The generalized orthocompletion and strongly projectable hull of a lattice ordered group

Richard N. Ball
1982 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
The central result is the existence and uniqueness for an arbitrary /-group G of two hulls, G and G u t 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 G oe . Two constructions of G and G oe are-given: an /-permutation construction leads to descriptive structural information, and a
more » ... tion, 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 existence 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-representable 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 multiplicative 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, complementation, 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.
doi:10.4153/cjm-1982-042-5 fatcat:r4li4hukrnh7zf7dhmy3vbnsau