ≥
⋀𝑦∈𝑌 (𝑅(𝑥, 𝑦) ↦ (𝑅(𝑥, 𝑦) ∗ 𝐴(𝑥))) ≥ 𝐴(𝑥).
In a similar way the second inequality can be proved.
Theorem 2. For every 𝐴 ∈ 𝐿𝑋 and every 𝐵 ∈ 𝐿𝑌
(1)𝐴♦■♦ = 𝐴♦ ; (3) 𝐵♦□♦ = 𝐵♦ ;
(2) 𝐴□♦□ = 𝐴□ ; (4) 𝐵■♦■ = 𝐵■ .
Proof. Applying Theorem 1 we have 𝐴♦■♦ = (𝐴♦■ )♦ ≥ 𝐴♦ ; 𝐴♦■♦ = (𝐴♦ )■♦ ≤ 𝐴♦ and
hence 𝐴♦■♦ = 𝐴♦ . On the other hand, again by Theorem 1, 𝐴□♦□ = (𝐴□♦ )□ ≤ 𝐴□ ; 𝐴□♦□ =
(𝐴□ )♦□ ≥ 𝐴□ and hence 𝐴□♦□ = 𝐴□ . The last two equalities can be proved in a similar
way.
From Theorem 2 we get
Corollary 1. For every 𝐴 ∈ 𝐿𝑋 and every 𝐵 ∈ 𝐿𝑌
(1) 𝐴♦■♦■ = 𝐴♦■ ; (3) 𝐵♦□♦□ = 𝐵♦□ ;
(2) 𝐴□♦□♦ = 𝐴□♦ ; (4) 𝐵■♦■♦ = 𝐵■♦ ,
and hence the operators 𝑅□♦ , 𝑅♦■ , 𝑅■♦ , 𝑅♦□ are idempotent.
Theorem 3. Operators 𝑅♦ ∶ 𝐿𝑋 → 𝐿𝑌 and 𝑅■ ∶ 𝐿𝑌 → 𝐿𝑋 form an adjoint pair (𝑅♦ , 𝑅■ ), that is
𝐴♦ ≤ 𝐵 ⟺ 𝐴 ≤ 𝐵■ for any 𝐴 ∈ 𝐿𝑋 , 𝐵 ∈ 𝐿𝑌 .
Operators 𝑅♦ ∶ 𝐿𝑌 → 𝐿𝑌 and 𝑅□ ∶ 𝐿𝑋 → 𝐿𝑌 form an adjoint pair (𝑅♦ , 𝑅□ ), that is 𝐵♦ ≤ 𝐴 ⟺
𝐵 ≤ 𝐴□ for any 𝐴 ∈ 𝐿𝑋 , 𝐵 ∈ 𝐿𝑌 .
Proof. To show that 𝐴♦ ≤ 𝐵 ⟺ 𝐴 ≤ 𝐵■ assume first that 𝐴♦ ≤ 𝐵. Then by Proposition 3(2)
𝐴♦■ ≤ 𝐵■ and by Theorem 1(1) 𝐴 ≤ 𝐴♦■ ≤ 𝐵■ . Conversely, assume that 𝐴 ≤ 𝐵■ . Then
𝐴♦ ≤ 𝐵■♦ by Proposition 3(1) and 𝐴♦ ≤ 𝐵■♦ ≤ 𝐵 by Theorem 1(2).
To show that 𝐵♦ ≤ 𝐴 ⟺ 𝐵 ≤ 𝐴□ assume first that 𝐵♦ ≤ 𝐴. Then by Proposition 3(2)
𝐵 ♦□ ≤ 𝐴□ and by Theorem 1(1) 𝐵 ≤ 𝐵♦□ ≤ 𝐴□ . Conversely, assume that 𝐵 ≤ 𝐴□ . Then
𝐵♦ ≤ 𝐴□♦ by Proposition 3(1) and 𝐵♦ ≤ 𝐴□♦ ≤ 𝐴 by Theorem 1(2).
4. Degree of conceptuality of fuzzy preconcepts in fuzzy rough
sets theory
Following the main lines of research in formal concept analysis, we start with a pair of sets
𝑋 and 𝑌 and a fuzzy relation 𝑅 ∶ 𝑋 × 𝑌 → 𝐿. Now the set 𝑋 is interpreted as the set of some
objects and the set 𝑌 as the set of some properties, the value 𝑅(𝑥, 𝑦) ≥ 𝛼 ∈ 𝐿 means that the
object 𝑥 has the property 𝑦 at least to the degree 𝛼. The quadruple (𝑋 , 𝑌 , 𝑅, 𝐿) is called a formal
fuzzy context.
