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sav08:interpretation_quotient_under_congruence [2009/05/14 14:06]
vkuncak
sav08:interpretation_quotient_under_congruence [2015/04/21 17:30] (current)
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===== Example: quotient on pairs of natural numbers ===== ===== Example: quotient on pairs of natural numbers =====

-Let ${\cal N} = \{0,​1,​2,​\ldots,​ \}$.  Consider a structure with domain $N^2$, with functions +Let ${\cal N} = \{0,​1,​2,​\ldots,​ \}$.  Consider a structure with domain $N^2$, with functions ​(plus and minus):
-$+\begin{equation*} p((x_1,​y_1),​(x_2,​y_2)) = (x_1 + x_2, y_1 + y_2) p((x_1,​y_1),​(x_2,​y_2)) = (x_1 + x_2, y_1 + y_2) -$ +\end{equation*}
-$+\begin{equation*} m((x_1,​y_1),​(x_2,​y_2)) = (x_1 + y_2, y_1 + x_2) m((x_1,​y_1),​(x_2,​y_2)) = (x_1 + y_2, y_1 + x_2) -$+\end{equation*}
Relation $r$ defined by Relation $r$ defined by
-$+\begin{equation*} r = \{((x_1,​y_1),​(x_2,​y_2)) \mid x_1 + y_2 = x_2 + y_1 \} r = \{((x_1,​y_1),​(x_2,​y_2)) \mid x_1 + y_2 = x_2 + y_1 \} -$ +\end{equation*}
-is a congruence with respect to operations $p$ and $m$.  ​+is a congruence with respect to operations $p$ and $m$.  ​Indded, we can check that, for example, if $r((x_1,​y_1),​(x'​_1,​y'​_1))$ and
+$r((x_2,​y_2),​(x'​_2,​y'​_2))$ then
+\begin{equation*}
+     ​r(p((x_1,​y_1),​(x_2,​y_2)),​ p((x'​_1,​y'​_1),​(x'​_2,​y'​_2)))
+\end{equation*}

Congruence is an equivalence relation. ​ What are equivalence classes for elements: Congruence is an equivalence relation. ​ What are equivalence classes for elements:
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In the resulting structure $([N^2], I_Q)$ we define operations $p$ and $m$ such that the following holds: In the resulting structure $([N^2], I_Q)$ we define operations $p$ and $m$ such that the following holds:
-$+\begin{equation*} \begin{array}{l} \begin{array}{l} ​I_Q(p)( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + x_2, y_1 + y_2)] \\ ​I_Q(p)( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + x_2, y_1 + y_2)] \\ ​I_Q(m)( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + y_2, y_1 + x_2)] ​I_Q(m)( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + y_2, y_1 + x_2)] \end{array} \end{array} -$+\end{equation*}
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. ++ 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. ++

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For each element $x \in D$, define For each element $x \in D$, define
-$+\begin{equation*} [x] = \{ y \mid (x,y) \in \alpha(eq) \} [x] = \{ y \mid (x,y) \in \alpha(eq) \} -$+\end{equation*}
Let Let
-$+\begin{equation*} [D] = \{ [x] \mid x \in D \} [D] = \{ [x] \mid x \in D \} -$+\end{equation*}
The constructed model will be $I_Q = ([D],​\alpha_Q)$ where  The constructed model will be $I_Q = ([D],​\alpha_Q)$ where
-$+\begin{equation*} \alpha_Q(R) = \{ ([x_1],​\ldots,​[x_n]) \mid (x_1,​\ldots,​x_n) \in \alpha(R) \} \alpha_Q(R) = \{ ([x_1],​\ldots,​[x_n]) \mid (x_1,​\ldots,​x_n) \in \alpha(R) \} -$+\end{equation*}
In particular, when $R$ is $eq$ we have In particular, when $R$ is $eq$ we have

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Functions are special case of relations: Functions are special case of relations:
-$+\begin{equation*} \alpha_Q(f) = \{ ([x_1],​\ldots,​[x_n],​[x_{n+1}]) \mid (x_1,​\ldots,​x_n,​x_{n+1}) \in \alpha(f) \} \alpha_Q(f) = \{ ([x_1],​\ldots,​[x_n],​[x_{n+1}]) \mid (x_1,​\ldots,​x_n,​x_{n+1}) \in \alpha(f) \} -$+\end{equation*}

Interpretation of variables is analogous to interpretation of constants: Interpretation of variables is analogous to interpretation of constants:
-$+\begin{equation*} ​\alpha_Q(x) = [\alpha(x)] ​\alpha_Q(x) = [\alpha(x)] -$+\end{equation*}

**Lemma 0:** For all $x_1,​\ldots,​x_n \in D$, **Lemma 0:** For all $x_1,​\ldots,​x_n \in D$,
-$+\begin{equation*} ([x_1],​\ldots,​[x_n]) \in \alpha_Q(R) \mbox{ iff } (x_1,​\ldots,​x_n) \in \alpha(R) ([x_1],​\ldots,​[x_n]) \in \alpha_Q(R) \mbox{ iff } (x_1,​\ldots,​x_n) \in \alpha(R) -$+\end{equation*}

**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 $x_1,​\ldots,​x_n \in D$, **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 $x_1,​\ldots,​x_n \in D$,
-$+\begin{equation*} \alpha_Q(f)([x_1],​\ldots,​[x_n]) = [\alpha(f)(x_1,​\ldots,​x_n)] \alpha_Q(f)([x_1],​\ldots,​[x_n]) = [\alpha(f)(x_1,​\ldots,​x_n)] -$+\end{equation*}

**Lemma 2:** For each term $t$ we have $e_T(t)(I_Q) = [e_T(t)(I)]$. **Lemma 2:** For each term $t$ we have $e_T(t)(I_Q) = [e_T(t)(I)]$.