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--- sav08:axioms_for_equality [2008/04/02 00:44]
vkuncak
+++ sav08:axioms_for_equality [2008/04/02 14:28]
vkuncak
@@ Line 33: / Line 33: @@
 is a congruence with respect to operations $p$ and $m$.
-Congruence is an equivalence relation.  What is the equivalence class for element $(1,1)$?
+Congruence is an equivalence relation.  What are equivalence classes for elements:
 $[(1,1)] = $ ++| $\{ (x,y) \mid x=y \}$++
@@ Line 41: / Line 41: @@
 $[(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:
+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 $([N^2], I_Q)$ we define operations $p$ and $m$ such that the following holds:
 \[
 \begin{array}{l}
-   p( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + x_2, y_1 + y_2)] \\
+   I_Q(p)( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + x_2, y_1 + y_2)] \\
-   m( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + y_2, y_1 + x_2)]
+   I_Q(m)( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + y_2, y_1 + x_2)]
 \end{array}
 \]
 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. ++
+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 consutrction to multiplication of strictly positive integers?
 ===== References =====
   * [[Calculus of Computation Textbook]], Section 3.1