4.1. Preconcepts and preconcept lattices
Viewing 𝐴 ∈ 𝐿𝑋 as a fuzzy set of potential objects and 𝐵 ∈ 𝐿𝑋 as a fuzzy set of properties we
interpret the pair (𝐴, 𝐵) as a potential fuzzy concept. These leads to the following definition:
Definition 3. [26]1 Given a formal fuzzy context (𝑋 , 𝑌 , 𝑅, 𝐿), a pair 𝑃 = (𝐴, 𝐵) ∈ 𝐿𝑋 × 𝐿𝑌 is
called a fuzzy preconcept.
To view the set 𝐿𝑋 × 𝐿𝑌 of all fuzzy preconcepts as a lattice we must first introduce an order
on it. To do this reasonably, we must take into account that an increase in the set of objects 𝐴
naturally leads to a decrease in the set 𝐵 of properties that these objects satisfy. Therefore we
introduce a partial order ⪯ as follows. Given 𝑃1 = (𝐴1 , 𝐵1 ) and 𝑃2 = (𝐴2 , 𝐵2 ), we set 𝑃1 ⪯ 𝑃2 if
and only if 𝐴1 ≤ 𝐴2 and 𝐵1 ≥ 𝐵2 . Let (ℙ, ⪯) be the set 𝐿𝑋 × 𝐿𝑌 endowed with this partial order.
Further, given a family of fuzzy preconcepts {𝑃𝑖 = (𝐴𝑖 , 𝐵𝑖 ) ∶ 𝑖 ∈ 𝐼 } ⊆ 𝐿𝑋 × 𝐿𝑌 , we define its join
(supremum) by ⊻𝑖∈𝐼 𝑃𝑖 = (⋁𝑖∈𝐼 𝐴𝑖 , ⋀𝑖∈𝐼 𝐵𝑖 ) and its meet (infimum) as ⊼𝑖∈𝐼 𝑃𝑖 = (⋀𝑖∈𝐼 𝐴𝑖 , ⋁𝑖∈𝐼 𝐵𝑖 ).
1
This notion of a fuzzy preconcept is not related to the notion of a preconcept as it is defined in [8]
Theorem 4. [26]. (ℙ, ⪯, ⊻, ⊼) is a complete lattice. Besides, if 𝐿 is an infinitely bi-distributive
lattice, then (ℙ, ⪯, ⊻, ⊼) is also an infinitely bi-distributive lattice.
4.2. Concept lattices in rough set theory.
Patterned after [29], and using our notations, we define an object-oriented fuzzy concept as the
pair (𝐴, 𝐵) ∈ ℙ such that 𝐴 = 𝐵♦ and 𝐴□ = 𝐵. Similarly, a property-oriented fuzzy concept is
defined as a pair (𝐴, 𝐵) ∈ 𝐿𝑋 × 𝐿𝑌 such that 𝐴 = 𝐵■ and 𝐴♦ = 𝐵. In what follows, we restrict to
considering object-oriented fuzzy concepts; the case of property-oriented fuzzy concepts can be
easily obtained just by replacing fuzzy relation 𝑅 ∶ 𝑋 × 𝑌 → 𝐿 by its inverse 𝑅−1 ∶ 𝑌 × 𝑋 → 𝐿.
Let 𝕆(𝑋 , 𝑌 ) ⊆ 𝐿𝑋 × 𝐿𝑌 be the family of object-oriented fuzzy concepts.
In order to explain the meaning of an object-oriented fuzzy concept, we consider the simplest
non-trivial case. Let 𝐿 be a lattice whose all elements are isolated from below, let 𝑅 ∶ 𝑋 × 𝑌 →
{0, 1} ⊆ 𝐿 and let 𝐴 ∈ 𝐿𝑋 , 𝐵 ∈ 𝐿𝑌 . We denote 𝐴𝛼 = {𝑥 ∈ 𝑋 ∣ 𝐴(𝑥) ≥ 𝛼}; 𝐵𝛼 = {𝑦 ∈ 𝑌 ∣ 𝐵(𝑦) ≥ 𝛼}.
Thus 𝐴𝛼 is the set of objects 𝑥 ∈ 𝑋 belonging to 𝐴 at least to degree 𝛼, respectively 𝐵𝛼 is the
set of properties 𝑦 ∈ 𝑌 belonging to 𝐵 at least to degree 𝛼. Now 𝐴□ □
𝛼 = {𝑦 ∣ 𝐴 (𝑦) ≥ 𝛼} can be
described as the set of properties 𝑦 ∈ 𝑌 such that if 𝑦 is related to some object 𝑥, then 𝑥 ∈ 𝐴𝛼 .
In turn, 𝐵𝛼♦ = {𝑥 ∈ 𝑋 ∣ 𝐵♦ (𝑥) ≥ 𝛼} is the set of objects 𝑥 ∈ 𝑋 which are related to some 𝑦 ∈ 𝐵𝛼 .
♦
Now (𝐴, 𝐵) ∈ 𝕆(𝑋 , 𝑌 ) means that 𝐴□ 𝛼 = 𝐵𝛼 and 𝐴𝛼 = 𝐵𝛼 for all 𝛼 ∈ 𝐿.
