Normal lattices

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Abstract
If L is a distributive lattice with 0 then it is shown that each prime ideal contains a unique minimal prime ideal if and only if, for any x and y in L, x ∧ y = 0 implies (x]*) ∨ (y]* L). A distributive lattice with 0 is called normal if it satisfies the conditions of this result. This terminology is appropriate for the following reasons. Firstly the lattice of closed subsets of a T1-space is normal if and only if the space is normal. Secondly lattices satisfying the above annihilator condition are sometimes called normal by those mathematicians interested in (Wallman-) compactications, for example see [2].

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