The Generalized Orthocompletion and Strongly Projectable Hull of a Lattice Ordered Group

The central result is the existence and uniqueness for an arbitrary /-group G of two hulls, G and Gut which in the representable case coincide with the orthocompletion and strongly protectable hull of G. This is done by introducing two notions of extension, written ^ and ^ w, and proving that each G has a maximal <! extension G and a maximal ^ w extension Gœ. Two constructions of G and Gœ are-given: an /-permutation construc­ tion leads to descriptive structural information, and a construction by ''consistent maps" leads to a strong universal mapping property. The notion of a strongly projectable hull has a lengthy history. The concept of an orthocompletion, together with the first proof of its exist­ ence and uniqueness, is due to Bernau [4]. Conrad summarized and extended all these results in an important paper [10]. The chief novelty of the present work is that the ideas apply to non-represent able as well as to representable /-groups. When specialized to the representable case, the construction of Section 2 is related to the nice constructions of Bleier in [6] and [7]. The notation, which is multiplicativ e even for the representable case, is standard. G is understood to be an /-group whose complete Boolean algebra of polars will be designated & G or simply 8P. The symbols V, A, -L, 0^, and 1& refer respectively to supremum, infimum, complemen­ tation, least element and greatest element in £P. The symbols V and A also refer to supremum and infimum of elements of G\ the reader must be prepared to distinguish the two meanings from context. 1. Extensions. The crucial concept is the following. For /-groups G and H define G ^ H to mean that G is an /-subgroup of H in which the polars of G and H are in one-to-one correspondence by intersection, and such that V {(hr1)^ g E G\ = l& for all h £ H. Similarly, for K a fixed infinite cardinal number, define G ^ K H to mean that G is an /-subgroup of H in which the polars of G and H are in one-toone correspondence by intersection, and such that for ev^ry h £ H there