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TODO: \refinerho missing... Properties of Refinement Operators

An \(La\) downward refinement operator \(rho\) is called
  • finite iff \(\rho(C)\) is finite for any concept \(\in \mathcal{C}(\mathcal{L})\)
  • redundant iff there exist two different \(\rho\) refinement chains from a concept C to a concept D.
  • proper iff for \( C,D\in \mathcal{C}(\mathcal{L}), C refinerho D \) implies \(C \not\equiv_T D \)
  • ideal iff it is finite, complete, and proper.
  • complete iff for \( C,D\in \mathcal{C}(La) with D \sqsubseteq_ T C there is a concept E with E \equiv_ T D and a refinement chain C refinerho \cdots refinerho E\)
  • weakly complete iff for any concept \(C\) with \(C \sqsubseteq_T \top\) we can reach a concept \(E\) with \(E \equiv_T C\) from \(\top\) by \(\rho\).
  • ideal = complete + proper + finite

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