A General Theory of Concept Lattice (I): Emergence of General Concept Lattice

TL;DR

This paper introduces a general concept lattice framework that unifies and extends traditional lattices like formal concept and rough set lattices, ensuring invariance of information content across different variable presentations.

Contribution

It develops the first part of a comprehensive theory for a general concept lattice that can encompass various categorization methods and guarantees invariance of information content.

Findings

Existence of a general concept lattice compatible with traditional lattices

The general lattice is invariant under different variable presentations

Formal concept and rough set lattices are special cases of the general lattice

Abstract

As the first part of the treatise on A General Theory of Concept Lattice (I-V), this work develops the general concept lattice for the problem concerning categorization of objects according to their properties. Unlike the conventional approaches, such as the formal concept lattice and the rough set lattice, the general concept lattice is designed to adhere to the general principle that the information content should be invariant regardless how the variables/parameters are presented. Here, one will explicitly demonstrate the existence of such a construction by a sequence of fulfillment compatible with the conventional lattice structure. The general concept lattice promises to be a comprehensive categorization for all the distinctive object classes according to whatever properties they are equipped with. It will be shown that one can always regain the formal concept lattice and rough set lattice from the general concept lattice.

Equations

$$X\subseteq G$$

$$Y\subseteq M$$

$$X\subseteq G$$

$$Y\subseteq M$$

$$X\subseteq G$$

$$Y\subseteq M$$

$$X^{III}=X^I,$$

$$X_1\subseteq X_2⟺(X_2{)}^I\subseteq (X_1{)}^I,$$

$$X^{c\square c}=X^{◊},X^{c◊c}=X^{\square },$$

$$\{\begin{array}{c}{X}^{\square }=Y\\ Y^{◊}=X\end{array}⟺\{\begin{array}{c}{X}^{cc\square c}=Y^c\\ Y^{cc◊c}=X^c\end{array}⟺\{\begin{array}{c}(X^c{)}^{◊}=(Y^c)\\ (Y^c{)}^{\square }=(X^c)\end{array},$$

$$\frac{[X_0^{\times }{]}_F}{[X_i^{\times }{]}_F}$$

$$\frac{[X_0^{+}{]}_F}{[X_j^{+}{]}_F}$$

$$(\neg \mu {)}^R={\left(\prod _{\alpha \in \psi }\neg \alpha \right)}^R={\left(\prod _{\alpha \in {\psi }^{\prime }}\alpha \right)}^R=X^c\text{mbox}but\forall \neg {\alpha }_1\in {\psi }^{\prime }{\left(\prod _{\alpha \in {\psi }^{\prime }\backslash \{\neg {\alpha }_1\}}\alpha \right)}^R\ne X^c,$$

$$X^I=\{m\in M\mid gRm,\forall g\in X\}$$

$$X^{\square }=\{m\in M\mid \forall g\in G,gRm⟹g\in X\}$$

$$X^{◊}=\{m\in M\mid \exists g\in G,(gRm,g\in X)\}$$

$$X\subseteq G$$

$$Y\subseteq M^{\ast }$$

$$X\subseteq G$$

$$Y\subseteq M^{\ast }$$

$$X\subseteq G$$

$$Y\subseteq M^{\ast }$$

$$\rho (X)=\sum _{X_0\subseteq X}\left(\sum [X_0^{\times }{]}_F\right)=\sum _{X_0\subseteq X}\sum \left(\bigcap _{X_0\subset X_i\subseteq X}\frac{[X_0^{\times }{]}_F}{[X_i^{\times }{]}_F}\right).$$

$$\eta (X)=\prod _{X_0\supseteq X}\left(\prod [X_0^{+}{]}_F\right)=\prod _{X_0\supseteq X}\prod \left(\bigcap _{X_0\supset X_j\supseteq X}\frac{[X_0^{+}{]}_F}{[X_j^{+}{]}_F}\right).$$

$$X_i\subset X_j⟺(X_j{)}^{I^{\ast }}\subset (X_i{)}^{I^{\ast }}$$

$$\displaystyle X_{i}\subset X_{j}\iff(X_{i})^{\Box^{\ast}}\subset(X_{j})^{\Box^{\ast}}\$$

$$\rho (X)=\sum _{X_0\subseteq X}\sum \left(\bigcap _{X_0\subset X_i\subseteq X}\frac{[X_0^{\times }{]}_F}{[X_i^{\times }{]}_F}\right)=\sum \left(\bigcup _{X_0\subseteq X}\left(\bigcap _{X_0\subset X_i\subseteq X}\frac{[X_0^{\times }{]}_F}{[X_i^{\times }{]}_F}\right)\right),$$

$$\eta (X)=\prod \left(\bigcup _{X_0\supseteq X}\left(\bigcap _{X_0\supset X_j\supseteq X}\frac{[X_0^{+}{]}_F}{[X_j^{+}{]}_F}\right)\right),$$

$$l_i\wedge l_j$$

$$l_i\vee l_j$$

$$l_k^{†}$$

$$\underset{l_0\in L_0}{\bigvee }{l}_0$$

$$\underset{l_0\in L_0}{\bigwedge }{l}_0$$

$$l_{sup}$$

$$l_{inf}$$

$$E_F^{rsl}=\left\{\bigcup _{m\in M_0}{m}^R\mid M_0\subseteq M\right\},E_F^{fcl}=\left\{\bigcap _{m\in M_0}{m}^R\mid M_0\subseteq M\right\}.$$

Full text

General Concept Lattice; Categorization; Formal Concept Lattice; Rough Set Lattice.

1 introduction

It is commonly accepted that the formal concept analysis (FCA) Wi82 ; GW99 ; Wi05 and the rough set theory (RST) Pa82 ; Pa91 are important approaches to the Big Data Analytics. One can superficially tell such importance from the rapid growth of interest in both fields, e.g., the inclusion of the FCA and/or RST topics in many international conferences and workshops. Although FCA and RST might have been motivated differently, it is believed that they are mutually expressible Ke96 ; DGO01 ; GD02 ; DG03 ; YY04 ; Wa05 . Historically, the formal concept lattice (FCL) is a native part for FCA since FCA had its origin in activities of restructuring mathematics, in particular, mathematics order and lattice theory Wi05 . On the other hand, the rough set lattice (RSL) has also been accomplished DG03 ; YY04 in terms of the modal logic operators which cope with RST in the binary version. Although the two concept lattices may be seen as dealing with different categorisations, one may articulate them of the same object collection $G$ and the same attribute set $M$ related in a unique information system, i.e. the same formal context. However, the systematic incorporation of the FCL and the RSL in the problem analysis remains unobserved.

The treatise “A General Theory of Concept Lattice (I$\sim$V)” is initiated by the general idea that the information content should be invariant regardless how the variables/parameters are presented. Notice the fact that the conventional employment of objects and attributes are intuitively different. Unlike the objects, which are individuals, the attributes as the object property description can overlap per conjunction. Thus, for general characterisation of logic problems it is essential also to take into accout the composite attributes based on the operations $\left\{\mbox{conjunction, disjunction, negation}\right\}$. There is no reason to stress the priority of the simple attributes in $M$ over the composite attributes since one may always rename the composite attributes into new simple (in contrast to composite) ones, which may render those originally simple attributes composite. Instructed by such a principle, the general concept lattice (GCL) thus speaks of the generalised attributes, which incorporate both the simple and the composite attributes constructable out of the members of $M$.

In this part (I) the existence of GCL subject to the formal context will be demonstrated, giving rise to a more informative structure than the conventional RSL and FCL. The GCL provides a comprehensive categorisation for whatever objects discernible by the formal context since in its construction one can exhaust all the generalised attributes, as will be clarified in the part (II) LLJD12-2 . By contrast, the FCL is devised for those attributes given as conjunctions of the members of $M$ and the RSL for those disjunctions. Remarkably, the comprehensive categorisation in the GCL then brings about a simple framework for all the implications extractable from the lattice structure. While these implications are considered between two generalised attributes, those implications deducible from the FCL and RSL can be regained when one restricts oneself to the attributes of conjunction type and disjunction type, respectively. The simple framework for all the implications enabled by the GCL in fact inspires the idea to represent any implication relation by one single attribute, thereby rendering the primary deduction system (PDS), see the part (III) LLJD12-3 , which is a simplified systems where logic statements only take into account the properties on a single subject by means of predicates. The PDS is algebraically manipulable in the sense that the deductions are solely Boolean algebras bypassing any of axioms. Note that being able to achieve an algebraically manipulable deduction is rather practical. One may then manage to resolve any logic problem mindlessly if it could be arranged according to the syntax of the algebraically manipulable deduction. Indeed, certain prevalent puzzles will be shown to be resolved in the PDS in this manner. Another point is that tautology in effect occurs whenever the algebraically manipulable deduction results in the Truth 1, which is essentially what all the axioms should end up. It can be shown that the PDS adheres to the classical logic rules since the Hilbert axioms all turns out to be tautology when restricted to the PDS.

In the parts (IV) and (V) of this treatise LLJD12-4 ; LLJD12-5 , efforts are devoted to render the algebraically manipulable deduction more realistic such that it can cope with the conventional reasoning. The following two points are of concern.

• •

Dealing with pure attribute-typed logic statements in the PDS is naïve since in reality every assertion, where a property assignment has to be ascribed to a definite referential attribute set based on which a collection of judgements altogether forms the statement.

• •

The PDS as concerned with the deduction on one sole subject (object class) cannot be sufficiently expressive in formalising a general logic statement; one is looking forward to the further extensions.

For the first point, various attribute-type logic statements, though ascribed to different referential attribute sets, may become the same one. In fact, such a problem reminds one about the conventional indifference of the set unions $X^{c}\cup X=Y^{c}\cup Y=(X^{c}\cap Y^{c})\cup(X\cap Y^{c})\cup(X^{c}\cap Y)\cup(X\cap Y)$ and so forth. Therefore, one argues LLJD12-4 that the notion of finite resolution enters the syntax of PDS and participates in the deduction as a novelty. The PDS incorporating the finite resolution will then be referred to as the resolvable PDS, which however exhibits anomalies for some of the Hilbert Axioms, entailing that the classical logic comprises certain counter-intuitiveness. As for the second point, the secondary deduction system (2DS) LLJD12-5 is designed to take into account those logic statements relating predicate forms among different subjects. Note that, despite involving the PDS syntax as its particular statements, the 2DS can be developed in parallel to the PDS. Thus, one does not anticipate different deduction rules from the 2DS than from the PDS. Moreover, the resolvable 2DS can likewise arise from the reasoning incorporating finite resolution, which can be traced back to the same origin that elicits the resolvable PDS.

To start with in this part (I), the GCL is a structure ordering lattice nodes marking the general concepts by means of Galois connections. The general concept takes the 2-tuple representation (general extent, general intent), (Ge,Gi) for short, which is similar to the formal concepts employed in FCL and in RSL. Below, one will demonstrate the existence of such GCL through the accomplishment of the following goals.

G1

The general extent runs over all the distinctive object classes recognised by the formal context. The general intent plays the role of properties of a definite object class and can be characterised by a pair of attributes referred to as the generalised rough set property (Grsp) and the generalised formal concept property (Gfcp). The 2-tuples (Ge, Grsp) and (Ge, Gfcp) in effect satisfy the generalisation of lattice conditions, respectively, employed for the RSL DG03 ; YY04 and for the FCL Wi82 ; GW99 ; Wi05 .

G2

The GCL constitutes a Hasse diagram with a power set structure, where the nodes in the Hasse diagram can be related by means of set inclusion relations among the object classes discernible by the formal context.

G3

Both RSL and FCL can be regained from the GCL as particular features.

G4

The GCL allows for deterministic construction based on a particular collection of general concepts to be described later.

G5

The GCL manifests a conjugate relation between its RSL- and FCL- counterparts and consequently emerges as a self-dual lattice.

The formulation of the work at hand differs from the traditional approaches (typically, Wi82 ; GW99 ; DG03 ; YY04 ) mainly by the employment of the generalised attributes. For concreteness, let $M$ be the attribute set under consideration then the GCL takes into account generalised attributes $M^{\ast}$ which are all the possible compositions of members from $M$ by means of all the basic Boolean operations. Thus, the information provided by the composite attributes $M^{\ast}\backslash M$ could be as significant as the information given in terms of simple (in contrast to the composite ones) attributes $M$. In Sec. 2 some preliminaries of the RST and FCA are revisited, where one attempts an equal-footing treatment for both theories based on their original conventions with minimal notation refinements. This is to establish a convenient framework above which one may differentiate the GCL with the RSL and FCL. In Sec. 3, it is shown that both the FCL- and RSL- intents could have been represented by single composite attributes without loss of generality. Hence, the consideration to generalise the attribute domain from $M$ to $M^{\ast}$ in the problem setting becomes intuitive. Moreover, the sign of the GCL is revealed from some inconsistency observed in the conventional FCL and RSL approach. In order to retain the consistency, one inevitably takes into account the full correspondence between the objects and generalised attribute set $M^{\ast}$, which then gives rise to the GCL. It turns out that the GCL ends up a comprehensive categorisation since its object classes range over all the possible object sets discernible by the formal context. In addition, the nodes on the GCL can be ordered as general concepts. While each object class is identified with a general extent, the corresponding general intent as its associate property may be acquired from the generalisation of FCL-intent and RSL-intent. Although all the general intents can be directly determined per read out from the formal context, as will be demonstrated in the next paper, one chooses here a constructive approach for them by means of certain irreducibility conditions. It is through the design of such irreducibility conditions that one can reveal how the traditional RSL and FCL being regained as part of GCL. In Sec. 4, results are summarised to affirm the accomplishments of goals G1 to G5. There are also discussions concerning further developments for the general theory of concept lattice.