Following the principles of concept analysis, the set (𝕆(𝑋 , 𝑌 ), ⪯) should be realized as a
complete lattice, where ⪯ is the relation induced from the partially ordered set (ℙ, ⪯, ⊼, ⊻), that
is (𝐴, 𝐵) ⪯ (𝐴′ , 𝐵′ ) iff 𝐴 ≤ 𝐴′ and 𝐵 ≥ 𝐵′ . Since the coordinate-wise infimum and supremum
of (even two) fuzzy concepts need not be a fuzzy concept, we must define infimum and
supremum operations on (𝕆(𝑋 , 𝑌 ), ≤) in a different way. We do it patterned after definitions
of such operations in case of “classical” (that is Wille-Ganter-Bĕlohlávek) concepts and concepts
in rough set theory, see [29]. Namely, given a family 𝔉 = {(𝐴𝑖 , 𝐵𝑖 ) ∣ 𝑖 ∈ 𝐼 } ⊆ 𝕆(𝑋 , 𝑌 ), we define
the infimum and the supremum of this family respectively by
□♦ ♦■
𝔉 = ((⋀𝑖∈𝐼 𝐴𝑖 ) , ⋀𝑖∈𝐼 𝐵𝑖 ) and 𝔉 = (⋁𝑖∈𝐼 𝐴𝑖 , (⋁𝑖∈𝐼 𝐵𝑖 ) ) .
c b
Theorem 5. (𝕆(𝑋 , 𝑌 ), ≤, , ) is a complete lattice.
□♦
Proof. We have to show that if 𝔉 = {(𝐴𝑖 , 𝐵𝑖 ) ∣ 𝑖 ∈ 𝐼 } ⊆ 𝕆(𝑋 , 𝑌 ), then ((⋀𝑖∈𝐼 𝐴𝑖 ) , ⋀𝑖∈𝐼 𝐵𝑖 ) and
♦■
(⋁𝑖∈𝐼 𝐴𝑖 , (⋁𝑖∈𝐼 𝐵𝑖 ) ) are object-oriented fuzzy concepts. However, we get it immediately by
applying statements (2) and (4) of Theorem 2 and recalling that 𝐴𝑖 = 𝐵𝑖♦ and 𝐴□
𝑖 = 𝐵𝑖 for every
𝑖 ∈ 𝐼.
4.3. Problems of practical applications of concept lattices.
Unfortunately the scope of practical applications of object-oriented (as well as property-oriented)
fuzzy concepts is rather restricted. For example, if 𝐴 ⊆ 𝑋, 𝐵 ⊆ 𝑌 are crisp sets and 𝑅 ∶ 𝑋 × 𝑌 → 𝐿,
then (𝐴, 𝐵) can be an object-oriented fuzzy concept only in case 𝐵♦ = 𝐴 and 𝐴□ = 𝐵. This
actually means that also 𝑅 must be a crisp relation and hence there is no even any ”flavour” of
fuzziness in this case. As a reasonable improvement of this situation we suggest to replace the
notion of an (object-oriented, property-oriented) fuzzy concept by a graded (object-oriented,
property oriented) fuzzy preconcept. We do it in the next subsection.
4.4. Graded fuzzy concept lattices in rough set theory
As above, let 𝑋 and 𝑌 be sets of objects and properties respectively and 𝑅 ∶ 𝑋 × 𝑌 → 𝐿 be a fuzzy
relation. We aim to determine for every pair (𝐴, 𝐵) ∈ 𝐿𝑋 × 𝐿𝑌 the extent, or the degree to which
this pair (𝐴, 𝐵) is a fuzzy object-oriented concept (resp. a fuzzy property-oriented concept).
This can be done along the lines of our article in [26], which proposes a graded approach in
“classical” fuzzy concept analysis.
Definition 4. The degree of (object-oriented) conceptuality of a pair (𝐴, 𝐵) ∈ 𝐿𝑋 × 𝐿𝑌 is defined
by Dob (𝐴, 𝐵) = (𝐴□ ≅ 𝐵) ∧ (𝐴 ≅ 𝐵♦ ).
The degree of (property-oriented) conceptuality of a pair (𝐴, 𝐵) ∈ 𝐿𝑋 × 𝐿𝑌 is defined by Dpr (𝐴, 𝐵) =
(𝐴♦ ≅ 𝐵) ∧ (𝐴 ≅ 𝐵■ ).
By changing pairs (𝐴, 𝐵) over 𝐿𝑋 × 𝐿𝑌 we obtain operators Dob ∶ 𝐿𝑋 × 𝐿𝑌 → 𝐿 and Dpr ∶
𝐿𝑋 × 𝐿𝑌 → 𝐿.
Definition 5. A graded fuzzy object-oriented concept lattice is the lattice ℙ endowed with operator
Dob ∶ 𝐿𝑋 × 𝐿𝑌 → 𝐿, that is the tuple (ℙ, ⪯, ⊻, ⊼, Dob ).
