Semi-modularity in Relatively Atomic, Upper Continuous Lattices
Usa Sasaki · Hiroshima Mathematical Journal · 1952
K. Menger [1] 1 > has introduced the following conditions to characterize the lattice of all subspaces of a finite dimensional affine space :( 71 ") If p is a point, then either p~a or avp covers a, for any element a. (;j) If h is covered by 1, then either a<h or a, covers ar.h, for any element a with a r.h=J=O.L. R. Wilcox [1] has shown that the lattice of all subspaces of an affine space is semi-modular in the sense that (A) (b, c)M, br.c=O imply (c, b)M,' and (B) br.c=J=O implies (b, c)M.The semi-modularity in this sense was used in my previous paper [1] to characterize the lattice of all subspaces of an affine space of arbitrary dimensions, noting that it might •be replaced by the following conditions :