Vol. 21, No. 3, 1967

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ISSN: 0030-8730
Order sums of distributive lattices

Raymond Balbes and Alfred Horn

Vol. 21 (1967), No. 3, 421–435

In this paper, the authors define the orde r sum of a family of distributive lattices which is indexed by a partially ordered set P. The order sum reduces to the free product when P is trivially ordered, and to the ordinal sum when P is simply ordered.

It is proved that the order sum of conditionally implicative lattices is conditionally implicative, and that every projective distributive lattice is conditionally implicative. The second half of the paper investigates conditions under which the order sum of projective lattices is projective. It is shown that if {Lα|α P} is a family of distributive lattices having largest and smallest elements, then the order sum of the family is projective if and only if each Lα is projective, and P is such that the order sum of the family {Mα|α P} of one-element lattices Mα is projective.

Mathematical Subject Classification
Primary: 06.50
Received: 22 August 1966
Published: 1 June 1967
Raymond Balbes
Alfred Horn