2 preliminaries and beneath

The formal context as stated in the FCA approach Wi05 is defined as a set structure $K:=(G,M,I)$, for which $G$ and $M$ are sets while $I$ is binary relation between $G$ and $M$, i.e. $I\subseteq G\times M$; the elements of $G$ and $M$ are called (formal) objects and (formal) attributes, respectively, and $gIm$, i.e. $(g,m)\in I$, is read: the object $g$ has the attribute $m$. However, for the RST theorists there is a different convention, governing similar functions, where the information system (IS) $(U,V,R)$ Pa82 is often employed as a synonym of formal context. Hence, in order to prevent from confusion caused by simultaneously treating both theories there is a compromise that preserves most of the original notations employed in FCA and RST as follows.

Definition 2.1**.**

A formal context is a set structure $K:=(G,M,R)$ (let it also be denoted by $F(G,M)$), for which $G$ and $M$ are sets while $R$ is binary relation between $G$ and $M$, i.e. $R\subseteq G\times M$; the elements of $G$ and $M$ are called formal objects and formal attributes, respectively, and $gRm$, i.e. $(g,m)\in R$, is read: the object $g$ has the attribute $m$.

Here, the notation $F(G,M)$ is to emphasise that $G$ and $M$ are related, conceived through the formal context $F$, which is instructive whenever more than one formal contexts are simultaneously treated. In particular, the definition given in Ref. Wi05 is modified into Definition 2.1 in the following manner.

• •

The original notation $I$ used in FCA is reserved for the derivation operator, which is typically a map from subset to subset. $R$ is a binary operation which relates a single object (attribute) to an attribute-set (object-set), denoted by $gRm$ Wi82 ; GW99 ; Wi05 where $g^{R}$ is an attribute-set and $m^{R}$ is an object-set.

• –

$m\in g^{R}$ ($g\in G\mapsto g^{R}\subseteq M$): the object $g$ has the attribute $m$,

• –

$g\in m^{R}$ ($m\in M\mapsto m^{R}\subseteq G$): $g$ is one of the objects that possess the attribute $m$.

Based on $R$, one may define all the derivation operators for both FCA and RST, i.e. $I$, $\Box$ and $\Diamond$, as will be clarified in Definition 2.2.

• •

The word formal is kept for the purpose to distinguish objects ($G$) and attributes ($M$), in contrast to the traditional approaches where only set operations are employed for both objects and attributes.

• –

Members of $G$ are to be treated formally like objects. In practice, there is a collection of individuals to be categorized into classes. Thus, the operations for objects are those which are apt to manipulate set relations, say intersection ($\cap$), union ($\cup$) and complementarity ($\{\}^{c}$).

• –

Members of $M$ are to be treated formally like attributes. It is natural to extend simple attributes into the composite attributes by means of the Boolean operations. Here,

the product “$\cdot$” ($\prod$) is employed for the conjunction, i.e. the logical AND,

the summation “$+$” ($\sum$) is employed for the disjunction, i.e. the logical OR,

the unary operation “$\neg$” is employed for the negation, i.e. the logical NOT.

A brief summary in terms of the above new notation for the traditional approaches in FCA and RST is in order.

Definition 2.2** (Wi82 ; GW99 ; Wi05 ; DG03 ; YY04 ).**

Given a formal context $F(G,M)$, the derivation operators are

[TABLE]

where, notably, $gRm$ is equivalently $mRg$ since $m\in g^{R}$ iff $g\in m^{R}$.

Subsequently, it is known that the maps in Eqs (1-3) manifest the following relations, for $X,X_{1},X_{2}\subseteq G$ and $X^{c}:=G\backslash X$,

[TABLE]

[TABLE]

where the same relations are also applied to $Y,Y_{1},Y_{2}\subseteq M$ and $Y^{c}:=M\backslash Y$. For convenience, let one employ the notions extent and intent Wi82 ; GW99 for both FCL and RSL such that the two theories can be handled on an equal footing.

Definition 2.3**.**

Consider $X\subseteq G$ and $Y\subseteq M$ subject to the formal context $F(G,M)$.

• •

The 2-tuple $(X,Y)$ is called an FCL concept if $X^{I}=Y$ and $Y^{I}=X$, where $X$ is the FCL extent, $Y$ is the FCL intent Wi82 ; GW99 .

• •

The 2-tuple $(X,Y)$ is called an RSL concept if $X^{\Box}=Y$ and $Y^{\Diamond}=X$, where $X$ is the RSL extent, $Y$ is the RSL intent.

Notably, the 2-tuple $(X,Y)$ with $\left\{\begin{smallmatrix}X^{\Box}=Y\\ Y^{\Diamond}=X\end{smallmatrix}\right.$ has been referred to as object-oriented RSL concept DG03 ; YY04 , in contrast to the case of property-oriented concept defined through $\left\{\begin{smallmatrix}X^{\Diamond}=Y\\ Y^{\Box}=X\end{smallmatrix}\right.$. However, one will ignore the consideration of $\left\{\begin{smallmatrix}X^{\Diamond}=Y\\ Y^{\Box}=X\end{smallmatrix}\right.$ since it can be derived from $\left\{\begin{smallmatrix}X^{\Box}=Y\\ Y^{\Diamond}=X\end{smallmatrix}\right.$ by means of interchanging the object and attribute. In addition, as a consequence of Eq. (10),

[TABLE]

telling YY04 that if $(X_{i},Y_{i})$ is a concept appropriate for $\left\{\begin{smallmatrix}X_{i}^{\Box}=Y_{i}\\ Y_{i}^{\Diamond}=X_{i}\end{smallmatrix}\right.$ then $(X_{i}^{c},Y_{i}^{c})$ is a concept appropriate for $\left\{\begin{smallmatrix}X_{i}^{\Box}=Y_{i}\\ Y_{i}^{\Diamond}=X_{i}\end{smallmatrix}\right.$ and vice versa. It is remarkable that Definition 2.3 provides fundamental ingredients for the original FCL and RSL GW99 ; YY04 . Based on Eq. (6), a $2$-tuple that fulfils $\left\{\begin{smallmatrix}X^{I}=Y\\ Y^{I}=X\end{smallmatrix}\right.$ is guaranteed to exist and can be written as $(X^{II},X^{I})$ or $(Y^{I},Y^{II})$, while a $2$-tuple that fulfils $\left\{\begin{smallmatrix}X^{\Box}=Y\\ Y^{\Diamond}=X\end{smallmatrix}\right.$ is guaranteed to be existent and can be written as $(X^{\Box\Diamond},X^{\Box})$ or $(Y^{\Diamond},Y^{\Diamond\Box})$. Moreover, the Galois connection composed by the two posets $(2^{G},\subseteq)$ and $(2^{M},\subseteq)$ can be accomplished in terms of the concepts by virtue of Eq. (9).

To proceed with the general theory of concept lattice, it is of crucial importance to clarify the notions contained in formal objects and formal attributes. Since a formal context discerns the objects based on the attributes they possess, the objects which possess the same attributes are grouped into an equivalent class.

Definition 2.4**.**

The discernible object classes in the perspective of $F(G,M)$ are definite. One will call these $F$-distinct subsets of $G$ and denote each such class with $D_{k}$ in the sense that $D_{k}$’s are regarded as distinct entities by $F(G,M)$. The total number of these $F$-distinct object classes $n_{F}$ is less than or equal to $|G|$, which is the cardinality of $G$. In effect, $\forall g\in D_{k}\ g^{R}=D_{k}^{I}$, and $D_{k}^{I}\neq D_{k^{\prime}}^{I}$ iff $D_{k}\neq D_{k^{\prime}}$. Hence, one has the collection of $F$-distinct object classes $G_{/R}=\{D_{k}\mid k=1\ldots n_{F}\}$ based on the equivalence relation “$g\approx_{R}g^{\prime}$ iff $g^{R}=g^{\prime R}$”, which means that $\{g,g^{\prime}\}\subseteq D_{k}$ for some $D_{k}\in G_{/R}$. Likewise, the $F$-codistinct subsets are gathered as $G^{/R}=\{G\backslash D_{k}\mid D_{k}\in G_{/R}\}$.

In contrast to the formal objects, the formal attributes can also emerge in a composite manner. One is free to construct various new attributes out of the given attributes by means of the standard Boolean operations, thereby forming an extended framework for the general consideration.

Definition 2.5**.**

Given a set $M$ of attributes, the set of generalised attributes over $M$ is defined as $M^{\ast}=\{b(M_{0})\mid M_{0}\subseteq M\}$, where $b$ iterates over all the possibilities of Boolean functions in which the concerned operations are “$\cdot$” (conjunction), “$+$” (disjunction) and “$\neg$” (negation). To be concrete, $M^{\ast}$ is meant to denote whatever one may construct out of $M$ by means of arbitrary compositions provided by the Boolean functions. In contrast to the members of $M^{\ast}$, which may be composite, the attributes in $M$ are then referred to as simple attributes.

Intuitively, $M^{\ast}$ also comprises the set $M$ itself as well as the conventional truth ${\bf 1}$ and falsity ${\bf 0}$. Moreover, by means of negation one may define several interesting subclasses of $M^{\ast}$.

Definition 2.6**.**

Let $M$ be certain collection of attributes of interest. A conjugate class $\Psi_{M}^{i}$ is referred to as one of the $2^{|M|}$ possibilities of $\{\alpha\mid\alpha=\mbox{either}\ a\ \mbox{or}\ \neg a,a\in M\}$. Accordingly, the set of all conjugate classes is denoted with ${\bf\Psi}_{M}=\{\Psi_{M}^{1}\ldots\Psi_{M}^{2^{|M|}}\}$. The symmetrised attribute set ${\breve{M}}$ is employed to extend $M$ by means of negation: ${\breve{M}}=M\cup\{\neg a\mid a\in M\}$.

One could also employ the set of generalised attributes as $M^{\ast}=({\Psi_{M}^{i}})^{\ast}=\{b(M_{0})\mid M_{0}\subseteq{\Psi_{M}^{i}}\}$ with any ${\Psi_{M}^{i}}\in{\bf\Psi}_{M}$. It is noteworthy that prescribing $M^{\ast}$, either by $({\Psi_{M}^{i}})^{\ast}$ or by Definition 2.5, may encounter abundant choices of Boolean function $b$. Technically, such abundance can be reduced by means of normal forms, where one is in particular interested in the conjunctive normal form (CNF) and the disjunctive normal form (DNF). The other choice to suppress the redundance is to consider $M^{\ast}=({\breve{M}})^{\ast}=\{b_{0}(M_{0})\mid M_{0}\subseteq{\breve{M}}\}$, where $b_{0}$ differs from $b$ by that no further negation is needed (cf. Definition 2.5). In what follows, the particular significance of the generalised attributes over $M$ to the structure of $F(G,M)$ will be presented step by step.

Attributes are considered to be further restrained through the attribute-object correspondence provided in the formal context $F(G,M)$.

Definition 2.7**.**

According to Definition 2.5 any generalised attribute $\mu\in M^{\ast}$ can be regarded as $\mu(M)$, namely that $\mu$ is a (Boolean) function of $M_{0}$, say $\mu=b(M_{0})$, for certain $b$ and for some $M_{0}\subseteq M$. Subject to $F(G,M)$, the contextual (Boolean) function $\mu_{F}(M)$ for any generalised attribute $\mu\in M^{\ast}$ is obtained from the expression of $\mu(M)$ by (1) replacing the concerned attribute $m$ with $m^{R}$ ($m^{R}\subseteq G$), (2) replacing the concerned negative attribute $\neg m$ with $(m^{R})^{c}$, (3) replacing the conjunction and disjunction operations with $\cap$ and $\cup$, respectively. Notably, the contextual function $\mu_{F}(M)$ is essentially a subset of $G$.

For the extreme case, one has “$m=m(M)\ \forall m\in M$” such that $m_{F}(M)=m^{R}\subseteq G\ \forall m\in M$. The contextual function can represent all the information contained in the formal context $F(G,M)$ since the contextual function assigns every attribute, say $m$, in $M$ with a definite object subset which possesses $m$ in common. Furthermore, such idea can be extended.

Lemma 2.8**.**

From $F(G,M)$ one can deduce the extended formal context $F^{\ast}(G,M^{\ast})$ without any additional assumption in the sense that $\forall\mu\in M^{\ast}$ $\mu^{R}$ can be determined by means of $\mu^{R}=\mu_{F}(M)$, which is called “$\mu^{R}$ is well defined in $M^{\ast}$”.

