Theory of Complete Lattices
A lattice is a partially ordered set in which any two elements have a greatest lower and least upper bound. A lattice is complete if and only if each set of lattice elements has a greatest lower and least upper bound. If <S,C> is a C-closure space, then <C,⊂> is a complete lattice in which ∩W is the greatest lower and Cl(UW) is the least upper bound of each subset W of C. It is convenient to let ‘∪ W’ and ‘(A∪B)’ designate Cl(∪ W) and Cl(A∪B), respectively. (∪ is the operation of closed union.) In this chapter, we use wellknown properties of lattices to derive some theorems about closure spaces.
KeywordsComplete Lattice Finite Subset Closure Space Finite Cover Proof Apply
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