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--- sav08:axioms_for_equality [2008/04/02 00:47]
vkuncak
+++ sav08:axioms_for_equality [2008/04/02 21:28]
vkuncak moved terminological note to semantics of FOL
@@ Line 7: / Line 7: @@
   * Congruence for function symbols: for $f \in {\cal L}$ function symbol with $ar(f)=n$, ++ |
 \[
-   \forall x_1,\ldots,x_n, y_1,\ldots,y_n.\ (\bigwedge_{i=1}^n x_i = y_i) \rightarrow f(x_1,\ldots,x_n) = f(y_1,\ldots,y_n)
+   \forall x_1,\ldots,x_n, y_1,\ldots,y_n.\ (\bigwedge_{i=1}^n eq(x_i,y_i)) \rightarrow eq(f(x_1,\ldots,x_n),f(y_1,\ldots,y_n))
 \]
 ++
   * Congruence for relation symbols: for $R \in {\cal L}$ relation symbol with $ar(R)=n$, ++ |
 \[
-   \forall x_1,\ldots,x_n, y_1,\ldots,y_n.\ (\bigwedge_{i=1}^n x_i = y_i) \rightarrow (R(x_1,\ldots,x_n) \leftrightarrow R(y_1,\ldots,y_n))
+   \forall x_1,\ldots,x_n, y_1,\ldots,y_n.\ (\bigwedge_{i=1}^n eq(x_i,y_i)) \rightarrow (R(x_1,\ldots,x_n) \leftrightarrow R(y_1,\ldots,y_n))
 \]
 ++
-Note: if an interpretation $(D,I)$ satisfies $AxEq$, then we call $e_F(I)(eq)$ (the interpretation of eq) a //congruence// relation for $(D,I)$.
+**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 $I$.
 === Example: quotient on pairs of natural numbers ===
@@ Line 51: / Line 51: @@
 \]
 This construction is an algebraic approach to construct from natural numbers one well-known structure.  Which one? ++| $({\cal Z}, + , -)$ where ${\cal Z}$ is the set of integers. ++
+Note: this construction can be applied whenever we have an associative and commutative operation $*$ satisfying the cancelation law $x * z = y * z \rightarrow x=y$.  It allows us to contruct a structure where operation $*$ has an inverse.  What do we obtain if we apply this construction to multiplication of strictly positive integers?
 ===== References =====
   * [[Calculus of Computation Textbook]], Section 3.1