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--- sav08:interpretation_quotient_under_congruence [2008/04/02 22:20]
vkuncak
+++ sav08:interpretation_quotient_under_congruence [2008/04/16 09:22]
maysam
@@ Line 1: / Line 1: @@
 ====== Quotient of an Interpretation under a Congruence ======
 ===== Example: quotient on pairs of natural numbers =====
@@ Line 35: / Line 36: @@
 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?
+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 construct a structure where operation $*$ has an inverse.  What do we obtain if we apply this construction to multiplication of strictly positive integers?
 ===== Definition of Quotient of an Interpretation =====
@@ Line 77: / Line 78: @@
 \]
-**Lemma 1:** $I_Q = ([D],I_Q)$ is a well-defined interpretation, that is,
+**Lemma 1:** For each function symbol $f$ with $ar(f)=n$, the relation $\alpha_Q(f)$ is a total function $[D]^n \to [D]$ and for all $x_1,\ldots,x_n \in D$,
-  * $[D]$ is non-empty;
+\[
-  * for each function symbol $f$ with $ar(f)=n$, the relation $\alpha_Q(f)$ is a total function $[D]^n \to [D]$.
+    \alpha_Q(f)([x_1],\ldots,[x_n]) = [\alpha(f)(x_1,\ldots,x_n)]
+\]
 **Lemma 2:** For each term $t$ we have $e_T(t)(I_Q) = [e_T(t)(I)]$.