C-polars in lattice-ordered groups

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1972

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Throughout this abstract G will denote a lattice-ordered group ("[cursive l]-group") and C will denote a convex [cursive l]-subgroup of G. This dissertation is primarily interested in studying the role that C-polars play in S,[cursive l]-groups. In Chapter I we introduce some definitions, notation and terminology that are frequently used in the literature and throughout this dissertation. In Chapter II we consider a representable [cursive l]-group G=(G[lowered t]|t[epsilon]T) and define the functions I[lowered C], Z[lowered C]. and the concept of a projection maximal subgroup ("p.m. subgroup") of G, Some of the properties of the functions [lowered C] and Z[lowered C] are discussed and examples are given to indicate that in most cases the strongest possible result has been stated. Moreover, we prove that every closed subgroup of the representable [cursive l]-group G=(G[lowered t]|t[epsilon]T) is a p.m. subgroup of G. If G=(G[lowered t]|t[epsilon]T) is a representable [cursive l]-group we are able to define a topology on the indexing set T by using the function Z[lowered C] defined in Chapter II. In Chapter III we investigate the resulting topological space which we denote by (T,T(C)). By using the device of a p.m. subgroup of G we are able to prove that the Boolean algebra of C-polars of G is nearly anti-isomorphic to the Boolean algebra of regular closed sets of the topological space (T,T(C)) and thus extend a similar result obtained by F. Sik concerning the Boolean algebra of polars of G. The main topic of discussion in Chapter IV is the set of all maximal ideals in the lattice of principal C-polars of G. We denote this set by M([Pi]raised ', lowered C). A topology is defined on M([Pi]raised ', lowered C) and a number of conditions are given that are equivalent to the resulting topological space being compact. In Chapter V we extend the definition of a prime subgroup of G to that of a C-prime subgroup of G and give a characterization of a minimal C-prime subgroup in terms of the principal C-polars of G. We then investigate the relationship between the C-polars of G and the polars of G/C where C is an [cursive l]-ideal of G and show that the Boolean algebra of C-polars of G is Boolean isomorphic to the Boolean algebra of polars of G/C. The remainder of the chapter is devoted to examining collections of C-prime 2-ideals of G whose intersection is C and investigating some of the properties that the resulting representations of G/C that they induce may possess. Chapter VI deals primarily with an investigation of minimal C-prime subgroups of G. We show that a minimal C-prime subgroup of G can be characterized as the union of a maximal ideal in the lattice of principal C-polars of G and hence extend a result due to F. Sik. Various other results concerning minimal C-prime subgroups are presented and the final part of the chapter is devoted to the definition and investigation of the idea of a strong C-subgroup of G. In Chapter VII the definition of a unit in an [cursive l]-group is extended by the definition of a C-unit in an £-group and we give twelve conditions that are equivalent to the condition that states that the set of elements of G that are not C-units of G form a strong C-subgroup of G, This result extends a similar result obtained by F. Sik for the set of units of an [cursive l]-group, Finally, in Chapter VIII we investigate the set of standard maximal ideals of the Boolean algebra of C-polars of G. We denote this set by M[lowered S](Klowered C). We inspect a number of different topologies on M[lowered S](Klowered C) and study some of their properties and their relationships to each other. This investigation extends the results obtained by F. Fiala in his study of the set of standard maximal ideals in the Boolean algebra of polars of G.

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