A graded fuzzy property-oriented concept lattice is the lattice ℙ endowed with operator Dpr ∶
𝐿𝑋 × 𝐿𝑌 → 𝐿, that is the tuple (ℙ, ⪯, ⊻, ⊼, Dpr ).
Going on with the study of graded fuzzy concept lattices in rough set theory, we shall restrict
to the case of an object-oriented lattice (ℙ, ⪯, ⊻, ⊼, Dob ), that is when Dob (𝐴, 𝐵) = (𝐴□ ≅
𝐵) ∧ (𝐴 ≅ 𝐵♦ ). The corresponding results for the property-oriented lattice Dpr (𝐴, 𝐵) = (𝐴□ ≅
𝐵) ∧ (𝐴 ≅ 𝐵♦ ) can be obtained in a similar way just by replacing fuzzy relation 𝑅 ∶ 𝑋 × 𝑌 → 𝐿
by the fuzzy relation 𝑅−1 ∶ 𝑌 × 𝑋 → 𝐿 defined by 𝑅−1 (𝑥, 𝑦) ∶= 𝑅(𝑦, 𝑥) for all 𝑥 ∈ 𝑋, for all
𝑦 ∈ 𝑌.
The computational complexity of the operator Dob ∶ 𝐿𝑋 × 𝐿𝑌 → 𝐿 laid in the definition of the
graded fuzzy object-oriented concept lattice (ℙ, ⪯, ⊻, ⊼, Dob ) and multidirectional components
involved in this definition are the reason for further study of these operators by dividing them
into several components.
4.5. Operators D1ob , D2ob , D3ob , D4ob ∶ 𝐿𝑋 × 𝐿𝑌 → 𝐿
Given a pair (𝐴, 𝐵) ∈ ℙ let D1ob (𝐴, 𝐵) = 𝐴□ ↪ 𝐵, D2ob (𝐴, 𝐵) = 𝐴□ ↩ 𝐵, D3ob (𝐴, 𝐵) = 𝐴 ↪ 𝐵♦ ,
D4ob (𝐴, 𝐵) = 𝐴 ↩ 𝐵♦ . Obviously,
Dob (𝐴, 𝐵) = D1ob (𝐴, 𝐵) ∧ D2ob (𝐴, 𝐵) ∧ D3ob (𝐴, 𝐵) ∧ D4ob (𝐴, 𝐵).
Next, we study the properties of the operators D1ob , D2ob , D3ob , D4ob separately.
Theorem 6. Properties of operator D1ob ∶ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿) → 𝐿.
1. Operator D1ob ∶ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿) → 𝐿 is upper semicontinuous, that is
D1ob (⋀𝑖∈𝐼 (𝐴𝑖 , 𝐵𝑖 )) ≥ ⋀𝑖∈𝐼 D1ob (𝐴𝑖 , 𝐵𝑖 ) ∀{(𝐴𝑖 , 𝐵𝑖 ) ∶ 𝑖 ∈ 𝐼 } ⊆ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿).
2. D1ob (𝐴, 1𝑌 ) = 1 for every 𝐴 ∈ 𝐿𝑋 . In particular D1ob (0𝑋 , 1𝑌 ) = 1. If 𝑅 is left connected, then
D1ob (0𝑋 , 𝐵) = 1 for every 𝐵 ∈ 𝐿𝑌 .
Proof. Referring to Proposition 6 and Proposition 2 we have
D1ob (⋀𝑖∈𝐼 (𝐴𝑖 , 𝐵𝑖 )) = D1ob (⋀𝑖∈𝐼 𝐴𝑖 , ⋁𝑖∈𝐼 𝐵𝑖 )) = (⋀𝑖∈𝐼 𝐴𝑖 )□ ↪ ⋁𝑖∈𝐼 𝐵𝑖 =
⋀𝑖∈𝐼 𝐴□ □ 1
𝑖 ↪ ⋁𝑖∈𝐼 𝐵𝑖 ≥ ⋀𝑖∈𝐼 (𝐴 ↪ 𝐵𝑖 ) ≥ ⋀𝑖∈𝐼 Dob (𝐴𝑖 , 𝐵𝑖 ).
From the definition it is clear that D1ob (𝐴, 1𝑌 ) = 𝐴□ ↪ 1 ≥ 1 ↦ 1 = 1. If 𝑅 is leftconnected, then
applying Proposition 5 we have D1ob (0𝑋 , 𝐵) = 0□ 𝑋 ↪ 𝐵 = 0𝑌 ↪ 𝐵 = ⋀𝑦∈𝑌 (0 ↦ 𝐵(𝑦)) = 1.
Theorem 7. Properties of operator D3ob ∶ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿) → 𝐿.
