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Finite Projective Distributive Lattices

Published online by Cambridge University Press:  20 November 2018

G. Grätzer  and
B. Wolk
G. Grätzer
Affiliation:
University of Manitoba, Winnipeg, Manitoba
B. Wolk
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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The theorem stated below is due to R. Balbes. The present proof is direct; it uses only the following two well-known facts: (i) Let K be a category of algebras, and let free algebras exist in K; then an algebra is projective if and only if it is a retract of a free algebra, (ii) Let F be a free distributive lattice with basis {xi | i ∊ I}; then ∧(xi | i ∊ J0) ≤ ∨(xi | i ∊ J1) implies J0∩J1≠ϕ. Note that (ii) implies (iii): If for J0 ⊆ I, a, b ∊ F, ∧(xi | i ∊ J0)≤a ∨ b, then ∧ (xi | i ∊ J0)≤ a or b.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 1 , March 1970 , pp. 139 - 140
Copyright
Copyright © Canadian Mathematical Society 1970

References

(2) Pacific J. Math. 21 (1967), 405-420.

(3) The map φ is by necessity the same as in R. Balbes, loc. cit.