Vol. 37, No. 3, 1971

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On point-free parallelism and Wilcox lattices

Shûichirô Maeda

Vol. 37 (1971), No. 3, 725–747

A Wilcox lattice L is constructed from a complemented modular lattice Λ, by deleting nonzero elements of some ideal of Λ and by introducing in the remains L the same order as Λ. The lattice Λ is called the modular extension of L. Using the theory of parallelism in atomistic lattices, it was proved that any affine matroid lattice is an atomistic Wilcox lattice, that is, an existence theorem of the modular extension in the atomistic case. The main purpose of this paper is to extend this result to the general case, by the use of arguments on point-free parallelism.

Mathematical Subject Classification
Primary: 06A30
Received: 25 May 1970
Revised: 3 March 1971
Published: 1 June 1971
Shûichirô Maeda