FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 1, PAGES 195-219

Algebraic interpretation of derivation axioms completeness

L. A. Pomortsev

Abstract

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The operation $ (X \to Y) \blacktriangleright (Z \to V) = X \cup (Z \setminus Y) \to (Y \cup V) $ is determined in the full set {X → Y | X,Y Í R} of F-dependences over a certain scheme R. Let F be an F-dependence, which follows from a set F of F-dependences. We prove that $ \Phi = \Phi _1 \blacktriangleright \Phi_2 \blacktriangleright \ldots \blacktriangleright \Phi_k
\blacktriangleright W \cdot \mathbf{F2} \cdot \mathbf{B3} $
for some F1, F2, ¼ , Fk Î F and W Í R, where $ \Phi_k \blacktriangleright W = \Phi_k \bigtriangleright (W \to W) $. The unary operations $ \cdot \mathbf {F2} $ and $ \cdot \mathbf {B3} $ correspond to axioms of derivation $ \mathbf {F2} $ (completion) and $ \mathbf {B3} $ (projectivity) pro tanto.

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Last modified: July 8, 2002