Direct Sums and Normal Ideals of Lattices
Fumitomo Maeda · Hiroshima Mathematical Journal · 1949
Let 1, be a lattice with 0 and 1.We say that/, is a direct sum of L(n,z 1 ) (i=l, .. ,11),::Z if and only if every element a of L is expressible uniquely in the formAnd in this case z 1 (i=l, .. ,n) are elements of the center of /,.In this paper, we shall extend this notion to the case where th~ existence of 1 is not assumed.And next, we shall define normal ideals in a modular lattice with some properties, and show that the set of all normal ideals acts as the center.And we shall apply this property to the general continuous geometry, that is, the continuous geometry without the assumption of the existence of 1, and show that the theories of dimension function and subdirect product representation hold also in the general continuous geometry as in the continuous geometry.§ 1. Direct Sum of Lattices.In this section, L is a lattice with the zero clement 0.DEFJNJTJ<l.N' 1-1.By avu, it is meant that anl,=0, and (a,1',b)V for every element :1:!L, that is, (aUJ:)nb=(anb)u(,rnb)=.rnli.If 8 is any subset of L, denote by 8 7 the