1. Operator D3ob ∶ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿) → 𝐿 is upper semicontinuous, that is
D3ob (⋀𝑖∈𝐼 (𝐴𝑖 , 𝐵𝑖 )) ≥ ⋀𝑖∈𝐼 D3ob (𝐴𝑖 , 𝐵𝑖 ) ∀{(𝐴𝑖 , 𝐵𝑖 ) ∶ ∈ 𝐼 } ⊆ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿).
2. D3ob (0𝑋 , 𝐵) = 1 for every 𝐵 ∈ 𝐿 in particular, D3ob (0𝑋 , 1𝑌 ) = 1. If 𝑅 is right connected, then
D3ob (𝐴, 1𝑌 ) = 1 for every 𝐴 ∈ 𝐿𝑋 .
Proof. Referring to Proposition 6, Proposition 2 we have
♦
D3ob (⋀𝑖∈𝐼 (𝐴𝑖 , 𝐵𝑖 )) = D3ob (⋀𝑖∈𝐼 𝐴𝑖 , ⋁𝑖∈𝐼 𝐵𝑖 )) = ⋀𝑖∈𝐼 𝐴𝑖 ↪ (⋁𝑖∈𝐼 𝐵𝑖 ) =
⋀𝑖∈𝐼 𝐴𝑖 ↪ ⋁𝑖∈𝐼 𝐵𝑖♦ ≥ ⋀𝑖∈𝐼 (𝐴𝑖 ↪ 𝐵𝑖♦ ) ≥ ⋀𝑖∈𝐼 D3ob (𝐴𝑖 , 𝐵𝑖 ).
Just from definitions we have D3ob (0𝑋 , 𝐵) = 0𝑋 ↪ 𝐵♦ = ⋀𝑥∈𝑋 (1 ↦ 𝐵♦ (𝑥)) ≥ 1 ↦ 1 = 1. In
case 𝑅 is right connected we apply Proposition 5 and have D3ob (𝐴, 1𝑌 ) = 𝐴 ↪ 1♦
𝑌 = 𝐴 ↪ 1𝑋 =
⋀𝑥∈𝑋 (𝐴(𝑥) → 1) = 1
Theorem 8. Properties of operator D2ob ∶ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿) → 𝐿.
1. Operator D2ob ∶ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿) → 𝐿 is lower semicontinuous, that is
D2ob (⋁𝑖∈𝐼 (𝐴𝑖 , 𝐵𝑖 )) ≥ ⋀𝑖∈𝐼 D2ob (𝐴𝑖 , 𝐵𝑖 ) ∀{(𝐴𝑖 , 𝐵𝑖 ) ∶ 𝑖 ∈ 𝐼 } ⊆ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿).
2. D2ob (𝐴, 0𝑌 ) = 1 for every 𝐴 ∈ 𝐿𝑋 and D2ob (1𝑋 , 𝐵) = 1 for every 𝐵 ∈ 𝐿𝑌 , in particular
D2ob (1𝑋 , 0𝑌 ) = 1.
Proof. Referring to Proposition 6, Proposition 2 we have
□
D2ob (⋁𝑖∈𝐼 𝐴𝑖 , ⋀𝑖∈𝐼 ) = (⋁𝑖∈𝐼 𝐴𝑖 ) ↩ ⋀𝑖∈𝐼 𝐵𝑖 ≥ ⋁𝑖∈𝐼 𝐴□ □
𝑖 ↩ ⋀𝑖∈𝐼 𝐵𝑖 ≥ ⋀𝑖∈𝐼 (𝐴𝑖 ↩ 𝐵𝑖 ) =
2
⋀𝑖∈𝐼 Dob (𝐴𝑖 , 𝐵𝑖 ).
D2ob (𝐴, 0𝑌 ) = 𝐴□ ↩ 0𝑌 = ⋀𝑦∈𝑌 (0 ↦ 𝐴□ (𝑦)) ≥ 0 ↦ 1 = 1. On the other hand D2ob (1𝑋 , 𝐵) =
1□ ↩ 𝐵 = ⋀𝑦∈𝑌 (𝐵(𝑦) ↦ 1) = 1.
Theorem 9. Properties of operator D4ob ∶ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿) → 𝐿.
1. Operator D4ob ∶ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿) → 𝐿 is lower semicontinuous, that is
D4ob (⋁𝑖∈𝐼 (𝐴𝑖 , 𝐵𝑖 )) ≥ ⋀𝑖∈𝐼 D4ob (𝐴𝑖 , 𝐵𝑖 ) ∀{(𝐴𝑖 , 𝐵𝑖 ) ∶ 𝑖 ∈ 𝐼 } ⊆ ℙ(𝑋 , 𝑌 , 𝑅, 𝐿).
