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sav08:axioms_for_equality [2008/04/02 14:51] vkuncak |
sav08:axioms_for_equality [2009/05/05 23:16] vkuncak |
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- | Note: if an interpretation $(D,I)$ satisfies $AxEq$, then we call $I(eq)$ (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 === | + | **Side remark:** Functions are relations. However, the condition above for function symbols is weaker than the condition for relation symbols. If $f$ is a function, then the relation $\{(x_1,\ldots,x_n,f(x_1,\ldots,x_n)) \mid x_1,\ldots,x_n \in D \}$ does not satisfy the congruence condition because it only has one result, namely $f(x_1,\ldots,x_n)$, and not all the results that are in relation eq with $f(x_1,\ldots,x_n)$. |
- | + | ||
- | 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 are equivalence classes for elements: | + | |
- | + | ||
- | $[(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 $([N^2], I_Q)$ we define operations $p$ and $m$ such that the following holds: | + | |
- | \[ | + | |
- | \begin{array}{l} | + | |
- | I_Q(p)( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + x_2, y_1 + y_2)] \\ | + | |
- | I_Q(m)( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + y_2, y_1 + x_2)] | + | |
- | \end{array} | + | |
- | \] | + | |
- | 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 construction to multiplication of strictly positive integers? | + | |
===== References ===== | ===== References ===== | ||
* [[Calculus of Computation Textbook]], Section 3.1 | * [[Calculus of Computation Textbook]], Section 3.1 | ||