ある semimodulated lattice について

その他

D. Sachs [4] has introduced the notion of a modulated lattice which has enough modular elements and given a characterization of partition lattices. In the previous paper [1], we showed that in some non-atomic modulated lattices, modular elements play a role instead of points.
In the present paper, we introduce the notion of a semimodulated lattice (Definition (3. 7)) and give a characterization of some semimodulated Wilcox lattice (Theorem (3. 10)). And moreover we show that some modulated lattice L and M which is the set of all modular elements in L have analogous properties (Theorem (4. 7)). By the above considerations, it seems that in some non-modular semimodulated lattice L, M plays a role in the same way as a Wilcox lattice L≡ ∧-S does in ∧ and that we obtain a generalization of modulated lattices.