Proof.

If $\mu^{R}$ is well defined in $M^{\ast}$ then one can achieve the extended formal context $F^{\ast}(G,M^{\ast})$ such that each attribute in $M^{\ast}$ is equipped with an object set which possess the attribute in common, similar to the case of $F(G,M)$. Here, the proof of “$\mu^{R}$ is well defined in $M^{\ast}$” can be carried out in two steps. It is to show that (1) ${\neg m}^{R}=({\neg m})_{F}\equiv(m^{R})^{c}$ (Definition 2.7) for $m\in M$ such that $\mu^{R}$ is well defined in ${\breve{M}}$ and (2) both $(\prod_{\mu}\mu)^{R}$ and $(\sum_{\mu}\mu)^{R}$ are well defined $\forall\mu\in{\breve{M}}$. Once these are done, one will end up with “$\mu^{R}$ is well defined in $M^{\ast}$” since $M^{\ast}$ can be obtained as ${\breve{M}}^{\ast}$ by means of “$+$” and “$\cdot$” (see the discussion after Definition 2.6).

• (1)

For $m\in M$ “$g\in(\neg m)^{R}$” can be interpreted as “$g\not\in m^{R}$”. In other words, $g\in G\backslash m^{R}=(m^{R})^{c}$, hence, $(\neg m)^{R}=(m^{R})^{c}=({\neg m})_{F}$, which gives rise to $m^{R}=((\neg m)^{R})^{c}$ after taking complementarity. Moreover, if the negated attributes are also taken into account, the same interpretation implies that $(\neg\neg m)^{R}=((\neg m)^{R})^{c}$. This is consistent since $(\neg\neg m)^{R}$ coincides with $m^{R}$: $gR(\neg\neg m)\iff g\not\in(\neg m)^{R}\iff g\in((\neg m)^{R})^{c}\iff g\in((m^{R})^{c})^{c}\iff gRm$.

• (2)

Since $\mu^{R}$ is well defined in ${\breve{M}}$, one may consider that $(\prod_{\mu}\mu)^{R}=\{g\mid gR(\prod_{\mu}\mu),g\in G\}=\bigcap_{\mu}\mu^{R}=(\prod_{\mu}\mu)_{F}$ in ${\breve{M}}$. Likewise, $(\sum_{\mu}\mu)^{R}=\{g\mid gR(\sum_{\mu}\mu),g\in G\}=\bigcup_{\mu}\mu^{R}=(\sum_{\mu}\mu)_{F}$ $\forall\mu\in{\breve{M}}$.

∎

Definition 2.9**.**

A contextual Venn diagram (or Euler diagram) ${\cal V}_{M}^{F}$ subject to $F(G,M)$ can illustrate the set relations among the contextual functions of attributes in $M$. One can achieve ${\cal V}_{M}^{F}$ by encircling the object-set $m^{R}$ within the object collection $G$, where the object-labellings are ignored. In contrast, $M$ may also have its own intrinsic logical structure in terms of the (conventional) Venn digram ${\cal V}_{M}$.

In practice, there are two types of ordering systems for attributes. As the first type, “$\geq,\leq$” is concerned with the intrinsic ordering of the entities in $M^{\ast}$. For instance, $\mu_{i}\leq\mu_{j}$ denotes that the region representing $\mu_{i}$ are included in the region representing $\mu_{j}$ on ${\cal V}_{M}$. The second type of ordering system “$\supseteq,\subseteq$” is employed for ${\cal V}_{M}^{F}$ because $\mu^{R}$s are in fact object sets. Notably, preserving the object labellings in the contextual Venn digram will restore all the information content of the formal context. The contextual Venn diagram with explicit object labellings turns out to be an alternative representation for the formal context. Once ${\cal V}_{M}^{F}$ is given, Lemma 2.8 becomes a rather intuitive result since one may read off every $\mu^{R}$ for $\mu\in M^{\ast}$. The other point is that the formal context does not per se suggest any intrinsic relation among members in $M$. Attributes in $M$ can be basically independent, hence, any two attributes have an intersection on the corresponding Venn diagram (${\cal V}_{M}={\cal V}_{M}^{0}$). However, it is also allowable that one imposes additional constraints on $M$ by removing some of the disjoint regions from ${\cal V}_{M}^{0}$.

Lemma 2.10**.**

Subject to the formal context $F(G,M),\ \forall\mu\forall\mu_{i}\forall\mu_{j}\in M^{\ast}$,

• •

$\mu^{R}\neq\emptyset\implies\mu\neq{\bf 0}$,

• •

$\mu_{i}=\mu_{j}\implies\ \mu_{i}^{R}=\mu_{j}^{R}$* but $\mu_{i}^{R}\neq\mu_{j}^{R}\implies\ \mu_{i}\neq\mu_{j}$,*

• •

$\mu_{i}<\mu_{j}\implies\ \mu_{i}^{R}\subseteq\mu_{j}^{R}$* but $\mu_{i}^{R}\subset\mu_{j}^{R}\implies\ \mu_{i}\not>\mu_{j}$.*

Proof.

• •

If $\mu={\bf 0}$ but $\mu^{R}=X\neq\emptyset$ then the existence of the object set $X$ is contradictory.

• •

$\mu_{i}=\mu_{j}\implies\ \mu_{i}^{R}=\mu_{j}^{R}$ since from $\mu_{i}(M)=\mu_{j}(M)$ it follows that $(\mu_{i})_{F}(M)=(\mu_{j})_{F}(M)$ (Lemma 2.8). If $\mu_{i}^{R}\neq\mu_{j}^{R}$ then $\mu_{i}^{Rc}\cap\mu_{j}^{R}\neq\emptyset$ and/or $\mu_{i}^{R}\cap\mu_{j}^{Rc}\neq\emptyset$ which means that $(\neg\mu_{i}\cdot\mu_{j})^{R}\neq\emptyset$ and/or $(\mu_{i}\cdot\neg\mu_{j})^{R}\neq\emptyset$. Consequently, $(\neg\mu_{i}\cdot\mu_{j})\neq{\bf 0}$ and/or $(\mu_{i}\cdot\neg\mu_{j})\neq{\bf 0}$, which contradicts $\mu_{i}=\mu_{j}$.

• •

For $\mu_{i}<\mu_{j}$ one may assume $\mu_{j}=\mu_{i}+\beta$ in which $\beta\in M^{\ast}$. Then, $\mu_{j}^{R}=\mu_{i}^{R}\cup\beta^{R}\supseteq\mu_{i}^{R}$ according to Lemma 2.8. Moreover, if $\mu_{i}^{R}\subset\mu_{j}^{R}$ but $\mu_{i}>\mu_{j}$, contraction occurs since $\mu_{i}>\mu_{j}$ implies $\mu_{i}^{R}\supseteq\mu_{j}^{R}$ which denies $\mu_{i}^{R}\subset\mu_{j}^{R}$. Thus, $\mu_{i}^{R}\subset\mu_{j}^{R}\implies\ \mu_{i}\not>\mu_{j}$.

∎

The above shows that the distinction between $\mu_{i}$ and $\mu_{j}$ is stronger than the pair $(\mu_{i}^{R},\mu_{j}^{R})$. More concretely, the number of disjoint regions on ${\cal V}_{M}^{0}$ is $2^{|M|}$, since ${\cal V}_{M}^{F}$ is less discernible than ${\cal V}_{M}^{0}$, thus $2^{|M|}\geq n_{F}$ 111 The formal context with $2^{|M|}=n_{F}$ will be named degenerate formal context LLJD12-2 , because the Grsp is then equivalent to Gfcp (Proposition 3.4) at each general extent. Although the degenerate formal context is not a practical example for the problem of categorisation, it provides a very useful theoretical tool for inspecting the GCL structure in every detail LLJD12-3 LLJD12-4 ..

Definition 2.11**.**

The contextual equivalent class of attribute subject to $F(G,M)$ is an attribute-set defined in the following manner. Given $\nu\in M^{\ast}$, one may collect all the attributes which have the same contextual function (Definition 2.7) with $\nu$ as $\{\nu\}_{F}=\{\mu\mid\mu^{R}=\nu^{R},\ \mu\in M^{\ast}\}$. Accordingly, since $\nu^{R}\subseteq G$, the idea can be employed with respect to definite object-set in the sense that $[X]_{F}=\{\mu\mid\mu^{R}=X,\ \mu\in M^{\ast}\}$ for $X\subseteq G$.

It is noteworthy that the construction $(X,[X]_{F})$ is in fact the precursor of the concept on the GCL. Here, the remaining condition lies on determining whether an arbitrary subset of $G$ can form a node of the GCL. Remarkably, a subset $X$ of $G$ with trivial contextual equivalent class, i.e. $[X]_{F}=\emptyset$, can hardly participate in the categorization provided by the formal context since no attribute can be found to label it. The point is that any of the $F$-distinct subsets of $G$ (Definition 2.7), say $D_{k}$, is readily a smallest possible subset equipped with non-trivial contextual equivalent attributes. In other words, the $F$-distinct subsets correspond to the smallest regions on the contextual Venn diagram (Definition 2.9), hence, there exists no attribute, say $\mu$, whose contextual function $\mu^{R}$ exactly encloses a region which is smaller than the region marked by $D_{k}$. Consequently, for $X$ that can be written as the union of $F$-distinct subsets ($X=\bigcup_{k_{X}}D_{k_{X}}$) one has non-trivial $[X]_{F}$, otherwise $[X]_{F}=\emptyset$.

A further instructive notion is the smallest yet non-trivial constituents of the generalised attribute set, for which one may develop two independent non-trivial minimisation aspects. Firstly, one may speak of the minimal attributes of $M^{\ast}$ which are not the Falsity ${\bf 0}$ and thus correspond to the smallest regions on the Venn diagram ${\cal V}_{M}$. Secondly, subject to $F(G,M)$, there are classes of irreducible attributes that determines the object classes which participate in the categorisation in a least possible manner. Here, the first non-trivial minimisation aspect is a prevalent issue which deserves clarification.

Lemma 2.12**.**

Given an attribute-set $M$, one may define the set of non-trivial infima $b_{inf}(M^{\ast}):=\{\tau\in M^{\ast}\mid\tau\succ_{M}{\bf 0}\}$ and the set of non-trivial suprema $b_{sup}(M^{\ast}):=\{\tau\in M^{\ast}\mid{\bf 1}\succ_{M}\tau\}$ for the corresponding generalised attribute-set $M^{\ast}$. In the case of $\mu\succ_{M}\nu$, one denotes that $\mu>\nu$ without any attribute $\chi\in M^{\ast}$ which fulfils $\mu>\chi>\nu$. Both $b_{inf}(M^{\ast})$ and $b_{sup}(M^{\ast})$ can be expressed in terms of ${\bf\Psi}_{M}$ (Definition 2.6):

• •

$b_{inf}(M^{\ast})=\left\{\prod\Psi_{M}^{j}\mid\Psi_{M}^{j}\in{\bf\Psi}_{M}\right\}$, $\prod\Psi_{M}^{j}$ is called $M^{\ast}$-atom, which is considered as an atom for $M^{\ast}$.

• •

$b_{sup}(M^{\ast})=\left\{\sum\Psi_{M}^{j}\mid\Psi_{M}^{j}\in{\bf\Psi}_{M}\right\}$, $\sum\Psi_{A}^{j}$ is called $M^{\ast}$-coatom.

Proof.

One may rewrite $\tau\succ_{M}{\bf 0}$ as “$\tau\cdot\beta\in\left\{\tau,{\bf 0}\right\}\ \forall\beta\in M^{\ast}$”.

• •

For any $\beta\in M^{\ast}$ one may write down the DNF as $\beta=\sum_{i_{\beta}}\prod\Psi_{M}^{i_{\beta}}$, where the range of $i_{\beta}$ is up to the given $\beta$. Hence, $\forall j\forall\beta\ \beta\cdot\prod\Psi_{M}^{j}=\left(\sum_{i_{\beta}}\prod\Psi_{M}^{i_{\beta}}\right)\cdot\prod\Psi_{M}^{j}\in\left\{\prod\Psi_{M}^{j},{\bf 0}\right\},\ \therefore\ \forall j\ \prod\Psi_{M}^{j}\in b_{inf}(M^{\ast})$. On the other hand, assume $\exists\chi\in b_{inf}(M^{\ast})$ but $\chi\neq\prod\Psi_{M}^{j}$ for $\Psi_{M}^{j}\in{\bf\Psi}_{M}$. However, attribute $\chi$ may take the form $\sum_{i_{\chi}}\prod\Psi_{M}^{i_{\chi}}$ in DNF, which apparently implies $\chi\not\succ_{M}{\bf 0}$. One concludes then $b_{inf}(M^{\ast})=\left\{\prod\Psi_{M}^{j}\mid\Psi_{M}^{j}\in{\bf\Psi}_{M}\right\}$.

