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In the first section of this
paper a characterization of the order sum of a family {Lα}α∈S 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}α∈S. The main result
here is that the m-order sum exists if the order sum L of {Lα}α∈S 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]).
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