2. D4ob (𝐴, 0𝑌 ) = 1 for every 𝐴 ∈ 𝐿𝑋 and D4ob (1𝑋 , 𝐵) = 1 for every 𝐵 ∈ 𝐿𝑌 . In particular
D4ob (1𝑋 , 0𝑌 ) = 1;
Proof. Referring to Proposition 6, Proposition 2 we have
♦
D4ob (⋁𝑖∈𝐼 𝐴𝑖 , ⋀𝑖∈𝐼 𝐵𝑖 ) = ⋁𝑖∈𝐼 𝐴𝑖 ↩ (⋀𝑖∈𝐼 𝐵𝑖 ) ≥ ⋁𝑖∈𝐼 𝐴𝑖 ↩ ⋀𝑖∈𝐼 𝐵𝑖♦ ≥ ⋀𝑖∈𝐼 (𝐴𝑖 ↩ 𝐵𝑖♦ ) =
⋀𝑖∈𝐼 D4ob (𝐴𝑖 , 𝐵𝑖 ).
Applying Proposition 5 D4ob (𝐴, 0𝑌 ) = 𝐴 ↩ 0♦ ♦
𝑌 = 0𝑌 ↪ 𝐴 = 0𝑋 ↪ 𝐴 = ⋀𝑥∈𝑋 (0 ↦ 𝐴(𝑥)) = 1.
In turn D4ob (1𝑋 , 𝐵) = 1𝑋 ↩ 𝐵♦ = 𝐵♦ ↪ 1𝑋 = ⋀𝑥∈𝑋 (𝐵♦ (𝑥) ↦ 1) = 1.
Corollary 2. If fuzzy relation 𝑅 is connected, then:
• Operators D1ob , D3ob ∶ 𝐿𝑋 ×𝐿𝑌 → 𝐿 are upper semicontinuous and D1ob (0𝑋 , 𝐵) = D3ob (0𝑋 , 𝐵) =
1 for every 𝐵 ∈ 𝐿𝑌 and D1ob (𝐴, 1𝑌 ) = D3ob (𝐴, 1𝑌 ) = 1 for every 𝐴 ∈ 𝐿𝑋 .
• Operators D2ob , D4ob ∶ 𝐿𝑋 × 𝐿𝑌 → 𝐿 are lower semicontinuous and D2ob (𝐴, 0𝑌 ) = D4ob (𝐴, 0𝑌 ) =
1 for any 𝐴 ∈ 𝐿𝑋 and D2ob (1𝑋 , 𝐵) = D4ob (1𝑋 , 𝐵) = 1 for any 𝐵 ∈ 𝐿𝑌
5. Examples of graded fuzzy concept lattices
In this section, we present two theoretical examples of graded fuzzy concept lattices. The first
of these examples as well as its generalizations (obtained by employing fuzzy sets 𝐴 and 𝐵 with
different number of values) of this example will be used for illustration of possible applications
outside “pure” mathematics. One of such applications is presented in the next subsection. The
second is a quite “artificial”, example. It is added in order to illustrate the fundamental role of the
fuzzy relation involved in the definition of graded fuzzy concepts and to stress the importance
of the reasonable choice of fuzzy relation when considering special situations.
2 2
Example 1. Let 𝑋 and 𝑌 be sets partitioned into three disjoint subsets 𝑋 = ⋃𝑖=0 𝑋𝑖 and 𝑌 = ⋃𝑖=0 𝑌𝑖 .
Further, let relation 𝑅 ∶ 𝑋 × 𝑌 → {0, 1} be defined by
1 if (𝑥, 𝑦) ∈ (𝑋0 × 𝑌0 ) ∪ (𝑋1 × 𝑌1 ) ∪ (𝑋2 × 𝑌2 )
𝑅(𝑥, 𝑦) = {
0 otherwise
We consider fuzzy sets 𝐴 ∶ 𝑋 → [0, 1], 𝐵 ∶ 𝑌 → [0, 1] and calculate 𝐴□ and 𝐵♦ :
0 if 𝑥 ∈ 𝑋0 0 if 𝑦 ∈ 𝑌0
𝐴(𝑥) = { 𝑎 if 𝑥 ∈ 𝑋1 𝐴□ (𝑦) = { 𝑎 if 𝑦 ∈ 𝑌1
1 if 𝑥 ∈ 𝑋2 1 if 𝑦 ∈ 𝑌2
0 if 𝑦 ∈ 𝑌0 0 if 𝑥 ∈ 𝑋0
𝐵(𝑦) = { 𝑏 if 𝑦 ∈ 𝑌1 𝐵♦ (𝑥) = { 𝑏 if 𝑥 ∈ 𝑋1
1 if 𝑦 ∈ 𝑌2 1 if 𝑥 ∈ 𝑋2
Thus we have:
𝐴□ ↪ 𝐵 = 𝑎 ↦ 𝑏, 𝐴□ ↩ 𝐵 = 𝑏 ↦ 𝑎 and 𝐴□ ≅ 𝐵 = 𝑎 ↦ 𝑏 ∧ 𝑏 ↦ 𝑎
𝐴 ↪ 𝐵♦ = 𝑎 ↦ 𝑏, 𝐴 ↩ 𝐵♦ = 𝑏 ↦ 𝑎 and 𝐴 ≅ 𝐵♦ = 𝑎 ↦ 𝑏 ∧ 𝑏 ↦ 𝑎.