• •

$\beta+\sum\Psi_{M}^{j}\equiv\beta\cdot\neg(\sum\Psi_{M}^{j})+\sum\Psi_{M}^{j}$ $\forall j\forall\beta$. Subsequently, $\beta\cdot\neg(\sum\Psi_{M}^{j})\in\left\{\neg(\sum\Psi_{M}^{j}),{\bf 0}\right\}$ based on the above result since $\neg(\sum\Psi_{M}^{j})$ can be identified as $\prod\Psi_{M}^{j^{\prime}}$ for some $\Psi_{M}^{j^{\prime}}\in{\bf\Psi}_{M}$, which implies that $\beta+\sum\Psi_{M}^{j}\in\left\{{\bf 1},\sum\Psi_{M}^{j}\right\}$, i.e. $\sum\Psi_{M}^{j}\in b_{sup}(M^{\ast})$. On the other hand, assume $\exists\chi\in b_{sup}(M^{\ast})$ but $\chi\neq\sum\Psi_{M}^{j}$ for $\Psi_{M}^{j}\in{\bf\Psi}_{M}$. However, attribute $\chi$ may take the form $\prod_{i_{\chi}}\sum\Psi_{M}^{i_{\chi}}$ in CNF, which apparently implies $\chi\not\prec_{M}1$. Therefore, $b_{sup}(M^{\ast})=\left\{\sum\Psi_{M}^{j}\mid\Psi_{M}^{j}\in{\bf\Psi}_{M}\right\}$.

∎

Corollary 2.13**.**

Whenever there are no intrinsic conditions pre-imposed on the attributes in $M$, the cardinalities are as follows.

• •

The numbers of non-trivial suprema and infima for $M^{\ast}$ are equivalent: $|b_{sup}(M^{\ast})|=|b_{inf}(M^{\ast})|=2^{|M|}$.

• •

There are $2^{2^{|M|}}$ distinct generalised attributes in $M^{\ast}$, i.e., $|M^{\ast}|=2^{2^{|M|}}$.

Proof.

Whenever there are no intrinsic conditions pre-imposed on the attributes in $M$, the elements of $b_{inf}(M^{\ast})$ can be represented by the $2^{|M|}$ disjoint regions on ${\cal V}_{M}^{0}$.

• •

Alternatively, by Definition 2.6 one may compute that $|b_{inf}(M^{\ast})|=|{\bf\Psi}_{M}|=2^{|M|}$. The elements of $b_{sup}(M^{\ast})$ amount to the complementary parts of the disjoint regions on ${\cal V}_{M}^{0}$, which demonstrates an one-to-one correspondence; $|b_{sup}(M^{\ast})|=|b_{inf}(M^{\ast})|$.

• •

Denote the members in $b_{inf}(M^{\ast})$ with $b_{k}$ where $1\leq k\leq 2^{|M|}$. An attribute in $M^{\ast}$, say $\mu$, can take the expression $\sum_{k_{\mu}}b_{k_{\mu}}$ in DNF. In other words, the attributes in $M^{\ast}$ may be implemented by $\sum_{k\in M_{1}}b_{k}$ where $M_{1}$ ranges over all the possibilities for $M_{1}\subseteq b_{inf}(M^{\ast})$. Counting all these possibilities is equivalent to enumerating the number of the power sets of a set with $2^{|M|}$ members, i.e., $|M^{\ast}|=2^{2^{|M|}}$.

∎

The second non-trivial minimisation aspect mentioned above is related to the formal context.

Definition 2.14**.**

A generalised attribute $\mu\in_{F+}M^{\ast}$ composed of the disjunction of attributes in ${\breve{M}}$ (Definition 2.6) is called an irreducible disjunction subject to $F(G,M)$ if eliminating any term in the disjunction from $\mu$, which results in $\mu^{\prime}$, always causes $(\mu^{\prime})^{R}\subset\mu^{R}$. Accordingly, an $X$-irreducible disjunction class can be defined as $[X^{+}]_{F}=\{\mu\in[X]_{F}\mid\mu\in_{F+}M^{\ast}\}$, see Definition 2.11. On the other hand, a generalised attribute $\mu\in_{F\times}M^{\ast}$ composed of the conjunction of attributes in ${\breve{M}}$ is called an irreducible conjunction in $M^{\ast}$ subject to $F(G,M)$ if eliminating any term in the conjunction from $\mu$, which results in $\mu^{\prime}$, always causes $(\mu^{\prime})^{R}\supset\mu^{R}$. The $X$-irreducible conjunction class is given as $[X^{\times}]_{F}:=\{\mu\in[X]_{F}\mid\mu\in_{F\times}M^{\ast}\}$.

In practice, $\mu\in_{F+}M^{\ast}$ can be written in terms of $\mu=\sum_{\alpha\in\psi}\alpha$ for some $\psi\subseteq\Psi_{M}^{j}$ (Definition 2.6) in which $\mu^{R}\neq\left(\sum_{\alpha\in\psi\backslash\{\alpha_{0}\}}\alpha\right)^{R}\ \forall\alpha_{0}\in\psi$. Likewise, $\mu\in_{F\times}M^{\ast}$ can be written in terms of $\mu=\prod_{\alpha\in\psi}\alpha$ for some $\psi\subseteq\Psi_{M}^{j}$ where $\mu^{R}\neq\left(\prod_{\alpha\in\psi\backslash\{\alpha_{0}\}}\alpha\right)^{R}\ \forall\alpha_{0}\in\psi$. Generally speaking, $[X^{+}]_{F}\cup[X^{\times}]_{F}\subseteq[X]_{F}$ and $[X^{+}]_{F}\cap[X^{\times}]_{F}\neq\emptyset$. The other useful expressions are

[TABLE]

Note that $\frac{[X_{0}^{\times}]_{F}}{[X_{i}^{\times}]_{F}}\neq[X_{0}^{\times}]_{F}$ only when $X_{i}\supset X_{0}$ and $\frac{[X_{0}^{+}]_{F}}{[X_{j}^{+}]_{F}}\neq[X_{0}^{+}]_{F}$ only when $X_{j}\subset X_{0}$. What follows is mainly concerned with regaining the FCL and RSL nodes on the GCL.

Lemma 2.15**.**

• •

$\forall{\breve{m}}\in{\breve{M}}\quad{\breve{m}}\in[({\breve{m}}^{R})^{+}]_{F}\cap[({\breve{m}}^{R})^{\times}]_{F}$* and, furthermore, $\forall X_{i}\forall X_{j}\neq{\breve{m}}^{R}\ \leavevmode\nobreak\ {\breve{m}}\in\frac{[({\breve{m}}^{R})^{+}]_{F}}{[X_{i}^{+}]_{F}}\cap\frac{[({\breve{m}}^{R})^{\times}]_{F}}{[X_{j}^{\times}]_{F}}$.*

• •

$\forall\mu\in M^{\ast}\quad\mu\in[X^{+}]_{F}\implies\neg\mu\in[\left(X^{c}\right)^{\times}]_{F}$* and $\mu\in[X^{\times}]_{F}\implies\neg\mu\in[\left(X^{c}\right)^{+}]_{F}$.*

Proof.

• •

Any attribute ${\breve{m}}\in{\breve{M}}$ (Definition 2.6) is non-composite and can be thus regarded as both an one-term disjunction and an one-term conjunction. Therefore, ${\breve{m}}$ is simultaneously an ${\breve{m}}^{R}$-irreducible disjunction and an ${\breve{m}}^{R}$-irreducible conjunction. Moreover, ${\breve{m}}\neq\nu_{1}\cdot\nu_{2}$ in which $\nu_{1}\in[X_{i}^{\times}]_{F}\ \mbox{or}\ \nu_{2}\in[X_{i}^{\times}]_{F}$ and ${\breve{m}}\neq\nu_{1}^{\prime}+\nu_{2}^{\prime}$ in which $\nu_{1}^{\prime}\in[X_{j}^{+}]_{F}\ \mbox{or}\ \nu_{2}^{\prime}\in[X_{j}^{+}]_{F}$.

• •

For $\mu\in[X^{+}]_{F}$, let $\mu=\sum_{\alpha\in\psi}\alpha$ where $\psi\subseteq\Psi_{M}^{j}$ for some $\Psi_{M}^{j}\in{\bf\Psi}_{M}$. Accordingly, $\mu^{R}=X$ but $\forall\alpha_{1}\in\psi$ $\left(\sum_{\alpha\in\psi\backslash\{\alpha_{1}\}}\alpha\right)^{R}\ \neq X$, which implies that $(\neg\mu)^{R}=X^{c}$ but $\forall\alpha_{1}\in\psi$ $\left(\sum_{\alpha\in\psi\backslash\{\alpha_{1}\}}\alpha\right)^{Rc}\neq X^{c}$. It then turns out that

[TABLE]

where $\psi^{\prime}=\{\neg\alpha\mid\alpha\in\psi\}\subseteq\Psi_{M}^{j^{\prime}}$ with $\Psi_{M}^{j^{\prime}}=\{\neg\alpha\mid\alpha\in\Psi_{M}^{j}\}\in{\bf\Psi}_{M}$. Therefore, $\mu\in[X^{+}]_{F}\implies\neg\mu\in[\left(X^{c}\right)^{\times}]_{F}$. Similarly, starting with $\mu^{\prime}=\sum_{\alpha\in\psi}\alpha\in[X^{\times}]_{F}$, one will end up with $\mu^{\prime}\in[X^{\times}]_{F}\implies\neg\mu^{\prime}\in[\left(X^{c}\right)^{+}]_{F}$.

∎

3 obtaining the general concept lattice

The general theory of concept lattice is capable of providing the categorisation for the formal objects $G$ according to members of $M^{\ast}$. First of all, one observes that one could have replaced both FCL and RSL intents in the original constructions in terms of single composite attributes without altering their information contents.

Lemma 3.1**.**

Let $X\subseteq G$, $Y\subseteq M$ for the formal context $F(G,M)$. The concepts for FCL and RSL (Definition 2.3) can be both expressed in terms of $(X,\mu)$ in the following manner.

• •

For FCL, use $\mu=\prod Y$, where $\mu$ is referred to as a formal concept property (fcp). Imposing $\left\{\begin{smallmatrix}X^{I}=Y\\ Y^{I}=X\end{smallmatrix}\right.$ on $(X,Y)$ is indistinguishable from imposing $\left\{\begin{smallmatrix}\prod X^{I}=\mu\\ \mu^{R}=X\end{smallmatrix}\right.$ on $(X,\mu)$.

• •

*For RSL, use $\mu=\sum Y$, where $\mu$ is referred to as a rough set property (rsp). Imposing $\left\{\begin{smallmatrix}X^{\Box}=Y\\ Y^{\Diamond}=X\end{smallmatrix}\right.$ on $(X,Y)$ is indistinguishable from imposing $\left\{\begin{smallmatrix}\sum X^{\Box}=\mu\\ \mu^{R}=X\end{smallmatrix}\right.$ on $(X,\mu)$. *

Proof.

• •

$X^{I}=Y\implies\prod X^{I}=\prod Y$. Reversely, since both $X^{I}$ and $Y$ consist of distinct members in $M$, $\prod X^{I}=\prod Y\implies X^{I}=Y$. Thus, $X^{I}=Y\iff\prod X^{I}=\prod Y$and it is appropriate to employ $\mu=\prod Y$. Moreover, by Eq. (1) $\mu^{R}=(\prod Y)^{R}=\bigcap_{y\in Y}y^{R}=\{g\in G\mid\ gRy,\ \forall y\in Y\}=Y^{I}$. Consequently, $\mu^{R}=X\iff Y^{I}=X$.

• •

$X^{\Box}=Y\implies\sum X^{\Box}=\sum Y$. Reversely, since both $X^{\Box}$ and $Y$ consist of distinct members in $M$, $\sum X^{\Box}=\sum Y\implies X^{\Box}=Y$. Thus, $X^{\Box}=Y\iff\sum X^{\Box}=\sum Y$ and it is appropriate to employ $\mu=\sum Y$. Moreover, by Eq. (2) and (3) $\mu^{R}=(\sum Y)^{R}=\bigcup_{y\in Y}y^{R}=\{g\in G\mid\ \exists y\in M,\ (gRy,\ y\in Y)\}=Y^{\Diamond}$. Consequently, $\mu^{R}=X\iff Y^{\Diamond}=X$.

∎

Since the above proof has invoked some of variants of derivation operators (see Eq. (1)-(3)), it is worthwhile to mention about all of such variants:

[TABLE]

where it is intuitive that interchanging the role of $X$ and $Y$ also gives rise to the other set of derivation operators. Another point is that there are particular object-classes which are fundamental in both the FCL and the RSL.

Proposition 3.2**.**

Given a formal context $F(G,M)$, $\forall m\in M$ the object-set $m^{R}$ is a common extent for FCL and RSL.

Proof.

