The vector lattice cover of certain partially ordered groups

Abstract

In this paper we introduce the notion of Riesz homomorphism on Archimedean directed partially ordered groups and use it to study the vector lattice cover of such groups. There are close relations between the theories of Boolean algebras, distributive lattices, abelian lattice ordered groups and the like on one hand and vector lattices (= Riesz spaces) on the other hand. In the case of Boolean algebras such a relation is made explicit by considering the elements of a Boolean algebra as so called place functions. The set of these place functions generates a vector lattice and this vector lattice can then be used to study the Boolean algebra (see Fremlin's book [8]).In the other cases mentioned, one simply "forgets" some of the structure of a Riesz space if one considers it as a lattice ordered group or a distributive lattice. Conrad in [6] introduced the notion of a vector lattice cover for Archimedean lattice ordered groups and indeed, in categorical language, he therefore studied "the adjoint of the forgetful functor". Shortly thereafter, Bleier in [3] proved a somewhat more general result than Conrad's: For every Archimedean lattice ordered group G there exists an (essentially) unique Archimedean Riesz space E(G) that contains G as a lattice ordered subgroup and which is minimal with that property; furthermore, such a G is automatically large in E{G) (that is, for every 0 < g e E(G) there exists

Riesz* homomorphisms on pre-Riesz spaces consisting of continuous functions

“…Using the structure of Riesz* homomorphisms on E we will investigate another type of homomorphism, namely the complete Riesz homomorphisms, which encompasses the Riesz* homomorphisms. First introduced and studied by Buskes and van Rooij [5], complete Riesz homomorphisms are exactly the operators between pre-Riesz spaces that extend to order continuous Riesz homomorphisms between the completions (see [11]). We aim to characterize the complete Riesz homomorphisms between order dense subspaces of C(X ) and at the same time characterize the order continuous Riesz homomorphisms between Riesz subspaces of C(X ).…”

Disjointness in Partially Ordered Vector Spaces

“…It contains a characterization of the partially ordered vector spaces that can be embedded as order dense subspaces in vector lattices (Riesz spaces). Alternatively, if one only considers Archimedean spaces, one could use the embedding in the Dedekind completion (see Vulikh [10,Section V.3]) or in the smaller enveloping Riesz space (see Buskes-Van Rooij [4]). The embedding maps are required to be linear and bipositive, which implies injectivity.…”

On the Biadjoint of Riesz-like homomorphisms on partially ordered vector spaces

“…For the sake of clarity, we shall now choose another name for what was called a Riesz homomorphism in [3]. Let F be a Riesz space.…”

“…The choice of an analogue for Riesz homomorphism in the setting of partially ordered vector spaces is not obvious at all. Somewhat surprisingly, the choice of homomorphisms that was made in [3] to describe the enveloping Riesz space does not work well for this duality problem as we shall show in the example following Theorem 10. Instead we have opted for a stronger type of homomorphism that is very close to Wickstead's definition in [14].…”

“…Their theorems are in the setting of partially ordered Banach spaces with the Riesz decomposition property and therefore do not generalise Kim's Theorem. Motivated by Arendt's proof of Kim's Theorem in [2] (also see [1,Theorem 7.4]) as well as by the notion of enveloping Riesz space in [3] (also see [5]), we arrive in this paper at general purely order theoretic results for partially ordered vector spaces. Thus, our Theorem 13 generalises Kim's Theorem.…”