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Strongly N-normal lattices. (English) Zbl 0547.06004
A distributive lattice L is said to be strongly (relatively)-n-normal if every prime filter is contained in at most n pairwise incomparable (prime) filters. The strongly-n-normal lattices are introduced and exhibited in the paper under review. The relatively-n-normal lattices were studied by W. H. Cornish [Proc. Am. Math. Soc. 45, 48-54 (1974; Zbl 0294.06008)].
Generalizing the concept of a relative pseudocomplement the author considers semi-Heyting algebras (L;\(\vee,\wedge,.,0,1)\) defined as follows: (L;\(\vee,\wedge,0,1)\) is a bounded distributive lattice satisfying the identities \(a\wedge (a.b)=a\wedge b\) and \(a.a=1\). The main results: (1) A semi-Heyting algebra satisfying the identity \((I_ n) \bigvee (x_ i.x_ j:\) \(i\neq j\), \(0\leq i,j\leq n)=1\) is strongly-n- normal. (2) A Heyting algebra is strongly-n-normal if and only if it satisfies \((I_ n)\). (3) A bounded strongly-1-normal lattice L is relatively-1-normal if and only if L is a reduct of some semi-Heyting algebra.
Reviewer: T.Katriňák

06D05 Structure and representation theory of distributive lattices
06D15 Pseudocomplemented lattices
06D20 Heyting algebras (lattice-theoretic aspects)