$(m^{R},(m^{R})^{I})$ coincides with $(Y^{I},Y^{II})$ and $(m^{R},(m^{R})^{\Box})$ coincides with $(Y^{\Diamond},Y^{\Diamond\Box})$, where in both cases $Y$ is identified as $\{m\}$.∎

Note that the result of Proposition 3.2 is rather intriguing, albeit true. Since the formal context conventionally develops a table, Proposition 3.2 in fact entails that the object-class $m^{R}$ is recognised as an extent for both the FCL and RSL just because $m$ is listed in the table. Now, assume that there is a formal concept $(X_{1},Y_{1})$ for FCL subject to $F(G,M)$. Then, by virtue of Proposition 3.1 one has $\mu_{1}^{R}=X_{1}$ with $\mu_{1}=\prod Y_{1}$. Consider $F^{\prime}(G,M\cup\{\mu_{1}\})$ which is obtained by explicitly including the given $(X_{1},\mu_{1})$ correspondence as an additional column in $F(G,M)$. By Lemma 2.8 $\mu_{1}^{R}=X_{1}$ is something deducible from $F(G,M)$, thus, $F^{\prime}(G,M\cup\{\mu_{1}\})$ should not provide different concept lattices. However, $\mu_{1}$ is now listed in the new table $F^{\prime}(G,M\cup\{\mu_{1}\})$, which implies that $X_{1}$ must also be an RSL-extent when it is known to be an FCL extent. This is denied by the original FCL and RSL (see the comparison in Ref. YY04 ), therefore, it is unnatural to neglect the role the composite attribute may play just because it is not simple (Definition 2.5). Hereafter, the general theory of concept lattice will proceed in a different way from the original ones by attempting a democratic consideration for all the members in $M^{\ast}$.

The concept lattice subject to $F(G,M)$ inevitably makes reference on its accompanied extended formal context, i.e. $F^{\ast}(G,M^{\ast})$. Since $\mu^{R}$ is well defined $\forall\mu\in M^{\ast}$ (Lemma 2.8), it is straightforward to manipulate $F^{\ast}(G,M^{\ast})$ in parallel to the conventional formal context.

Definition 3.3**.**

Following from Definition 2.2, the derivation operators appropriate for $F^{\ast}(G,M^{\ast})$ are given as

[TABLE]

Obviously, Eq. (6) to (10) remain valid after the substitutions “$I\ \mbox{by}\ I^{\ast}$, $\Box\ \mbox{by}\ \Box^{\ast}$, $\Diamond\ \mbox{by}\ \Diamond^{\ast}$”.

Proposition 3.4**.**

Subject to $F^{\ast}(G,M^{\ast})$, the general concept $(X,\rho(X),\eta(X))$ can be obtained as a generalisation of the original FCL and RSL concepts (Lemma 3.1) as follows. If $\rho(X)=\sum X^{\Box^{\ast}},\ \eta(X)=\prod X^{I^{\ast}}$ and $(\rho(X))^{R}=(\eta(X))^{R}=X$ then $X$ is called the general extent, where the allowable general extents are collected as $E_{F}:=\{X\subset G\mid[X]_{F}\neq\emptyset\}=\left\{\mu^{R}\mid\ \mu\in M^{\ast}\right\}$. In addition, $\rho(X)$ is the generalised rough set property (Grsp) and $\eta(X)$ is the generalised formal concept property (Gfcp).

Proof.

The general concept here is the consequence of applying Proposition 3.2 to $F^{\ast}(G,M^{\ast})$, which is telling that $\forall\mu\in M^{\ast}\ \mu^{R}$ is a common extent of FCL and RSL in the perspective of $F^{\ast}(G,M^{\ast})$. In practice, $(X,\rho(X),\eta(X))$ represents the dual use of $(X,\mu)$ in Proposition 3.1 generalised in the sense of Definition 3.3. Accordingly, $\left\{\begin{smallmatrix}(X,\rho(X))=(X,\sum X^{\Box^{\ast}})\\ (X,\eta(X))=(X,\prod X^{I^{\ast}})\end{smallmatrix}\right.$ as well as $(\rho(X))^{R}=X=(\eta(X))^{R}$.∎

The general extent can $X$ then take the form of $\mu^{R}$ with $\mu=\rho(X)$ or $\mu=\eta(X)$, thereby being referred to as an object class with non-trivial contextual equivalent attributes (Definition 2.11).

Proposition 3.5**.**

Let $E_{F}=\{X\subset G\mid[X]_{F}\neq\emptyset\}\equiv\left\{\mu^{R}\mid\ \mu\in M^{\ast}\right\}$ denote the collection of all the object classes with non-trivial contextual equivalent classes of attributes.

• •

The set $E_{F}$ is the sigma algebra $\sigma(\left\{m^{R}\mid m\in M\right\})$.

• •

$E_{F}=\sigma({G_{/R}})=\left\{\bigcup_{D\in E^{0}}D\mid E^{0}\subseteq G_{/R}\right\}$* is only concerned with unions of the members in ${G_{/R}}$ and $|E_{F}|=2^{n_{F}}$. *

• •

Given $\left\{X_{1},X_{2}.\ldots\right\}\subseteq E_{F}$ then $\sigma(\left\{X_{1},X_{2}.\ldots\right\})\subseteq E_{F}$.

Proof.

The significance of the sigma algebra generated by a collection of sets is that it exhausts whatever constructable out of the collection by means of iterating various steps of set operations such as intersection, union and complementarity. The sigma algebra is essentially closed under intersection, union and complementarity.

• •

$E_{F}=\{\mu^{R}\mid\ \mu\in M^{\ast}\}$ (Proposition 3.4), where $\mu^{R}=\mu_{F}\ \forall\mu\in M^{\ast}$ is well defined according to Lemma 2.8. Hence, $E_{F}$ can be obtained from $\left\{m^{R}\mid m\in M\right\}$ by means of intersection, union and complementarity (Definition 2.7). Moreover, upon representing the member of $E_{F}$ as $\mu_{i}^{R}$ for some $\mu_{i}\in M^{\ast}$, the closure relations are as follows.

$\mu_{1}^{R}\cap\mu_{2}^{R}=(\mu_{1}\cdot\mu_{2})^{R}\in E_{F}$ since $\mu_{1}\cdot\mu_{2}\in M^{\ast}$,

$\mu_{1}^{R}\cup\mu_{2}^{R}=(\mu_{1}+\mu_{2})^{R}\in E_{F}$ since $\mu_{1}+\mu_{2}\in M^{\ast}$,

$(\mu_{3}^{R})^{c}=(\neg\mu_{3})^{R}\in E_{F}$ since $\neg\mu_{3}\in M^{\ast}$.

Therefore, $E_{F}$ is closed under intersection, union and complementarity.

• •

Firstly, $E_{F}\supseteq G_{/R}$ in that $\forall D_{k}\in G_{/R}\ \exists\mu_{k}$ s.t. $\mu_{k}^{R}=D_{k}$. For instance, $\mu_{k}^{R}=D_{k}$ is satisfied by taking $\mu_{k}=\prod\left\{m\mid m\in D_{k}^{I}\subseteq M\right\}\cdot\prod\left\{\neg m\mid m\in M\backslash D_{k}^{I}\right\}$. Moreover, $E_{F}\supseteq\sigma(G_{/R})$ due to the fact that $E_{F}$ is itself a sigma algebra. On the other hand, $\sigma(G_{/R})\supseteq\left\{m^{R}\mid m\in M\right\}$ since, e.g., $X_{0}:=m^{R}$ can be expressed in terms of $\bigcup_{k_{X_{0}}}D_{k_{X_{0}}}$. Likewise, $\sigma(G_{/R})\supseteq\sigma(\left\{m^{R}\mid m\in M\right\})=E_{F}$ since $\sigma(G_{/R})$ is a sigma algebra. Therefore, $E_{F}=\sigma(G_{/R})$. Secondly, $\sigma(G_{/R})=\left\{\bigcup_{D\in E^{0}}D\mid E^{0}\subseteq G_{/R}\right\}$ because $G_{/R}$ only comprises the disjoint members “$D_{1},D_{2}\ldots,D_{n_{F}}$” where $\bigcup_{k=1}^{n_{F}}D_{k}=G$. Accordingly, one obtains that $|E_{F}|=|\sigma(G_{/R})|=2^{n_{F}}$.

• •

$\sigma(\left\{X_{1},X_{2}.\ldots\right\})$ is the smallest collection that is closed under intersection, union and complementarity and contains $\left\{X_{1},X_{2}.\ldots\right\}$, therefore, $\left\{X_{1},X_{2}.\ldots\right\}\subseteq\sigma(\left\{X_{1},X_{2}.\ldots\right\})\subseteq E_{F}$ because $E_{F}$ is a sigma algebra.

∎

Proposition 3.6**.**

*The requirement for the general concept $(X,\rho(X),\eta(X))$ in Proposition 3.4 is satisfied if $X\in E_{F}$.

$E_{F}$ is in effect the full collection of general extents for the GCL.*

• •

$\forall X\in E_{F},\ \rho(X)=\sum_{X_{0}\subseteq X}\left(\sum[X_{0}]_{F}\right)$* and $\eta(X)=\prod_{X_{0}\supseteq X}\left(\prod[X_{0}]_{F}\right)$ where in both cases $X_{0}\in E_{F}$.*

• •

Equivalently, $\rho(X)=\sum[X]_{F}$ and $\eta(X)=\prod[X]_{F}$.

Proof.

It is remarkable for any subset $O\subseteq G$ that $O\in E_{F}$ implies $[O]_{F}\neq\emptyset$ otherwise $[O]_{F}=\emptyset$. In what follows, $X$ is reserved for the use of $X\in E_{F}$ such that $\sum[X]_{F}$ and $\prod[X]_{F}$ can be defined.

• •

According to Definition 3.3,

$\rho(X)=\sum X^{\Box^{\ast}}=\sum\left\{\mu\in M^{\ast}\mid\forall g\in G,\ gR\mu\implies g\in X\right\}=\sum\left(\bigcup_{O\subseteq X}[O]_{F}\right)=\sum_{X_{0}\subseteq X}\left(\sum[X_{0}]_{F}\right)$,

$\therefore\ \rho(X)^{R}=\left(\sum_{X_{0}\subseteq X}\left(\sum[X_{0}]_{F}\right)\right)^{R}=\bigcup_{X_{0}\subseteq X}(\bigcup_{\mu\in[X_{0}]_{F}}\mu^{R})=X$, which satisfies Proposition 3.4,

$\eta(X)=\prod X^{I^{\ast}}=\prod\left\{\mu\in M^{\ast}\mid gR\mu,\ g\in X\right\}=\prod\left(\bigcup_{O\supseteq X}[O]_{F}\right)=\prod_{X_{0}\supseteq X}\left(\prod[X_{0}]_{F}\right)$,

$\therefore\ \eta(X)^{R}=\left(\prod_{X_{0}\supseteq X}\left(\prod[X_{0}]_{F}\right)\right)^{R}=\bigcap_{X_{0}\supseteq X}(\bigcap_{\mu\in[X_{0}]_{F}}\mu^{R})=X$, ditto.

• •

If $\mu\in[X]_{F}$ then $\mu+\sum_{X_{0}\subset X}[X_{0}]_{F}\in[X]_{F}$ as well as $\mu\cdot\prod_{X_{0}\supset X}[X_{0}]_{F}\in[X]_{F}$.

Therefore, $\sum[X]_{F}=\sum_{\mu\in[X]_{F}}\mu=\sum_{\mu\in[X]_{F}}\left(\mu+\sum_{X_{0}\subset X}[X_{0}]_{F}\right)=\sum_{X_{0}\subseteq X}\left(\sum[X_{0}]_{F}\right)$

and $\prod[X]_{F}=\prod_{\mu\in[X]_{F}}\mu=\prod_{\mu\in[X]_{F}}\left(\mu\cdot\prod_{X_{0}\supset X}[X_{0}]_{F}\right)=\prod_{X_{0}\supseteq X}\left(\prod[X_{0}]_{F}\right)$.

∎

The GCL then comprises $2^{n_{F}}$ nodes which correspond to $2^{n_{F}}$ general concepts (Proposition 3.5). Specifically, there is no more need to search for the object classes that satisfy the requirements for general concepts. The general extents are nothing but the object classes equipped with non-trivial contextual equivalent classes of attributes. The GCL thus provides a categorisation for whatever object sets that can be explicitly labelled by attributes. Subsequently, the following conjugate relation considerably reduces the complexity for deducing GCL.

Proposition 3.7**.**

$(X,\rho(X),\eta(X))$* is a general concept if only if $(X^{c},\rho(X^{c}),\eta(X^{c}))$ is a general concept,

where $\rho(X^{c})=\neg\eta(X)$ and $\eta(X^{c})=\neg\rho(X)$.*

Proof.

Since $E_{F}$ is a sigma algebra by Proposition 3.5, $X\in E_{F}\iff X^{c}\in E_{F}$. Moreover, with Proposition 3.6,

$\neg{\rho}(X)=\neg\sum[X]_{F}=\prod\left\{\mu\mid\ \neg\mu\in[X]_{F}\right\}=\prod\left\{\mu\mid\ \mu\in[X^{c}]_{F}\right\}=\prod[X^{c}]_{F}=\eta(X^{c})$. Likewise,

$\neg{\eta}(X)=\neg\prod[X]_{F}=\sum[X^{c}]_{F}=\rho(X^{c})$.∎

Interestingly, although the above result has a similar appearance to Eq. (11), it possesses a completely different meaning. The GCL is self-dual in the sense that the general concepts always appear pairwise. Subsequently, in order to furnish the lattice structure, the way to order concepts as nodes on the lattice remains in question. We now proceed to resolve the question.

