A representation for a class of lattice ordered groups
Franklin D. Pedersen · Transactions of the American Mathematical Society · 1969
In a lattice ordered group (/-group), the set of regular subgroups forms a root system in the complete lattice of all convex /-subgroups.Conrad, Harvey, and Holland [4] have shown that an abelian /-group can be represented as an /-group of real-valued functions on any plenary subset of its root system.This paper is concerned with further investigation of the root system leading to a representation for a class of /-groups based on the decomposition of plenary subsets into connected parts (Definition 1).To accomplish this, the concept of T-indecomposable /-groups (Definition 8) is introduced.The major theorem (Theorem 15) then presents necessary and sufficient conditions that an /-group be representable as a full subdirect sum of a cardinal sum of T-indecomposable /-groups of the first kind.1.Here we present some of the basic notation, definitions, and theorems relative to the study of/-groups.The uninitiated reader might also want to refer to either [1 ] or [5] whereas a person knowledgeable in this field might prefer to skip this section, (i) The positive elements of an /-group G are denoted by G+.From [5, p. 70], it can be deduced that for g, he G+, there exist g and h with g t\h=0 such that g = ghh + g and h = gAh + h.(ii) C(A) denotes the convex /-subgroup generated by a nonvoid subset AqG.For convenience, C({g}) = C(g).(iii) r(C) denotes the lattice of all convex /-subgroups of G.By [3, Introduction], r(G) forms a complete distributive sublattice of the lattice of all subgroups of G.Normal convex /-subgroups are called l-ideals.A regular subgroup is an element of F(G) which is maximal with respect to not containing some O^geG.For each nonzero g e G, the completeness of F(G) assures the existence of at least one regular subgroup maximal without containing g [3, Proposition 3.3].Similarly, if x ^ K e T(G), then there exists H e T(G) which is maximal without containing x and such that K^H.Thus, as noted in [3], the regular subgroups of G generate the lattice T(G).(iv) Given M e T(G), M # G, M is a prime subgroup if M satisfies any one of the following equivalent conditions [3, Theorem 3.2]:(a) If A, Be T(G) such that AnBçM, then A^M or B^M.(b) If a, be G+ and a, b $ M, then aAb$M.