# A Fast Algorithm for Three-Way Object-Oriented Concept Acquisition
## 2 Preliminaries
\(\square\)  
Based on Propositions 4 and 5, it can be concluded that \(Ext(C_{OEO}^{k})=Ext(C_{Candi}^{k}\cup B_{OEO}^{k})\). Therefore, we can group the candidate k-granularity OEO-concepts \(C_{Candi}^{k}\) and the k-granularity basic OEO-concepts \(B_{OEO}^{k}\) with the same extent. And, then, the corresponding intents of these extents are confirmed as follows:  
**Definition 8** (Equivalence class of extent). : _Given a formal context (U, V, R), if \(c\in C_{Candi}^{k}\cup B_{OEO}^{k}\) and the set \([c]_{Ext}\) meets the condition_  
\[=\{y\in V|Ry\subseteq X\} \tag{5}\]3 \[=\{y\in V|Ry\subseteq X\} \tag{5}\]4  
_then the set \([c]_{Ext}\) is called the equivalence class of the extent of \(c\)._  
The equivalence class of the extent of \(c\) is used to obtain the intents of a new OEO-concept through the union operations of the intents of all concepts in the equivalence class. The process is defined as the function \(\varphi\). **Definition 9** (Function \(\varphi\)). : _Given a formal context (U, V, R), the \(\varphi\) function is a mapping \(C_{Candi}^{k}\) to \(C_{OEO}^{k}\). The definition is as follows:_  
\[=\{y\in V|Ry\subseteq X\} \tag{5}\]5 \[=\{y\in V|Ry\subseteq X\} \tag{5}\]6  
_where \(c=(X,(Y,Z))\in C_{Candi}^{k}\)._  
As \(B_{OEO}^{k}\subseteq B_{OEO}\) has been already generated by \(\lambda\) (Definition 3), \(\varphi\) needs to only map each concept \(c\in C_{Candi}^{k}\) to an OEO-concept in \(C_{OEO}^{k}\). Now, Proposition 6 is provided to prove that \(\varphi(c)\) is an OEO-concept. **Proposition 6**. : _For a formal context (U, V, R), if \(c_{1}=(X_{1},Y_{1},Z_{1})\in C_{Candi}^{k}\), then \(\varphi(c_{1})=(X_{1},(Y_{max},Z_{max}))\) is an OEO-concept._  
Proof. : If we want to prove that \(\varphi(c_{1})\) is an OEO-concept, it needs to satisfy \(Y_{\max}=X_{1}^{\square},Z_{\max}=X_{1}^{\square}\) and \(X_{1}=Y_{\max}^{\lozenge}\cup Z_{\max}^{\lozenge}\) based on the definition of OEO-concepts. 1. First, we prove that \(Y_{\max}=X_{1}^{\square}\). 1. \(\forall b\in Y_{\max}\), if \(g(b)\subseteq X_{1}\); then, by the definition of the necessary operator, \(b\in X_{1}^{\square}\). Given the arbitrariness of \(b\), it follows that \(Y_{\max}\subseteq X_{1}^{\square}\). 2. Assume that \(\exists b\in X_{1}^{\square}\) but \(b\
otin Y_{\max}\), i.e., \(b\in X_{1}^{\square}\setminus Y_{\max}\); then, let \(c_{2}=(g(b)^{\square\lozenge}\cup g(b)^{\overline{\lozenge}},(g(b)^{\square}, g(b)^{\overline{\lozenge}}))\). \(c_{2}\) is then an OEO-concept based on Proposition 1. Because \(g(b)\subseteq X_{1},g(b)^{\square\lozenge}\cup g(b)^{\overline{\lozenge}} \subseteq X_{1}\). If \(g(b)^{\square\lozenge}\cup g(b)^{\overline{\lozenge}}=X_{1}\), then we get \(c_{2}\in[c_{1}]_{\text{Ext}}\). This implies that \(b\in Y_{\max}\). Therefore, given the arbitrariness of \(b\), we conclude that \(X_{1}^{\square}\subseteq Y_{\max}\). In the case \(g(b)^{\square\lozenge}\cup g(b)^{\overline{\lozenge}}\subset X_{1}\), let \(c_{3}=(X_{1},(X_{1}^{\square},X_{1}^{\overline{\lozenge}}))\). Because \(c_{1}\in C_{Candi}^{k}\) and \(X_{1}\) is the extent of an OEO-concept, \(c_{3}\) is an OEO-concept and \(c_{2}<c_{3}\). According to Proposition 4, there must exist an OEO-concept \(c_{4}\leq c_{3}\) such that \(M(c_{2},c_{4})\in[c_{1}]_{Ext}\). This means that \(b\in Y_{Max}\). Given the arbitrariness of \(b\), we get \(X_{1}^{\square}\subseteq Y_{\max}\) as well. 3. According to above 1. and 2., we get \(Y_{\max}=X_{1}^{\square}\). 2. Second, it can similarly be proved that \(Z_{\max}=X_{1}^{\overline{\gamma}}\). 3. Third, \(Y_{\max}^{\diamondsuit}\cup Z_{\max}^{\overline{\gamma}}=X_{1}^{\squaresuit}\cup X _{1}^{\overline{\squaresuit}\overline{\zeta}}\subseteq X_{1}\). Moreover, \((X_{1},(X_{1}^{\squaresuit},X_{1}^{\overline{\squaresuit}}))\) is an OEO-concept. Therefore, \(X_{1}=X_{1}^{\squaresuit}\cup X_{1}^{\overline{\squaresuit}\overline{\zeta}}\). Thus, \(Y_{\max}^{\diamondsuit}\cup Z_{\max}^{\overline{\zeta}}=X_{1}\). From the above proofs, the proposition has been proven.