Proposition 3.8**.**

Given a formal context $F(G,M)$, $\forall X\in E_{F}$

• •

if $X\not\in G_{/R}$ and $X\neq\emptyset,\ \rho(X)=\sum_{X_{0}\subset X}\rho(X_{0})$,

• •

if $X\not\in G^{/R}$ and $X\neq G,\ \eta(X)=\prod_{X_{0}\supset X}\eta(X_{0})$.

Proof.

• •

$\rho(X)=\sum_{X_{0}\subseteq X}\left(\sum[X_{0}]_{F}\right)=\sum[X]_{F}+\sum_{X_{0}\subset X}\left(\sum[X_{0}]_{F}\right)=\rho(X)+\sum_{X_{0}\subset X}\rho(X_{0})$ by Proposition 3.6, hence, $\rho(X)\geq\sum_{X_{0}\subset X}\rho(X_{0})$. In addition, if one can prove that $\sum_{X_{0}\subset X}\rho(X_{0})\geq\rho(X)$ then the statement is established. Since $X\not\in G_{/R}\cup\left\{\emptyset\right\}$, there exists some subset $X_{1}\subset X$ s.t. $X\backslash X_{1}\neq\emptyset$ and $X_{1}\in E_{F}$. Consider then $\sum_{X_{0}\subset X}\rho(X_{0})=\sum_{X_{0}\subset X}\sum[X_{0}]_{F}=\sum\left(\bigcup_{X_{0}\subset X}[X_{0}]_{F}\right)$. Clearly, for any $\tau_{1}\in[X_{1}]_{F}$ one has $\rho(X)\cdot\tau_{1}\in[X_{1}]_{F}\subseteq\bigcup_{X_{0}\subset X}[X_{0}]_{F}$ because $(\rho(X)\cdot\tau_{1})^{R}=\rho(X)^{R}\cap\tau_{1}^{R}=X_{1}$. Moreover, $(\rho(X)\cdot\neg\tau_{1})^{R}=\rho(X)^{R}\cap\tau_{1}^{Rc}=X\backslash X_{1}$, which implies that $\rho(X)\cdot\neg\tau_{1}\in[X\backslash X_{1}]_{F}\subseteq\bigcup_{X_{0}\subset X}[X_{0}]_{F}$, where notably $[X\backslash X_{1}]_{F}\neq\emptyset$ for $[X_{1}]_{F}\neq\emptyset$ (Proposition 3.5). Consequently, $\bigcup_{X_{0}\subset X}[X_{0}]_{F}\supseteq\left\{\rho(X)\cdot\tau_{1},\ \rho(X)\cdot\neg\tau_{1}\right\}$, which implies that $\sum_{X_{0}\subset X}\rho(X_{0})\equiv\sum\left(\bigcup_{X_{0}\subset X}[X_{0}]_{F}\right)\geq(\rho(X)\cdot\tau)+(\rho(X)\cdot{\neg\tau})\equiv\rho(X)$.

• •

The above result implies that $\neg\rho(X)=\neg\sum_{X_{0}\subset X}\rho(X_{0})=\prod_{X_{0}\subset X}\neg\rho(X_{0})$ for $X\in E_{F}\backslash G_{/R}\backslash\left\{\emptyset\right\}$. Then, by Lemma 3.7, $\eta(X^{c})=\prod_{X_{0}\subset X}\eta(X_{0}^{c})$ for $X\in E_{F}\backslash G_{/R}\backslash\left\{\emptyset\right\}$ which is equivalent to $\eta(X)=\prod_{X_{0}\supset X}\eta(X_{0})$ for $X\in E_{F}\backslash G^{/R}\backslash\left\{G\right\}$.

∎

All the general concepts, i.e. $(X,\rho(X),\eta(X))\ \forall X\in E_{F}$, can be deduced in a definite manner based on a few fundamental attributes.

Proposition 3.9**.**

Subject to the formal context $F(G,M)$ $\forall X\in E_{F}$

• •

if $X\neq\emptyset$ then $\rho(X)=\sum_{k\in K}\rho(D_{k})$ for some $K\subseteq\left\{1,\ldots n_{F}\right\}$,

• •

if $X\neq G$ then $\eta(X)=\prod_{{k}\not\in K}\eta(D_{k}^{c})$ for some $K\subseteq\left\{1,\ldots n_{F}\right\}$, where $D_{k}^{c}\equiv G\backslash D_{k}$.

Proof.

Since $X\in E_{F}$, Proposition 3.5 entails that $X=\bigcup_{k\in K}D_{k}=\bigcap_{{k}\not\in K}D_{k}^{c}$ for some $K\subseteq\left\{1,\ldots n_{F}\right\}$.

• •

If $X\not\in G_{/R}$ then $\rho(X)=\sum_{X_{0}\subset\bigcup_{k\in K}D_{k}}\rho(X_{0})=\sum_{k\in K}\rho(D_{k})$ (Definition 2.4, Lemma 3.8). Meanwhile, $X\in G_{/R}$ entails that $X=D_{k}$ and $\rho(X)\equiv\rho(D_{k})$ for some $k\in\left\{1,\ldots n_{F}\right\}$.

• •

If $X\not\in G^{/R}$ then $\eta(X)=\prod_{X_{0}\supset\bigcap_{{k}\not\in K}D_{k}^{c}}\eta(X_{0})=\prod_{{k}\not\in K}\eta(D_{k}^{c})$. Meanwhile, $X\in G^{/R}$ entails that $X=D_{k}^{c}$ and $\eta(X)\equiv\eta(D_{k}^{c})$ for some $k\in\left\{1,\ldots n_{F}\right\}$.

∎

Thus, once ${\cal P}_{\rho}:=\left\{\rho(\emptyset)\right\}\cup\left\{\rho(D_{k})\mid 1\leq k\leq n_{F}\right\}$ or ${\cal P}_{\eta}:=\left\{\eta(G)\right\}\cup\left\{\eta(D_{k}^{c})\mid 1\leq k\leq n_{F}\right\}$ is obtained, Proposition 3.9 in practice ensures that the full Grsp’s and Gfcp’s can be unambiguously determined. However, the other instructive issue is that certain particular attributes can be determined without ${\cal P}_{\rho}$ or ${\cal P}_{\eta}$.

Proposition 3.10**.**

Given a formal context $F(G,M)$, $\eta(D_{k})$ is an $M^{\ast}$-atom and $\rho(D_{k}^{c})$ an $M^{\ast}$-coatom (Lemma 2.12) for $1\leq k\leq n_{F}$. It can be identified that $\eta(D_{k})=\prod\Psi^{k}$ and $\rho(D_{k}^{c})=\sum({\breve{M}}\backslash\Psi^{k})$ with $\Psi_{M}=\{m\in M\mid m\in D_{k}^{I}\}\cup\{\neg m\mid m\not\in D_{k}^{I},m\in M\}$, where $\Psi^{k}\in{\bf\Psi}_{M}$ (Definition 2.6).

Proof.

Since $g^{R}=D_{k}^{I}\ \forall g\in D_{k}$, if $\Psi^{k}=\{m\in M\mid m\in D_{k}^{I}\}\cup\{\neg m\mid m\not\in D_{k}^{I},m\in M\}$ then $(\prod\Psi^{k})^{R}=D_{k}$. It turns out that $(\prod\Psi^{k})\in[D_{k}]_{F}$, and $\eta(D_{k})=\prod[D_{k}]_{F}\leq\prod\Psi^{k}$. Meanwhile, $\prod\Psi^{k}\succ_{M}{\bf 0}$ by Lemma 2.12, telling that $\forall\mu\in M^{\ast}\ \mu\not<\prod\Psi^{k}$ unless $\mu={\bf 0}$. Note that $\eta(D_{k})\neq{\bf 0}$ for $1\leq k\leq n_{F}$ since ${\bf 0}^{R}$ is the empty object set $\emptyset$. Therefore, $\eta(D_{k})=\prod\Psi^{k}$, where $\Psi^{k}=\{m\in M\mid m\in D_{k}^{I}\}\cup\{\neg m\mid m\not\in D_{k}^{I}\}$. On the other hand, $\rho(D_{k}^{c})=\neg\eta(D_{k})=\neg\prod\Psi^{k}=\sum({\breve{M}}\backslash\Psi^{k})$ by Proposition 3.7, where notably ${\breve{M}}\backslash\Psi^{k}\in{\bf\Psi}_{M}$.∎

Notably, Proposition 3.10 is concerned with $\eta(D_{k})$ and $\rho(D_{k}^{c})$ rather than $\eta(D_{k}^{c})$ and $\rho(D_{k})$ respectively required in ${\cal P}_{\rho}$ and ${\cal P}_{\eta}$. It is also remarkable that based on the results of Proposition 3.10 one may further resolve a truly feasible and realistic construction for GCL LLJD12-2 . However, to recover the RSL- and FCL- nodes in terms of the general theory of concept lattice remains the concern for the moment. Subject to this goal, it is more convenient to adopt the irreducible expressions of Definition 2.14 to find the elements of ${\cal P}_{\rho}$ and ${\cal P}_{\eta}$, which is described in order.

Proposition 3.11**.**

Subject to the formal context $F(G,M)$ the Grsp and Gfcp can be expressed in terms of irreducible attribute classes:

• •

Expressed in terms of DNF, the Grsp can be simplified (Eq. (12)) as

[TABLE]

• •

Expressed in terms of CNF, the Gfcp can be simplified as

[TABLE]

Proof.

• •

Written in DNF, $\rho(X)=\sum_{k_{\rho}}\psi_{k_{\rho}}$ in which $\psi_{k_{\rho}}$ is a product of members of ${\breve{M}}$ and $\psi_{k_{\rho}}^{R}\subseteq X$ since $\rho(X)^{R}=X$. Thus, $\rho(X)=\sum_{X_{0}\subseteq X}\left(\sum\varphi_{X_{0}}\right)$ (cf. Proposition 3.6), where $\varphi_{X_{0}}$ is a subset of $[X_{0}]_{F}$ which comprises only the attributes given in terms of a product of members of ${\breve{M}}$. For the members of $\varphi_{X_{0}}$, if $\psi_{0}\not\in_{F\times}M^{\ast}$ then there must be $\psi_{1}\in\varphi_{X_{0}}$ such that $\psi_{1}>\psi_{0}$ ($\psi_{1}+\psi_{0}=\psi_{1}$) and $\psi_{1}\in_{F\times}M^{\ast}$ (Definition 2.14). Therefore, $\sum\varphi_{X_{0}}=\sum[X_{0}^{\times}]_{F}$, hence, $\rho(X)=\sum_{X_{0}\subseteq X}\left(\sum[X_{0}^{\times}]_{F}\right)$.

Moreover, consider a part of expression of $\sum_{X_{0}\subseteq X}\left(\sum[X_{0}^{\times}]_{F}\right)$ which emerges as $\sum[X_{1}^{\times}]_{F})+\sum[X_{2}^{\times}]_{F}$ with $X_{1}\subset X_{2}$. For $\mu\in[X_{1}^{\times}]_{F}$, if $\mu=\alpha\cdot\nu$ with $\alpha\in[X_{2}^{\times}]_{F}$ or $\nu\in[X_{2}^{\times}]_{F}$ then $\mu$ is absent in $\left(\sum[X_{1}^{\times}]_{F}\right)+\left(\sum[X_{2}^{\times}]_{F}\right)$ since $\mu+\left(\sum[X_{2}^{\times}]_{F}\right)=\left(\sum[X_{2}^{\times}]_{F}\right)$. Consequently, $\sum_{X_{0}\subseteq X}\left(\sum[X_{0}^{\times}]_{F}\right)=\sum_{X_{0}\subseteq X}\sum\left(\bigcap_{X_{0}\subset X_{i}\subseteq X}\frac{[X_{0}^{\times}]_{F}}{[X_{i}^{\times}]_{F}}\right)$.

• •

By Proposition 3.7, $\eta(X)=\neg\rho(X^{c})=\neg\sum[X^{c}]_{F}=\prod\left(\bigcup_{X_{0}^{c}\subseteq X^{c}}\left\{\mu\mid\ \neg\mu=[\left(X_{0}^{c}\right)^{\times}]_{F}\right\}\right)$. Subsequently, $\neg\mu=[\left(X_{0}^{c}\right)^{\times}]_{F}$ iff $\mu=[\left(X_{0}\right)^{+}]_{F}$ by Lemma 2.15. Therefore, $\eta(X)=\prod\left(\bigcup_{X_{0}\supseteq X}\left\{\mu\mid\ \mu=[\left(X_{0}\right)^{+}]_{F}\right\}\right)=\prod_{X_{0}\supseteq X}\left(\prod[X_{0}^{+}]_{F}\right)$.

