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sav08:axioms_for_equality [2008/04/02 21:40]
vkuncak
sav08:axioms_for_equality [2009/05/05 23:16]
vkuncak
Line 17: Line 17:
  
 **Definition:​** if an interpretation $I = (D,\alpha)$ the axioms $AxEq$ are true, then we call $\alpha(eq)$ (the interpretation of eq) a //​congruence//​ relation for interpretation $I$. **Definition:​** if an interpretation $I = (D,\alpha)$ the axioms $AxEq$ are true, then we call $\alpha(eq)$ (the interpretation of eq) a //​congruence//​ relation for interpretation $I$.
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 +**Side remark:** Functions are relations. ​ However, the condition above for function symbols is weaker than the condition for relation symbols. ​ If $f$ is a function, then the relation $\{(x_1,​\ldots,​x_n,​f(x_1,​\ldots,​x_n)) \mid x_1,​\ldots,​x_n \in D \}$ does not satisfy the congruence condition because it only has one result, namely $f(x_1,​\ldots,​x_n)$,​ and not all the results that are in relation eq with $f(x_1,​\ldots,​x_n)$.
  
 ===== References ===== ===== References =====
   * [[Calculus of Computation Textbook]], Section 3.1   * [[Calculus of Computation Textbook]], Section 3.1