Order sums of distributive lattices
Raymond Balbes, Alfred Horn · Pacific Journal of Mathematics · 1967
HORN trivially ordered, in the sense that a <^ β only when a = β, then the order sum reduces by very definition to the free product [2]. DEFINITION 1.3. Let {L a \aeP}be a family of pairwise disjoint lattices indexed by a chain P. The ordinal sum L of the family is the set \JaβpL a with the following partial order: if xe L a and y e L β , then x ^ y if and only if either a < β, or a = β and x ^ y in the original order of L a .If the lattices L a are not pairwise disjoint, then the ordinal sum of the L a is defined to be the ordinal sum of pairwise disjoint lattices L a such that L a ~La .If P is the chain {0,1} with 0 < 1, then the ordinal sum of {L o , L x } is denoted by L o 0 L x .THEOREM 1.4.If P is a chain, then the ordinal sum L of {L a I a 6 P} is the order sum Σ aep L a .