Vol. 27, No. 3, 1968

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On (J,M,m)-extensions of order sums of distributive lattices

Raymond Balbes

Vol. 27 (1968), No. 3, 441–451

In the first section of this paper a characterization of the order sum of a family {Lα}αS of distributive lattices is given which is analogous to the characterization of a free distributive lattice as one generated by an independent set. We then consider the collection Q of order sums obtained by taking different partial orderings on S. A natural partial ordering is defined on Q and its maximal and minimal elements are characterized.

Let J and M be collections of nonempty subsets of a distributive lattice L, and m a cardinal. We define a (J,M,m)-extension (ψ,E) of L, where E is a m-complete distributive lattice and ψ : L E is a (J,M)-monomorphism. In the last section we define a m-order sum of a family of distributive lattices {Lα}αS. The main result here is that the m-order sum exists if the order sum L of {Lα}αS has a (J,M,m)-extension, where J and M are certain collections of subsets of L. These results are analogous to R. Sikorski’s work in Boolean algebras (e.g., [6]).

Mathematical Subject Classification
Primary: 06.50
Received: 10 July 1967
Published: 1 December 1968
Raymond Balbes