# A Fast Algorithm for Three-Way Object-Oriented Concept Acquisition
## 2 Preliminaries
Therefore, \(Y\) satisfies the definition of \(X^{\square}\) in the formal context \((U,V,R)\), i.e., Equation (5). Thus, \(Y=X^{\square}\). 2. Second, we prove that \(Z=X^{\square}\). We know that \(Z=\cup\mathfrak{a}_{h}\), where \(\forall a_{h}\in V,g(a_{h})\subseteq X\). Thus, \(Z=X^{\square}\) satisfies the definition of \(X^{\square}\) in the complement context \((U,V,R^{c})\). Hence, \(Z=X^{\square}\). 3. Finally, we prove \(X=Y^{\diamond}\cup X^{\overline{\diamond}}\). Based on (1), (2), and the property in Equation (9), we obtain \(Y^{\diamond}\cup Z^{\overline{\diamond}}=X^{\square\lozenge}\cup X^{\overline{ \lozenge}\overline{\lozenge}}\subseteq X\). Assuming that \(X^{\square\lozenge}\cup X^{\overline{\lozenge}\overline{\lozenge}}\subset X\), we have \(X^{\square\lozenge}\subset X\) and \(X^{\overline{\lozenge}\overline{\lozenge}}\subset X\). As \((X,a_{j})\in C_{s}\), this means that \(g(a_{j})=X\) and \(a_{j}\in X^{\square}\). Thus, we obtain \(X^{\square\lozenge}\supseteq X\). This leads to a contradiction with the assumption. Thus, we obtain \(X=Y^{\diamond}\cup Z^{\overline{\lozenge}}\). Based on (1), (2) and (3), the proposition has been proven. According to Proposition 2, we can obtain OEO-concepts with the function \(\lambda\) for a formal context \((U,V,R)\). Furthermore, these OEO-concepts that are derived by the function \(\lambda\) can be used to obtain the other remaining OEO-concepts by using the steps below. For convenience of expression, we provide the following definitions:  
Definition 4 (Basic OEO-concept): For a formal context \((U,V,R)\), the concept derived from \(\lambda((X,a_{i}))\), where \(X\subset U\) and \((X,a_{i})\in C_{s}\cup C_{s}^{c}\), is called a basic OEO-concept. The set of all basic OEO-concepts is denoted by \(B_{OEO}\). Definition 5 (\(k\)-granularity OEO-concept): Given an OEO-concept \(c\) of a formal context \((U,V,R)\), if the cardinality of the extent of \(c\) is equal to \(k\), i.e., \(|\mathit{Ext}(c)|\)=\(k\), then the concept \(c\) is called a \(k\)-granularity OEO-concept, and it is denoted by \(c_{OEO}^{k}\). For a formal context \((U,V,R)\), the set of all \(k\)-granularity OEO-concepts is denoted by \(C_{OEO}^{k}\), while the set of all \(k\)-granularity basic OEO-concepts is denoted by \(B_{OEO}^{k}\). Moreover, all OEO-concepts can be categorized into \(C_{OEO}^{0},C_{OEO}^{1},\ldots,C_{OEO}^{|U|}\) by their \(k\)-granularity. Thus, we can get the following equation:  
\[X^{\square} =\{y\in V|\forall x\in U(xRy\Rightarrow x\in X)\}\]9  
Proposition 3: _Any 1-granularity OEO-concept \(c_{OEO}^{1}\in C_{OEO}^{1}\) can be obtained from the basic OEO-concepts, i.e., \(C_{OEO}^{1}=B_{OEO}^{1}\)._  
Proof: For a 1-granularity OEO-concept \(c_{OEO}^{1}=(X,(Y,Z))\), we can obtain \(|X|=1\). \(\forall a_{i}\in Y,g(a_{i})=X\) and \(\forall a_{h}\in Z,\overline{g}(a_{h})=X\). Thus, \((X,a_{i})\) and \((X,a_{h})\) are single-attribute concepts based on Definition 2. Furthermore, according to Definition 4, we can obtain \(\lambda(X,a_{i})=c_{OEO}^{1}\), which is a basic OEO-concept. Thus, any 1-granularity OEO-concept can be obtained from basic OEO-concepts, i.e., \(C_{OEO}^{1}=B_{OEO}^{1}\). Now, we can obtain \(C_{OEO}^{0}\) by Proposition 1, i.e., \((\varnothing,(\varnothing^{\square},\varnothing^{\overline{\lozenge}}))\), and can obtain \(C_{OEO}^{1}\) based on Proposition 3. In order to obtain the remaining OEO-concepts, i.e., \(C_{OEO}^{2},\ldots,C_{OEO}^{|U|}\), it is necessary to perform merge operations on the generated OEO-concepts. Definition 6 (Concept merging): For a formal context \((U,V,R)\), if \(c_{1}=(X_{1},(Y_{1},Z_{1}))\) and \(c_{2}=(X_{2},(Y_{2},Z_{2}))\) are OEO-concepts, then the merge operation of \(c_{1}\) and \(c_{2}\) is defined as follows:  
\[=\{y\in V|Ry\subseteq X\} \tag{5}\]0  
where \(X_{1},X_{2}\subseteq U,Y_{1},Y_{2}\subseteq V\), and \(Z_{1},Z_{2}\subseteq V\),For any two OEO-concepts \(c_{1}\) and \(c_{2}\), a new pair can be obtained by using \(M(c_{1},c_{2})\).