l-群的凸l-子群格的极小条件

Minimal condition of the lattice of all convex l-subgroups of a lattice-ordered group

设 G是 l-群 ,C(G)是 G的凸 l-子群格 .称 C(G)满足极小条件 ,如果 C(G)中每个元均包含一个原子元 .本文将 C(G)的链条件 (见文 [1 ])推广到极小条件 ,主要结果是 :C(G)满足极小条件且 C(G)中每个原子元均是 G的基数直和项当且仅当∑λ∈ΛRλ G λ∈ΛRλ(其中每个 Rλ≌实数加群 R的某个子群 ) .

let G be an l group, C(G) denotes the lattice of all convex l subgroups of G.C(G) is said to satisfy minimal condition if each element of C(G) contains an atom.The main result of this paper is the following: C(G) satisfies minimal condition and each atom of C(G) is a cardinal summand of G if and only if ∑λ∈ΛR λGλ∈ΛR λ(R λ≌ some subgroup of R,λ∈Λ) .