Similar to the case for $\rho(X)$, consider $\prod[X_{1}^{+}]_{F}\cdot\prod[X_{2}^{+}]_{F}$ in which $X_{1}\supset X_{2}$ within the expression $\prod_{X_{0}\supseteq X}\left(\prod[X_{0}^{+}]_{F}\right)$. For $\mu^{\prime}\in[X_{1}^{+}]_{F}$, if $\mu^{\prime}=\alpha^{\prime}+\nu^{\prime}$ with $\alpha^{\prime}\in[X_{2}^{+}]_{F}$ or $\nu^{\prime}\in[X_{2}^{+}]_{F}$ then $\mu^{\prime}\cdot\left(\prod[X_{2}^{+}]_{F}\right)=\left(\prod[X_{2}^{+}]_{F}\right)$. Consequently, $\prod_{X_{0}\supseteq X}\left(\prod[X_{0}^{+}]_{F}\right)=\prod_{X_{0}\supseteq X}\prod\left(\bigcap_{X_{0}\supset X_{j}\supseteq X}\frac{[X_{0}^{+}]_{F}}{[X_{j}^{+}]_{F}}\right)$.

∎

Apparently, the above results are consequences of Proposition 3.6.

Proposition 3.12**.**

Given a formal context $F(G,M)$, one can obtain the contextual truth and falsity $(1_{\eta}$ and $0_{\rho})$, in contrast to the conventional truth and falsity $({\bf 1}$ and ${\bf 0})$ in the following way,

• •

$\eta(\emptyset)={\bf 0}$* and $\rho(G)={\bf 1}$,*

• •

$\rho(\emptyset)+\rho(X)=\rho(X)$* and $\rho(\emptyset)\cdot\rho(X)=\rho(\emptyset)$,*

$\eta(G)\cdot\eta(X)=\eta(X)$* and $\eta(G)+\eta(X)=\eta(G)$, $\forall X\in E_{F}$.*

Accordingly, $0_{\rho}:=\rho(\emptyset)$ since it behaves like the falsity for Grsp, while $1_{\eta}:=\eta(G)$ since it behaves like the truth for Gfcp. In addition, from Proposition 3.7, it follows that $1_{\eta}=\neg 0_{\rho}$. Moreover, for the truth of Grsp $1_{\rho}\equiv{\bf 1}$, while for the falsity of Gfcp $0_{\eta}\equiv{\bf 0}$.

Proof.

• •

$\eta(\emptyset)=\prod_{X\neq\emptyset}\eta\left(X\right)\leq\prod_{k=1}^{n_{F}}\eta(D_{k})={\bf 0}$ by Proposition 3.8 and 3.10 since $\eta(D_{k})\eta(D_{k^{\prime}})={\bf 0}$ for $k\neq{k^{\prime}}$.

On the other hand, $\rho(G)=\neg{\eta}(\emptyset)=\neg{\bf 0}={\bf 1}$ by Proposition 3.7.

• •

On employing Proposition 3.6,

$\rho(\emptyset)+\rho(X)=\sum[\emptyset]_{F}+\sum_{X_{0}\subseteq X}\left(\sum[X_{0}]_{F}\right)=\sum_{X_{0}\subseteq X}\left(\sum[X_{0}]_{F}\right)=\rho(X)$,

$\rho(\emptyset)\cdot\rho(X)=\sum[\emptyset]_{F}\cdot\sum_{X_{0}\subseteq X}\left(\sum[X_{0}]_{F}\right)=\sum[\emptyset]_{F}=\rho(\emptyset)$,

$\eta(G)\cdot\eta(X)=\prod[G]_{F}\cdot\prod_{X_{0}\supseteq X}\left(\prod[X_{0}]_{F}\right)=\prod_{X_{0}\supseteq X}\left(\prod[X_{0}]_{F}\right)=\eta(X)$,

$\eta(G)+\eta(X)=\prod[G]_{F}+\prod_{X_{0}\supseteq X}\left(\prod[X_{0}]_{F}\right)=\prod[G]_{F}=\eta(X)$.

∎

Corollary 3.13**.**

• •

$\rho(\emptyset)\equiv 0_{\rho}=\sum[\emptyset^{\times}]_{F}$* and $\eta(G)\equiv 1_{\eta}=\prod[G^{+}]_{F}$.*

• •

$\rho(D_{k})=\rho_{0}(D_{k})+0_{\rho}$* and $\eta(D_{k}^{c})=\eta_{0}(D_{k}^{c})\cdot 1_{\eta}$,*

where $\rho_{0}(D_{k})=\sum[D_{k}^{\times}]_{F}$ and $\eta_{0}(D_{k}^{c})=\prod[(D_{k}^{c})^{+}]_{F}$ for $1\leq k\leq n_{F}$.

Proof.

These are direct consequences of Proposition 3.12, upon using the formulae in Proposition 3.11.

• •

$\rho(\emptyset)=\sum[\emptyset]_{F}=\sum[\emptyset^{\times}]_{F}$ and $\eta(G)=\prod[G]_{F}=\prod[G^{+}]_{F}$.

• •

$\rho(D_{k})=\sum[D_{k}^{\times}]_{F}+\sum[\emptyset^{\times}]_{F}=\rho_{0}(D_{k})+0_{\rho}$ and $\eta(D_{k}^{c})=\prod[\left(D_{k}^{c}\right)^{+}]_{F}\cdot\prod[G^{+}]_{F}=\eta_{0}(D_{k}^{c})\cdot 1_{\eta}$.

∎

Hence, one may also replace ${\cal P}_{\rho}$ and ${\cal P}_{\eta}$ by ${\cal P}^{0}_{\rho}:=\left\{0_{\rho}\right\}\cup\left\{\rho_{0}(D_{k})\mid 1\leq k\leq n_{F}\right\}$ and ${\cal P}^{0}_{\eta}:=\left\{1_{\eta}\right\}\cup\left\{\eta_{0}(D_{k}^{c})\mid 1\leq k\leq n_{F}\right\}$, respectively. Here, one may recover the full general concepts via Proposition 3.8 in terms of the irreducible attribute classes given in Definition 2.14. According to Proposition 3.9 and based on ${\cal P}^{0}_{\rho}$ and ${\cal P}_{\eta}$, one then obtains Grsp’s in DNF and the Gfcp’s in CNF. We now proceed to consider the corresponding lattice structure.

The ordering of general concepts can be constructed in an unambiguous way, thereby forming the desired Galois connection.

Proposition 3.14**.**

• •

For $X_{i},X_{j}\in E_{F},\ X_{i}\neq X_{j}$ iff $\rho(X_{i})\neq\rho(X_{j})$ and $\eta(X_{i})\neq\eta(X_{j})$.

• •

*For $X_{i},X_{j}\in E_{F},\ X_{i}\subset X_{j}$ iff $\rho(X_{i})<\rho(X_{j})$ and $\eta(X_{i})<\eta(X_{j})$. *

Proof.

• •

By Lemma 2.10, $\forall\mu_{i}\forall\mu_{j}\ \mu_{i}^{R}\neq\mu_{j}^{R}\implies\ \mu_{i}\neq\mu_{j}$. Here, $X_{i}\neq X_{j}$ means $\left\{\begin{smallmatrix}\rho(X_{i})^{R}\neq\rho(X_{j})^{R}\\ \eta(X_{i})^{R}\neq\eta(X_{j})^{R}\end{smallmatrix}\right.$, which implies $\left\{\begin{smallmatrix}\rho(X_{i})\neq\rho(X_{j})\\ \eta(X_{i})\neq\eta(X_{j})\end{smallmatrix}\right.$. Thus, $X_{i}\neq X_{j}\implies\left\{\begin{smallmatrix}\rho(X_{i})\neq\rho(X_{j})\\ \eta(X_{i})\neq\eta(X_{j})\end{smallmatrix}\right.$. On the other hand, if $X_{i}=X_{j}$ then $\left\{\begin{smallmatrix}\rho(X_{i})=\rho(X_{j})\\ \eta(X_{i})=\eta(X_{j})\end{smallmatrix}\right.$. Consequently, if $\rho(X_{i})\neq\rho(X_{j})$ or $\eta(X_{i})\neq\eta(X_{j})$ then $X_{i}\neq X_{j}$, i.e., $\left\{\begin{smallmatrix}\rho(X_{i})\neq\rho(X_{j})\implies X_{i}\neq X_{j}\\ \eta(X_{i})\neq\eta(X_{j})\implies X_{i}\neq X_{j}\end{smallmatrix}\right.$. Moreover, $\rho(X_{i})\neq\rho(X_{j})\implies X_{i}\neq X_{j}\implies\left\{\begin{smallmatrix}\rho(X_{i})\neq\rho(X_{j})\\ \eta(X_{i})\neq\eta(X_{j})\end{smallmatrix}\right.$ and $\eta(X_{i})\neq\eta(X_{j})\implies X_{i}\neq X_{j}\implies\left\{\begin{smallmatrix}\rho(X_{i})\neq\rho(X_{j})\\ \eta(X_{i})\neq\eta(X_{j})\end{smallmatrix}\right.$. Therefore, $X_{i}\neq X_{j}\iff\left\{\begin{smallmatrix}\rho(X_{i})\neq\rho(X_{j})\\ \eta(X_{i})\neq\eta(X_{j})\end{smallmatrix}\right.$.

• •

With Definition 3.3, one has the ordering rules

$X_{i}\subseteq X_{j}\iff(X_{j})^{I^{\ast}}\subseteq(X_{i})^{I^{\ast}}$ and $X_{i}\subseteq X_{j}\iff(X_{i})^{\Box^{\ast}}\subseteq(X_{j})^{\Box^{\ast}}$,

which is the consequence of Eq. (9) extended to $F^{\ast}(G,M^{\ast})$. Therefore,

[TABLE]

by Proposition 3.4.

∎

Proposition 3.15**.**

The conventional FCL and RSL nodes can always be recovered from the GCL:

• •

The criterion for $X$ being an RSL extent, which is $Y_{rsl}^{\Diamond}=X$ by $Y_{rsl}:=X^{\Box}$, can be satisfied with

$X\in\left\{\bigcup_{m\in M_{0}}m^{R}\mid M_{0}\subseteq M\right\}$.

The criterion for $X$ being an FCL extent, which is $Y_{fcl}^{I}=X$ by $Y_{fcl}:=X^{I}$, can be satisfied with

$X\in\left\{\bigcap_{m\in M_{0}}m^{R}\mid M_{0}\subseteq M\right\}$.

• •

Whenever $X\in E_{F}$ is an FCL and/or RSL extent,

*the corresponding RSL intent $Y_{rsl}$ can be faithfully read out from $\rho(X)$, *

the corresponding FCL intent $Y_{fcl}$ can be faithfully read out from $\eta(X)$.

Proof.

• •

The criteria for $X$ being the extent can be rearranged by means of Eq. (13).

For $Y_{rsl}=X^{\Box}$, it turns out that $\left(X^{\Box}\right)^{\Diamond}\equiv\bigcup_{m\in X^{\Box}}m^{R}=X$. Note that $\forall M_{0}\subseteq M$ if $X=\bigcup_{m\in M_{0}}m^{R}$ then $X^{\Box}=M_{0}$, thus, $\bigcup_{m\in X^{\Box}}m^{R}=\bigcup_{m\in M_{0}}m^{R}\equiv X$. Therefore, $\left(X^{\Box}\right)^{\Diamond}=X$ for $X\in\left\{\bigcup_{m\in M_{0}}m^{R}\mid M_{0}\subseteq M\right\}$. On the other hand, for $Y_{fcl}=X^{I}$, one requires that $\left(X^{I}\right)^{I}\equiv\bigcap_{m\in X^{I}}m^{R}=X$. Likewise, if $X=\bigcap_{m\in M_{0}}m^{R}$ then $M_{0}$, thus, $\bigcap_{m\in X^{I}}m^{R}=\bigcap_{m\in M_{0}}m^{R}\equiv X$. Consequently, $\left(X^{I}\right)^{I}=X$ for $X\in\left\{\bigcap_{m\in M_{0}}m^{R}\mid M_{0}\subseteq M\right\}$.

• •

By Proposition 3.11,

[TABLE]

where $\bigcup_{X_{0}\subseteq X}\left(\bigcap_{X_{0}\subset X_{i}\subseteq X}\frac{[X_{0}^{\times}]_{F}}{[X_{i}^{\times}]_{F}}\right)\supset\bigcup_{X_{0}\subseteq X}\left\{m\in M\mid m^{R}=X_{0}\right\}$ according to Lemma 2.15. Moreover, $\bigcup_{X_{0}\subseteq X}\left\{m\in M\mid m^{R}=X_{0}\right\}=\left\{m\in M\mid m^{R}\subseteq X\right\}=X^{\Box}$ by Eq. (13). Therefore, if $m\in X^{\Box}\equiv Y_{rsl}$ then $m\in\bigcup_{X_{0}\subseteq X}\left(\bigcap_{X_{0}\subset X_{i}\subseteq X}\frac{[X_{0}^{\times}]_{F}}{[X_{i}^{\times}]_{F}}\right)$. Similarly,

[TABLE]

where $\bigcup_{X_{0}\supseteq X}\left(\bigcap_{X_{0}\supset X_{j}\supseteq X}\frac{[X_{0}^{+}]_{F}}{[X_{j}^{+}]_{F}}\right)\supset\left\{m\in M\mid m^{R}\supseteq X\right\}=X^{I}$ by Lemma 2.15 and Eq. (13). Therefore, if $m\in X^{I}\equiv Y_{fcl}$ then $m\in\bigcup_{X_{0}\supseteq X}\left(\bigcap_{X_{0}\supset X_{j}\supseteq X}\frac{[X_{0}^{+}]_{F}}{[X_{j}^{+}]_{F}}\right)$.

