Notes on lattices
Koichi Atsumi · Proceedings of the Japan Academy Series A Mathematical Sciences · 1958
Let L be a lattice with an inclusion relation , meet a,-,b and join ab.L.M. Blumenthal and D. O. Ellis [2 showed that the follwing three relations (G), (G*) and (G**) are equivalent in modular lattices, and that they are also equivalent to metric betweeness for normed lattices.G (ac)(bc)-c-(ac).(bc) G* ) (ac)(bc)-c-c(ab) (G**) (ac)(bc)-c-c(ab) Recently, Y. Matsushima 3 introduced for any lattice L three kinds of sets in L as follows: *) J(a, b) a, b).He gave among others a characterization of distributive lattices by using B(a, b), and a characterization of modular lattices by using B(a, b) and B*(a, b) in 3, 4.