(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 $\left(X \to Y\right) \blacktriangleright \left(Z \to V\right) = X \cup \left(Z \setminus Y\right) \to \left(Y \cup V\right)$ is determined in the full set $\left\{X \to Y | X,Y$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 $\Phi_k \blacktriangleright W = \Phi_k \bigtriangleright \left(W \to W\right)$. The unary operations $\cdot \mathbf \left\{F2\right\}$ and $\cdot \mathbf \left\{B3\right\}$ correspond to axioms of derivation $\mathbf \left\{F2\right\}$ (completion) and $\mathbf \left\{B3\right\}$ (projectivity) pro tanto.

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