On minimal absorbent sets for some types of tolerance relations
Let \(\tau\) be a tolerance (i.e., reflexive and symmetric) relation on a set \(X\). Define \(\tau(x)= \{y\in X: x\tau y\}\), for every \(x\in X\). Then \(\tau\) is called non-Archimedean if for each \(x\in X\), \(\tau(x)\) intersects finitely many sets \(\tau(y)\), \(y\in X\). A subset \(Y\subseteq X\) is said to be \(\tau\)-absorbent if \(Y\cap \tau(x)\neq \emptyset\) for all \(x\in X\). A \(\tau\)-absorbent extension of a set \(Y\subseteq X\) is a \(\tau\)-absorbent set \(Y'\subseteq X\) such that \(Y\subseteq Y'\). Main result: if \(\tau\) is a non-Archimedean tolerance relation, then every subset \(Y\subseteq X\) has a minimal \(\tau\)-absorbent extension. If \(\mathcal T\) is a family of tolerance relations on \(X\), a \({\mathcal T}\)-absorbent extension of a set \(Y\subseteq X\) is a set \(Y'\) which is a \(\tau\)-absorbent extension of \(Y\) for each \(\tau\in {\mathcal T}\). Supposing the set \(X\) is finite, the author gives an algorithm which provides a minimal \({\mathcal T}\)-absorbent extension of smallest cardinality.
scientific article; zbMATH DE number 832190