Factorizing formal contexts from closures of necessity operators

Roberto G. Aragón; Jesús Medina; Eloísa Ramírez-Poussa

cs.AIFeb 25, 2026

Factorizing formal contexts from closures of necessity operators

Authors: Roberto G. Aragón, Jesús Medina, Eloísa Ramírez-Poussa

Organizations: Department of Mathematics, University of Cádiz. Spain

Abstract

Factorizing datasets is an interesting process in a multitude of approaches, but many times it is not possible or efficient the computation of a factorization of the dataset. A method to obtain independent subcontexts of a formal context with Boolean data was proposed in~\cite{dubois:2012}, based on the operators used in possibility theory. In this paper, we will analyze this method and study different properties related to the pairs of sets from which a factorization of a formal context arises. We also inspect how the properties given in the classical case can be extended to the fuzzy framework, which is essential to obtain a mechanism that allows the computation of independent subcontexts of a fuzzy context.

Figures & tables

$R$	$b_{1}$	$b_{2}$	$b_{3}$	$b_{4}$	$b_{5}$	$b_{6}$
$a_{1}$	0	1	1	1	0	0
$a_{2}$	0	0	0	1	0	0
$a_{3}$	1	0	0	0	0	0
$a_{4}$	0	0	0	0	1	1
$a_{5}$	0	0	1	0	0	0
$a_{6}$	0	0	0	0	1	0

Table 1: Relation of the formal context of Example 12 .

Panel of Figure 1 — Figure 1: List of concepts and concept lattice of Example 12 .

Panel of Figure 2 — Figure 2: List of elements and lattice of $\mathcal{C}_{N}$ of Example 12 .

$R$	$b_{1}$	$b_{2}$	$b_{3}$	$b_{4}$	$b_{5}$	$b_{6}$	$b_{7}$
$a_{1}$	0	1	1	1	0	0	0
$a_{2}$	0	0	0	1	0	0	0
$a_{3}$	1	0	0	0	0	0	0
$a_{4}$	0	0	0	0	0	1	1
$a_{5}$	0	1	1	0	0	0	0
$a_{6}$	0	0	0	0	1	1	0

Table 2: Relation of the formal context of Example 17 .

Panel of Figure 3 — Figure 3: List of concepts and concept lattice of Example 17 .

$R$	$b_{1}$	$b_{2}$
$a_{1}$	0.5	0
$a_{2}$	0	0.75

Table 3: Fuzzy relation

R

of Example 19 .

Figure 4: Fuzzy relation $R_{1}$ and concept lattice of Example 21 .

Figure 5: Fuzzy relation $R_{2}$ and concept lattice of Example 21 .

(g_{\bot},f_{\bot})=(\{b_{1}/0,b_{2}/0,b_{3}/0\},\{a_{1}/0,a_{2}/0,a_{3}/0\})

(g_{1},f_{1})=(\{b_{1}/0,b_{2}/1b_{3}/0\},\{a_{1}/0,a_{2}/1,a_{3}/0\})

(g_{2},f_{2})=(\{b_{1}/0.25,b_{2}/0,b_{3}/0.25\},\{a_{1}/0.25,a_{2}/0,a_{3}/0.25\})

(g_{3},f_{3})=(\{b_{1}/0.25,b_{2}/0,b_{3}/0.5\},\{a_{1}/0.5,a_{2}/0,a_{3}/0.25\})

(g_{4},f_{4})=(\{b_{1}/0.5,b_{2}/0,b_{3}/0.25\},\{a_{1}/0.25,a_{2}/0,a_{3}/0.5\})

(g_{5},f_{5})=(\{b_{1}/0.5,b_{2}/0,b_{3}/1\},\{a_{1}/1,a_{2}/0,a_{3}/0.5\})

(g_{6},f_{6})=(\{b_{1}/0.5,b_{2}/1,b_{3}/0.25\},\{a_{1}/0.25,a_{2}/1,a_{3}/0.5\})

Table 4: Some pairs of

\mathcal{F}_{N}

in the context

(A,B,R_{2})

of Example 21 .

