Vol. 20, No. 3, 1967

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Complete distributivity in lattice-ordered groups

Richard Dowell Byrd

Vol. 20 (1967), No. 3, 423–432
Abstract

Throughout this note let G be a lattice-ordered group (1-group”). G is said to be representable if there exists an 1-isomorphism of G into a cardinal sum of totally ordered groups ( 0-groups”). The main result of §3 establishes five conditions in terms of certain convex 1-subgroups each of which is equivalent to representability. In §4 it is shown that there is an 1-isomorphism of G onto a subdirect product of 1-groups where each 1-group is a transitive l-subgroup of all o-permutations of a totally ordered set and that this 1-isomorphism preserves all joins and meets if and only if G possesses a collection of closed prime subgroups whose intersection contains no nonzero l-ideal. Both theorems lead to results concerning complete distributivity.

Mathematical Subject Classification
Primary: 06.75
Secondary: 20.00
Milestones
Received: 2 May 1966
Published: 1 March 1967
Authors
Richard Dowell Byrd