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 $\$\; (X\; \backslash to\; Y)\; \backslash blacktriangleright\; (Z\; \backslash to\; V)\; =\; X\; \backslash cup\; (Z\; \backslash setminus\; Y)\; \backslash to\; (Y\; \backslash cup\; V)\; \$$ is determined in the full set $\{X\; \to \; Y\; |\; X,Y$Í R} of F-dependences over a certain scheme $R$. Let $$F be an F-dependence, which follows from a set $F$ of F-dependences. We prove that $\$\; \backslash Phi\; =\; \backslash Phi\; \_1\; \backslash blacktriangleright\; \backslash Phi\_2\; \backslash blacktriangleright\; \backslash ldots\; \backslash blacktriangleright\; \backslash Phi\_k$
\blacktriangleright W \cdot \mathbf{F2} \cdot \mathbf{B3} $ for some $$F₁, F₂, ¼ , F_k Î F and $W$Í R, where $\$\; \backslash Phi\_k\; \backslash blacktriangleright\; W\; =\; \backslash Phi\_k\; \backslash bigtriangleright\; (W\; \backslash to\; W)\; \$$. The unary operations $\$\; \backslash cdot\; \backslash mathbf\; \{F2\}\; \$$ and $\$\; \backslash cdot\; \backslash mathbf\; \{B3\}\; \$$ correspond to axioms of derivation $\$\; \backslash mathbf\; \{F2\}\; \$$ (completion) and $\$\; \backslash mathbf\; \{B3\}\; \$$ (projectivity) pro tanto.

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Last modified: July 8, 2002