The k-Normal Completion of Function Lattices

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Abstract
A subset G of a non-empty partially ordered set C is called normal if it coincides with the set of all upper bounds of the set of lower bounds of G. This is equivalent to stipulating that G be the set of all upper bounds of some subset of C called a set of generators for G. When ordered by inclusion, the family of all normal subsets of C forms a complete lattice with maximum C and minimum empty or singleton. The meet operation is simply point set intersection; whence, the meet of a family Gi of normal subsets is the set of upper bounds of ∪ Fi where Fi generates Gi for each i. A normal subset is called proper if it is neither void nor C, and the proper normal subsets. of C form a boundedly complete lattice.

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