Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
sav08:axioms_for_equality [2008/04/02 21:25] vkuncak |
sav08:axioms_for_equality [2008/04/02 21:40] vkuncak |
||
---|---|---|---|
Line 17: | Line 17: | ||
**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 $I$. | **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 $I$. | ||
- | |||
- | **A terminological note:** in algebra, an interpretation is often called a //structure//. Instead of using $\alpha$ mapping language ${\cal L} = \{ f_1,\ldots,f_n, R_1,\ldots,R_m\}$ to interpretation of its symbols, the structure is denoted by a tuple $(D,\alpha(f_1),\ldots,\alpha(f_n),\alpha(R_1),\ldots,\alpha(R_n))$. For example, an interpretation with domain $D$, with one binary operation $+$ and one binary relation $\leq$ can be written as a pair $(D,+,\leq)$. This avoids writing $\alpha$. | ||
- | |||
- | === 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 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 | ||