Lateral and Dedekind completions of strongly projectable lattice ordered groups

Constructive completions of ordered sets, groups and fields

The Lattice of Completions of an Ordered Set

For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind-MacNeille completion of P. The lattice K(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of clos...

Convex $L$-lattice subgroups in $L$-ordered groups

In this paper, we have focused to study convex $L$-subgroups of an $L$-ordered group. First, we introduce the concept of a convex $L$-subgroup and a convex $L$-lattice subgroup of an $L$-ordered group and give some examples. Then we find some properties and use them to construct convex $L$-subgroup generated by a subset $S$ of an $L$-ordered group $G$ . Also, we generalize a well known result a...

Amalgamations of Lattice Ordered Groups

The author considers the problem of determining whether certain classes of lattice ordered groups (¿-groups) have the amalgamation property. It is shown that the classes of abelian totally ordered groups (ogroups) and abelian /-groups have the property, but that the class of ¿-groups does not. However, under certain cardinality restrictions one can find an ¿-group which is the "product" of /-gr...

Arithmetic of Dedekind cuts of ordered Abelian groups

We study Dedekind cuts on ordered Abelian groups. We introduce a monoid structure on them, and we characterise, via a suitable representation theorem, the universal part of the theory of such structures. MSC: 06F05; 06F20