Hence, in particular, for the most important 𝑡-norms:
- for the Łukasiewicz 𝑡-norm Dob (𝐴, 𝐵) = 1− ∣ 𝑎 − 𝑏 ∣;
- in case of the product 𝑡-norm Dob (𝐴, 𝐵) = min{ 𝑎𝑏 , 𝑎𝑏 };
- in case of the minimum 𝑡-norm Dob (𝐴, 𝐵) = min{𝑎, 𝑏}.
2 2
Example 2. Let 𝑋 and 𝑌 be sets partitioned into three disjoint subsets 𝑋 = ⋃𝑖=0 𝑋𝑖 and 𝑌 = ⋃𝑖=0 𝑌𝑖
and let 𝐴 ∶ 𝑋 → [0, 1], 𝐵 ∶ 𝑌 → [0, 1] be defined as in the previous example. Further, let
relation 𝑅 ∶ 𝑋 × 𝑌 → {0, 1} be defined by
𝑐 if (𝑥, 𝑦) ∈ 𝑋0 × 𝑌1
𝑅(𝑥, 𝑦) ∶= { 1 if (𝑥, 𝑦) ∈ 𝑋1 × 𝑌2 ∪ 𝑋2 × 𝑌0
0 otherwise
Then
1 if 𝑦 ∈ 𝑌0
𝐴□ (𝑦) = { 𝑐 ↦ 0 if 𝑦 ∈ 𝑌1 𝑐∗𝑏 if 𝑥 ∈ 𝑋0
𝑎 if 𝑦 ∈ 𝑌2 𝐵♦ (𝑥) = { 1 if 𝑥 ∈ 𝑋1
0 if 𝑥 ∈ 𝑋2
By an easy calculation we get
- D1ob (𝐴, 𝐵) = 𝐴□ ↪ 𝐵 = 0, D2ob (𝐴, 𝐵) = 𝐴□ ↩ 𝐵 = 𝑏 ↦ (𝑐 ↦ 0) ∧ 𝑎
- D3ob (𝐴, 𝐵) = 𝐵♦ ↪ 𝐴 = (𝑐 ∗ 𝑏 ↦ 0) ∧ 𝑎, D1ob (𝐴, 𝐵) = 𝐵♦ ↩ 𝐴 = 0,
and hence Dob (𝐴, 𝐵) = 0.
6. Application of graded fuzzy concept lattices in assessment of
country risks
We consider the imposition of international sanctions (embargoes) to different countries for
illustration of possible application of graded fuzzy concept lattices. These sanctions are in-
troduced either by international organizations (like the United Nations) or different countries
(including the European Union) to restrict economic activities of particular countries posing
different global risks via prohibiting the trading of certain goods or provision of particular
services, limiting the access to financial resources, etc. Sanctions are also applied to different
private individuals and legal entities. Consequently, several countries and their residents are
subject to restrictive measures of lower or higher degree which can be mathematically illustrated
using fuzzy sets. Meanwhile only North Korea is considered as fully sanctioned country, but
the recent emergence of the war in Ukraine has induced a global wave of new sanctions against
Russia making it as almost fully sanctioned country with only few leftovers for very limited
non-restricted trades with commodities, e.g. natural gas.
In order to set up a conceptual lattice reflecting this matter we consider the countries as objects
and the sanctions as properties. The contents of this lattice can be visualized, e.g. using data
from the site www.sanctionsmap.eu comprising information on sanctions imposed by the Euro-
pean Union and the United Nations, and these data can be further enriched by different other
sanctions imposed by countries with major global influence, like the United States of America.
