A Priestley-type Method for Generating Free l-Groups
A Priestley-type topological representation is developed for the class of partially ordered groups which are p. o. subgroups of lattice-ordered groups. The appropriated analog of the spectrum of prime lattice filters is the class of increasing subsets P satisfying ab ∈ P and cd ∈ P imply ad ∈ P or cb ∈ P. In developing this representation, we give new embedding theorems for these groups. In particular, we give a necessary and sufficient condition for a p. o. group in this class to be embedded in an l-group of sets in such a way that the embedding preserves meets and joins that already exist in the group. Our construction gives also an alternative and very natural way to obtain the free l-group generated by a p. o. group in this class.
KeywordsDistributive Lattice Positive Cone Opposite Inclusion Complete Family Priestley Space
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