$R_{1}$	$b_{1}$	$b_{2}$	$b_{3}$
$a_{1}$	0.5	0.25	0
$a_{2}$	0.5	1.0	0
$a_{3}$	0	0	0.75

Table 5: Fuzzy relation

R_{1}

of Example 31 .

$R_{2}$	$b_{1}$	$b_{2}$	$b_{3}$
$a_{1}$	1	0.25	0
$a_{2}$	0.5	1	0
$a_{3}$	0	0	1

Table 6: Fuzzy relation

R_{2}

of Example 31 .

(g_{\bot},f_{\bot})=(\{b_{1}/0,b_{2}/0,b_{3}/0\},\{a_{1}/0,a_{2}/0,a_{3}/0\})

(g_{1},f_{1})=(\{b_{1}/0,b_{2}/0,b_{3}/0.5\},\{a_{1}/0,a_{2}/0,a_{3}/0.5\})

(g_{2},f_{2})=(\{b_{1}/0.75,b_{2}/0.5,b_{3}/0\},\{a_{1}/0.75,a_{2}/0.5,a_{3}/0\})

(g_{3},f_{3})=(\{b_{1}/1,b_{2}/0.75,b_{3}/0\},\{a_{1}/1,a_{2}/0.75,a_{3}/0\})

(g_{4},f_{4})=(\{b_{1}/0.75,b_{2}/0.5,b_{3}/0.5\},\{a_{1}/0.75,a_{2}/0.5,a_{3}/0.5\})

(g_{5},f_{5})=(\{b_{1}/0,b_{2}/0,b_{3}/0.5\},\{a_{1}/0,a_{2}/0,a_{3}/0.5\})

(g_{\top},f_{\top})=(\{b_{1}/1,b_{2}/1,b_{3}/1\},\{a_{1}/1,a_{2}/1,a_{3}/1\})

Table 7: Some pairs of

\mathcal{F}_{N}

in the context

(A,B,R_{2})

of Example 31 .

Formal Concept Analysis

Formal Concept Analysis organizes object–attribute data into concept lattices and uses their structure for reasoning and interpretation.

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Formal Concept Analysis with Three Types of Negation

Zhenghua Pan

School of Mathematics and Data Science, Jiangnan University, Wuxi 214122, China

Formal Concept Analysis

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Classic Formal Concept Analysis (FCA) primarily focuses on the positive relationships between objects and attributes and does not have mechanisms for handling negation.To overcome this limitation, we introduce three types of negation concepts (contradictory negation, opposite negation, intermediary negation) into FCA.Based on the set SCOI and logic LCOI+PLCOI with these three types negation, we define formal context, Galois connection operators, formal concept and concept lattice with three types of negation,this leads to the proposal of a FCACOI: Formal Concept Analysis with contradictory negation, opposite negation and intermediary negation.For the reasoning in FCACOI, this paper focuses on attribute implication reasoning. Based on the logic LCOI+PLCOI and its semantics, we introduce the notion of ICOI-entailment as the semantic implication for attribute implication reasoning in FCACOI. Through ICOI-entailment, a connection is established between attribute implication reasoning in FCACOI and inference in the logic LCOI+PLCOI, it indicate that formally proven inference rules (theorems) in LCOI+PLCOI are valid in the attribute implication reasoning of FCACOI, LCOI+PLCOI provides a logical foundation for attribute implication reasoning in FCACOI. To illustrate the capability of attribute implication reasoning in FCACOI, we discuss its application in a concrete example. Moreover, we explore attribute reduction of the formal context in FCACOI, propose two research frameworks for attribute reduction from different perspectives, and compare their characteristics.We believe that, based on richer logic and semantics, FCACOI elevates FCA from a theory that describes affirmations to one that can describe affirmations and its contradiction(either this or that), opposition(extreme negation) and intermediary (transitional states between oppositions).

Abstract

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