Naturally, based on our Example 1 provided in the previous section all non-sanctioned countries
are included in the subset 𝑋0 , all fully sanctioned countries are included in the subset 𝑋2 , and
all partly sanctioned countries are included in the subset 𝑋1 thus creating the set of objects. For
the sake of clarity we should admit that terms ”partly” and ”fully” are conditional and depend
on the total number of particular sanctions applied to corresponding countries. Furthermore
we introduce the set of properties by applying different importance to particular sanctions. In
such case the subset 𝑌0 is introduced as subset not containing any sanctions, subset 𝑌2 contains
the sanctions imposing the strongest possible restrictions, e.g. prohibition of any financial
transactions, and subset 𝑌1 contains all other types of sanctions targeting particular sectors of
economic activities. For the purposes of our example we have chosen the value of 𝑎 equal to 0.4
and the value of 𝑏 equal to 0.6. We obtain the following fuzzy sets 𝐴 and 𝐵 and corresponding
values for 𝐴□ and 𝐵♦ :
0 if 𝑥 ∈ 𝑋0 0 if 𝑦 ∈ 𝑌0
𝐴(𝑥) = { 0.4 if 𝑥 ∈ 𝑋1 𝐴□ (𝑦) = { 0.4 if 𝑦 ∈ 𝑌1
1 if 𝑥 ∈ 𝑋2 1 if 𝑦 ∈ 𝑌2
0 if 𝑦 ∈ 𝑌0 0 if 𝑥 ∈ 𝑋0
𝐵(𝑦) = { 0.6 if 𝑦 ∈ 𝑌1 𝐵♦ (𝑥) = { 0.6 if 𝑥 ∈ 𝑋1
1 if 𝑦 ∈ 𝑌2 1 if 𝑥 ∈ 𝑋2
The meaning of 𝐴□ is explained as relation of particular subsets of sanctions to corresponding
countries while the meaning of 𝐵♦ is explained as relation of particular subsets of countries
to corresponding sanctions. Furthermore we obtain the following values of Dob (𝐴, 𝐵): 0.8 in
case of Łukasiewicz 𝑡-norm, 0.67 in case of product 𝑡-norm and 0.4 in case of minimum 𝑡-norm.
These results basically demonstrate that the level of country sanctioning, i.e. ”partial” or ”full”
cannot exceed the importance of sanctions applied to respective countries. This is also the
mathematical reasoning for, e.g. difficulties in taking decision on disconnection of all Russian
financial institutions from the global network of the Society for Worldwide Interbank Financial
Telecommunication (SWIFT) while still enabling of at least limited trades with commodities
(e.g. natural gas). This particular example was used to demonstrate the applicability of concept
lattice for analysis of international sanction in a very simplified way. It is obvious that in a
more realistic scenario we must refine both the partitioning of the set of objects (countries)
and the set of properties (sanctions). As a result we would be dealing with more granulated
𝑚 𝑛
setup, disjoint subsets 𝑋 = ⋃𝑖=0 𝑋𝑖 and 𝑌 = ⋃𝑗=0 𝑌𝑗 and corresponding values of 𝑎1 , ..., 𝑎𝑚 and
𝑏1 , ..., 𝑏𝑛 . These values of 𝑎𝑖 and 𝑏𝑗 are still finite and in practice do not exceed the number of
officially recognized countries (currently there are 197 countries) and the number of meaningful
sanctions. The analysis of such more granulated landscape is subject to our further research.
7. Conclusion
Operators D1ob , D2ob , D3ob and D4ob ∶ 𝐿𝑋 × 𝐿𝑌 → 𝐿 were introduced in order to evaluate various
aspects of the correspondence between a fuzzy set of (potential) objects and a fuzzy set of
(potential) properties. Based on these operators, we defined graded fuzzy concept lattices and
studied their main properties. Our next research will focus on using these operators for the
construction of so called, conceptional hulls of fuzzy preconcepts. Given a fuzzy preconcept
(𝐴, 𝐵) ∈ 𝐿𝑋 × 𝐿𝑌 and 𝛼 ∈ 𝐿, the triple (𝐴, 𝐵↑ , 𝐵↓ ), where 𝐵↓ ≤ 𝐵 ≤ 𝐵↑ , is called the concept 𝛼-hull
of a fuzzy preconcept (𝐴, 𝐵), if
1 (𝐴, 𝐵 ↑ ) ∧ 𝐷 3 (𝐴, 𝐵 ↑ ) ≥ 𝛼 and 𝐷 2 (𝐴, 𝐵 ↓ ) ∧ 𝐷 4 (𝐴, 𝐵 ↓ ) ≥ 𝛼
𝐷𝑜𝑏 𝑜𝑏 𝑜𝑏 𝑜𝑏
and in addition 𝐵↑ , 𝐵↓ are the “optimal” fuzzy sets satisfying these conditions. We shall carry
out the study of the construction 𝐿𝑋 × 𝐿𝑌 ⇒ 𝐿𝑋 × 𝐿𝑌 × 𝐿𝑌 assigning to each (𝐴, 𝐵) ∈ 𝐿𝑋 × 𝐿𝑌
its conceptional 𝛼-hull (𝐴, 𝐵↑ , 𝐵↓ ). Specifically we plan to investigate algebraic (in particular
categorical) and topological properties of this construction. In addition, we intend to apply
this construction for practical purposes, in particular, for the choice of strategy developed in
response to the risks related to global political and economic developments.
The second interesting direction, kindly indicated us by a referee, is the study of the relations
between graded fuzzy concept lattices and pattern structures described by certain semi-lattices,
see, e.g. [9], [18]. In such structures patterns are used instead of attributes (properties), allowing
thus a certain gradation of concepts.
Acknowledgments
The first and the third named authors are grateful for the partial financial support from the
project No. Lzp-2020/2-0311 by the Latvian Council of Science. All authors thank the anonymous
referees for reading the paper carefully and making useful comments.
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