# An Intelligent System to Ensure Interoperability for the Dairy Farm Business Model
## 4 Technical Instrumentation  
FCA (([PERSON] et al., 2017)) yields a hierarchy of conceptual abstractions, referred to as a lattice of concepts, that are used to analyse objects. FCA encodes binary data into a formal context. A formal context \(K\) is a 3-tuple \((G,\ M,\ I)\), where \(G\) is a set of objects, \(M\) is a set of attributes, and \(I\subseteq G\times M\) is a binary relation that specifies which objects have which attributes. For instance, Table 2 shows a formal context \(K=(G,\ M,\ I)\). The rows are the objects \(G=\{DF1,\ DF2,\ DF3,\ DF4,\ DF5,\ DF6\}\); the columns are the attributes \(M=\{NF,\ GS,\ PM,\ UPM,\ NtIA,\ NtF\}\), where NF, GS, PM, UPM, NtIA, and NtF stand for _natural_forage_, _growth_stimulant_, _pasteurised_milk_, _unpasteurised_milk_, _nextTo_industrialAreas_, and _nextTo_Forests_, respectively; and a cross in a cell, identified by a pair, e.g., \((DF1,\ NF)\in I\), signifies that object \(DF1\in G\) has the attribute \(NF\in M\). Two derivation operators, both denoted \({}^{\prime}\), are defined for \(X\subseteq G\) and \(Y\subseteq G\) as follows:  
\[{}^{\prime}:P(G)\ \to P(M),\ X^{\prime}=\{m\in M|\forall g\in X,(g,m)\in I\},\]  
\[{}^{\prime}:P(M)\ \to P(G),\ Y^{\prime}=\big{\{}g\in G|\forall m\in Y,(g,m)\in I \big{\}}.\]  
The \({}^{\prime}\) operators define a Galois connection between \(P(G)\) and \(P(M)\). The set \(X^{\prime}\) is the set of all attributes in \(M\) shared by the objects in \(X\). Similarly, \(Y^{\prime}\) is the set of all objects in \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(K\) & **NF** & **GS** & **PM** & **UPM** & **NtIA** & **NtF** \\ \hline DF1 & \(\times\) & & \(\times\) & & \(\times\) & \(\times\) \\ \hline DF2 & \(\times\) & \(\times\) & \(\times\) & & & \(\times\) \\ \hline DF3 & & \(\times\) & & \(\times\) & \(\times\) & \\ \hline DF4 & \(\times\) & & \(\times\) & \(\times\) & & \(\times\) \\ \hline DF5 & & \(\times\) & & \(\times\) & \(\times\) & \\ \hline DF6 & \(\times\) & & \(\times\) & \(\times\) & \(\times\) & \(\times\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The formal context \(K\) built from the raw data given in Table 1. Each column name is abbreviated (i.e., NF, GS, PM, UPM, NtIA, and NtF stand for _natural_forage_, _growth_stimulant_, _pasteurised_milk_, _unpasteurised_milk_, _nextTo_industrialAreas_, and _nextTo_Forests_, respectively). that have the attributes in \(Y\). The composition operators \({}^{\prime\prime}\) are closure operators since they are idempotent, monotonous, and extensive [17]. When \(X=X^{\prime\prime}\) and \(Y=Y^{\prime\prime}\), then \(X\) and \(Y\) are closed sets. For example, relying on \(K\) given in Table 2, if \(X_{1}=\{DF3,\,DF5\}\), then \(X_{1}{}^{\prime}=\{GS,\,UPM,\,NIIA\}\), and \(X_{1}{}^{\prime\prime}=\{DF3,\,DF5\}\). Since \(X_{1}=X_{1}{}^{\prime\prime}\), \(X_{1}\) is a closed set. In contrast, \(X_{2}=\{DF1,\,DF2\}\) is not closed because \(X_{2}{}^{\prime\prime}=\{DF1,\,DF2,\,DF4,\,DF6\}\
eq X_{2}\). A formal concept \(c\) derived from the context \(K=(G,\,M,\,I)\) is a pair \((X,Y)\), where \(X\subseteq G\) and \(Y\subseteq M\), s.t. \(X^{\prime}=Y\) and \(Y^{\prime}=X\). The set of objects \(X\) is called the extent of \(c\), while the set of attributes \(Y\) is called the intent of \(c\). Let \(c_{1}=(X_{1},Y_{1})\) and \(c_{2}=(X_{2},Y_{2})\) be two concepts from \(C_{K}\). The generalisation order \(\preceq_{K}\) is defined by \(c_{1}\preceq_{K}\)\(c_{2}\) iff \(X_{1}\subseteq\ X_{2}(\Leftrightarrow Y_{2}\subseteq\ Y_{1})\). In this case, \(c_{1}\) is called a sub-concept of \(c_{2}\), and \(c_{2}\) is a super-concept of \(c_{1}\). If \(c_{1}\preccurlyeq_{K}\)\(c_{2}\) and there is no \(c_{3}\) s.t. \(c_{1}\preccurlyeq_{K}\)\(c_{3}\preccurlyeq_{K}\)\(c_{2}\), then \(c_{1}\) is a lower neighbour of \(c_{2}\) and \(c_{2}\) is an upper neighbour of \(c_{1}\)[38]. The concept lattice \(L_{K}\) of \(K=(G,\,M,\,I)\) is the partial order \((C_{K},\preceq_{K})\).