∎

Remarkably, the roles the RSL and FCL play within the GCL can be understood easily. The components of an RSL intent are those simple attributes (Definition 2.5) found from the Grsp in DNF, whereas the components of an FCL intent are simple attributes found from Gfcp in CNF. Moreover, the RSL is only able to categorise those object classes which are expressible in terms of a union of $m^{R}$’s for some $m$’s in $M$, whereas the FCL is only able to categorise those object classes which are expressible in terms of an intersection of $m^{R}$’s for some $m$’s in $M$.

One may construct the GCL in terms of lattice theory Wi82 ; YY04 ; BH00 .

Proposition 3.16**.**

Based on the collection of all the general concepts $L_{F}=\left\{(X,\rho(X),\eta(X))\mid\ X\in E_{F}\right\}$, the GCL exhibits a lattice structure $\Gamma_{F}:=(L_{F},\wedge,\vee,\dagger)$, where $\wedge$ (meet) and $\vee$ (join) are binary operators and the unary operator $\dagger$ provides a kind of duality:

• •

$L_{F}$* defines the partially ordered set (poset) $(L_{F},\leq)$ with the unambiguous order $\leq$.*

• •

The lattice operations $\wedge$, $\vee$ and $\dagger$ are well defined.

$\Gamma_{F}$* is self-dual in the sense that $l\in L_{F}$ iff $l^{\dagger}\in L_{F}$. $\dagger$ reverses the ordering of lattice nodes in the sense that $l_{i}\leq l_{j}$ iff $l_{i}^{\dagger}\geq l_{j}^{\dagger}$. *

• •

$\Gamma_{F}$* is complete in the sense that the supremum and infimum are found to be $l_{sup}=(G,{\bf 1},1_{\eta})$ and $l_{inf}=(\emptyset,0_{\rho},{\bf 0})$, respectively, where $G$ is the full collection of objects.*

Proof.

For all $l_{k}\in L_{F}$, let $l_{k}=(X_{k},\rho(X_{k}),\eta(X_{k}))$.

• •

Because $X_{i}\subseteq X_{j}$ iff $\left\{\begin{smallmatrix}\rho(X_{i})\leq\rho(X_{j})\\ \eta(X_{i})\leq\eta(X_{j})\end{smallmatrix}\right.$ (Proposition 3.14), it is natural to define $l_{i}\leq l_{j}$ via $\left\{\begin{smallmatrix}X_{i}\subseteq X_{j}\\ \rho(X_{i})\leq\rho(X_{j})\\ \eta(X_{i})\leq\eta(X_{j})\end{smallmatrix}\right.$.

• •

$\forall l_{i}\forall l_{j}\forall l_{k}\in L_{F}$, one has

[TABLE]

where $(X_{k}^{c},\rho(X_{k}^{c}),\eta(X_{k}^{c}))=(X_{k}^{c},\neg\eta(X_{k}),\neg\rho(X_{k}))$ is based on Proposition 3.7.

Moreover, $l_{i}\leq l_{j}$ iff $l_{i}^{\dagger}\geq l_{j}^{\dagger}$ as $\left\{\begin{smallmatrix}X_{i}\subseteq X_{j}\\ \rho(X_{i})\leq\rho(X_{j})\\ \eta(X_{i})\leq\eta(X_{j})\end{smallmatrix}\right.\iff\left\{\begin{smallmatrix}X_{i}^{c}\supseteq X_{j}^{c}\\ \rho(X_{i}^{c})\geq\rho(X_{j}^{c})\\ \eta(X_{i}^{c})\geq\eta(X_{j}^{c})\end{smallmatrix}\right.$ or $\left\{\begin{smallmatrix}X_{i}^{c}\supseteq X_{j}^{c}\\ \neg\rho(X_{i})\geq\neg\rho(X_{j})\\ \neg\eta(X_{i})\geq\neg\eta(X_{j})\end{smallmatrix}\right.$.

• •

Given $L_{0}\subseteq L_{F}$, let $l_{0}=(X_{0},\rho(X_{0}),\eta(X_{0}))\in L_{0}$ in which $X_{0}\in E_{0}\subseteq E_{F}$. Then,

[TABLE]

Since $\bigvee_{l_{0}\in L_{0}}l_{0}\geq\ l\ \geq\bigwedge_{l_{0}\in L_{0}}l_{0}\quad\forall l\in L_{0}$, it turns out that

[TABLE]

by Proposition 3.8, 3.12.

∎

It is also noteworthy that within the expression $(X,\rho(X),\eta(X))$ one may regard $(\rho(X),\eta(X))$ as a prescription for the general intent corresponding to the general extent $X$. Such a prescription is convenient for the discussion concerning the RSL and FCL in relation to the GCL.

4 discussion

One has proved the existence of the general concept lattice above the conventional framework of RSL and FCL, following the anticipated accomplishments G1-G5 proposed in Sec. 1. The formal context $F(G,M)$ automatically induces the extended formal context $F^{\ast}(G,M^{\ast})$ (Lemma 2.8). In effect, based on $F(G,M)$ there are intrinsic relations between the objects in $G$ and the members of the full generalised attribute set $M^{\ast}$, which all pertain to the indispensable structure of the content information. It is by virtue of $F^{\ast}(G,M^{\ast})$ that one can assure the 2-tuple general concept expression in terms of $(\mbox{general extent},\mbox{general intent})$ for G1, where the general extents emerge as the object classes allowable within GCL and the general intents their corresponding property descriptions. From Proposition 3.4 and 3.6 one notices that the general intent can be represented by the pair 222 This is a fact that is further supported by Proposition 3.14, which entails that both the generalised rough set property and generalised formal concept property are ordered simultaneously.

(generalised rough set property, generalised formal concept property) , i.e. (Grsp,Gfcp).

Moreover, the general extents are found to be the collection $E_{F}=\sigma(\{m^{R}\mid m\in M\})$ in view of Proposition 3.5 and Proposition 3.6, where in particular $X\in E_{F}$ iff $X^{c}\in E_{F}$. Note that G3 is achieved as follows. Since $E_{F}$ in effect comprises whatever distinctive object classes, making the GCL constitute a Hasse diagram with $2^{|\sigma(G_{/R})|}=2^{n_{F}}$ nodes (G2), it is certain on the GCL that one can always find all the nodes corresponding to the object classes categorised according to the RSL and the FCL, respectively. Indeed, Proposition 3.15 also provides systematic ways to regain both the RSL intent and FCL intent from the irreducible expressions of Grsp and Gfcp, respectively. Another point is that the GCL permits deterministic construction (G4), as is the method proposed by Lemma 3.8, 3.11 and Corollary 3.13. Though to complete the general concept construction in such a manner turns out to be inefficient, the method is rather elucidative for the GCL in relation to the conventional lattices. Especially, one may notice that the freedom involved in the formal context can be significantly reduced. The GCL in fact emerges as a self-dual lattice bearing the structure of Proposition 3.16 based on the one-to-one correspondence $\forall X\in E_{F}\left\{\begin{smallmatrix}\rho(X^{c})=\neg\eta(X)\\ \eta(X^{c})=\neg\rho(X)\end{smallmatrix}\right.$ demonstrated in Lemma 3.7, which is the conjugateness relation needed by G5. Obviously, the self-duality halves the complexity for completing the general intents.

Being an original construction, the present approach of GCL has thus far focused on its comparabibility with the existing approaches. To this end, one found the employment of Grsp in DNF and Gfcp in CNF particularly instructive for carrying out such comparisons.

• •

$\rho(X)$ and $\eta(X)$ can be regarded respectively as the generalisation for RSL intent and the generalisation for FCL intent (Proposition 3.1, 3.4):

Without altering its theoretical content, one may replace the RSL intent $X^{\Box}$ by its disjunction $\sum X^{\Box}$ while $\rho(X)=\sum X^{\Box^{\ast}}$, which can be intuitively arranged into an expression in DNF.

The FCL intent $X^{I}$ can be represented by its conjunction $\prod X^{I}$ while $\eta(X)=\prod X^{I^{\ast}}$, which can be arranged into an expression in CNF.

• •

One has acquired more precise descriptions for the properties of the object class through $\rho(X)$ and $\eta(X)$ than through the RSL intent and FCL intent:

Observe that $\rho(X)=\sum X^{\Box^{\ast}}=\sum X^{\Box}+(\ldots)\geq\sum X^{\Box}$ since obviously $X^{\Box^{\ast}}\supseteq X^{\Box}$. If $X$ is an RSL extent, $\rho(X)$ suggests that there are in fact more properties peculiar to the class $X$ to be considered, in contrast to $\sum X^{\Box}$.

Likewise, $\eta(X)=\prod X^{I^{\ast}}\cdot(\ldots)\leq\prod X^{I}$ ($\because\ X^{I^{\ast}}\supseteq X^{I}$). Namely, if $X$ is an FCL extent then $\eta(X)$, in contrast to $\prod X^{I}$, in fact incorporates more properties the objects in the class $X$ can possess in common.

Technically, the general concept construction reflecting the above conventional wisdom of the traditional FCL and RSL can be accomplished via Lemma 3.8, 3.11 and Corollary 3.13, which requires the determination of various irreducible attributes (Definition 2.14). Unfortunately, such a method remains tedious and impractical even though the irreducible attributes are constructable. However, things turn out drastically different when one instead considers the Gfcp in DNF and the Grsp in CNF. In Ref. LLJD12-2 it will be shown that the Gfcp can be identified as components of the contextual truth $1_{\eta}$ and $\rho(X)$ the contextual falsity $0_{\rho}$ defined in Proposition 3.12. Moreover, the determination of the general intents can be implemented by binary representations which label the object classes. Noteworthy is also that the contextual truth (or falsity) also plays the crucial role for the logical deduction in the frame of classical logics. In practice, all the rules of implication concerned with the attributes will be shown to be derivable based on $\mu\rightarrow\mu\cdot 1_{\eta}\ \forall\mu$ (or, equivalently, from $\mu+0_{\rho}\rightarrow\mu$). We shall present the tractability in both obtaining the full general concepts and determining all the accompanied rules of logic deduction for the general concept lattice in the next paper LLJD12-2 .

In conclusion, the GCL provides the comprehensive categorisation for whatever distinctive objects according to whatever properties one can refer to in the formal context. The general extents are collected as $E_{F}=\left\{\mu^{R}\mid\mu\in M^{\ast}\right\}$ as opposed to Proposition 3.15, which states 333More precisely, $E_{F}^{rsl}=\left\{\bigcup_{m\in M_{0}}m^{R}\mid M_{0}\subseteq M\right\}\cup\left\{\emptyset\right\}$ and $E_{F}^{fcl}=\left\{\bigcap_{m\in M_{0}}m^{R}\mid M_{0}\subseteq M\right\}\cup\left\{G\right\}$, as will be further clarified in Ref. LLJD12-2 . Note that in their constructions $\emptyset$ (the empty object set) is always included as an RSL extent and $G$ as an FCL extent in view of the completness of lattice Wi82 ; YY04 . that

[TABLE]

Clearly, $E_{F}^{rsl}\cup E_{F}^{fcl}\subset E_{F}$ $E_{F}$ is also referred to as $\left\{\bigcup_{D\in E^{0}}D\mid E^{0}\subseteq G_{/R}\right\}$ (Proposition 3.5), meaning that the categorisation the GCL can accomplish ranges over all the subsets of $G$ that are distinct from the perspective of formal context. It is also remarkable that the general intent is more adequately represented by the equivalent class of attributes $[X]_{F}=\left\{\mu\mid\mu^{R}=X,\ \mu\in M^{\ast}\right\}$ meaning that $[X]_{F}$ refers to all the attributes the objects in $X$ possess in common. Here, $\eta(X)$ and $\rho(X)$ simply appear as the bounds of $[X]_{F}$. It can be further shown LLJD12-2 that every general intent is distinctive in the sense that $[X_{i}]_{F}\cap[X_{j}]_{F}=\emptyset$ iff $X_{i}\neq X_{j}$ and $M^{\ast}=\bigcup_{X\in E_{F}}[X]_{F}$. Thus, the GCL not only provides the categorisation for all the discernible subsets of $G$ but also exhausts the full generalised attribute set $M^{\ast}$.

acknowledgements

The authors are grateful to Dr. Arthur Chen of Tamkang University who asked for clarifying some mathematical definitions from the FCA theory in the year 2012, a crucial issue that inspired the present treatise. This paper is partially supported by Ministry of Science and Technology, Taiwan (Grant number: MOST 105-2633-E-001-001).