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--- sav08:axioms_for_equality [2008/04/02 00:43]
vkuncak
+++ sav08:axioms_for_equality [2008/04/02 21:40]
vkuncak
@@ 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 ===
-Let ${\cal N} = \{0,1,2,\ldots, \}$.  Consider a structure with domain $N^2$, with functions
-\[
-    p((x_1,y_1),(x_2,y_2)) = (x_1 + x_2, y_1 + y_2)
-\]
-\[
-    m((x_1,y_1),(x_2,y_2)) = (x_1 + y_2, y_1 + x_2)
-\]
-Relation $r$ defined by
-\[
-   r = \{((x_1,y_1),(x_2,y_2)) \mid x_1 + y_2 = x_2 + y_1  \}
-\]
-is a congruence with respect to operations $p$ and $m$.
-Congruence is an equivalence relation.  What is the equivalence class for element $(1,1)$?
-$[(1,1)] = $ ++| $\{ (x,y) \mid x=y \}$
-$[(10,1)] = $ ++| $\{ (x,y) \mid x=y+9 \}$
-$[(1,10)] = $ ++| $\{ (x,y) \mid x+9=y \}$
-Whenever we have a congruence in an interpretation, we can shrink the structure to a smaller one by merging elements that are in congruence.  In the resulting structure we can define operations $p$ and $m$ such that the following holds:
-\[
-\begin{array}{l}
-   p( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + x_2, y_1 + y_2)] \\
-   m( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + y_2, y_1 + x_2)]
-\end{array}
-\]
-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. ++
 ===== References =====
   * [[Calculus of Computation Textbook]], Section 3.1