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# Accession Number:

## AD0608396

# Title:

## NEARLY MODULAR ORTHOCOMPLEMENTED LATTICES,

# Descriptive Note:

# Corporate Author:

## BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH

# Report Date:

## 1964-07-01

# Pagination or Media Count:

##
32.0

# Abstract:

## Let L be a complete, weakly modular, orthocomplemented lattice. A modular element in L is an element a such that O, a is a modular lattice and x,a is a modular pair for all x. The lattice L is nearly modular if it is semi-modular and every element in L is the join of modular elements. The following is a key result. Theorem. If L is nearly modular a is a modular element, and b is perspective to a, then b is modular. Using this theorem and some continuity theorems involving modular elements the following theorems are proved. Theorem. L is nearly modular if an only if L is a locally finite dimension lattice. Theorem. L is semi-modular and contains a minimal element a with central cover 1 if and only if L is a type I dimension lattice. A simple construction, the horizontal sum, is defined and used to show that neither of the conditions in the definition of nearly modular can be dropped. Author

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#