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Normal lattices

Published online by Cambridge University Press:  09 April 2009

William H. Cornish
William H. Cornish
Affiliation:
The Flinders University of South Australia, Bedford Park, South Australia 5042
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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].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 2 , September 1972 , pp. 200 - 215
Copyright
Copyright © Australian Mathematical Society 1972

References

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