-normal lattices

Author:
William H. Cornish

Journal:
Proc. Amer. Math. Soc. **45** (1974), 48-54

MSC:
Primary 06A35

MathSciNet review:
0340133

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Abstract | References | Similar Articles | Additional Information

Abstract: An -normal lattice is a distributive lattice with 0 such that each prime ideal contains at most minimal prime ideals. A relatively -normal lattice is a distributive lattice such that each bounded closed interval is an -normal lattice.

The main results of this paper are:

(1) a distributive lattice with 0 is -normal if and only if for any such that for any ,

(2) a distributive lattice is relatively -normal if and only if for any incomparable prime ideals .

**[1]**William H. Cornish,*Normal lattices*, J. Austral. Math. Soc.**14**(1972), 200–215. MR**0313148****[2]**George Grätzer,*Lattice theory. First concepts and distributive lattices*, W. H. Freeman and Co., San Francisco, Calif., 1971. MR**0321817****[3]**Neil Hindman,*Minimal 𝔫-prime ideal spaces*, Math. Ann.**199**(1972), 97–114. MR**0321926****[4]**K. B. Lee,*Equational classes of distributive pseudo-complemented lattices*, Canad. J. Math.**22**(1970), 881–891. MR**0265240**

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DOI:
https://doi.org/10.1090/S0002-9939-1974-0340133-6

Keywords:
Distributive lattice,
minimal prime,
-prime ideal,
-normal lattice

Article copyright:
© Copyright 1974
American Mathematical Society