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:interpretation_quotient_under_congruence [2008/04/02 22:36] vkuncak |
sav08:interpretation_quotient_under_congruence [2009/05/14 14:06] vkuncak |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ====== Quotient of an Interpretation under a Congruence ====== | ====== Quotient of an Interpretation under a Congruence ====== | ||
| + | |||
| ===== Example: quotient on pairs of natural numbers ===== | ===== 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. ++ | 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 ===== | ===== Definition of Quotient of an Interpretation ===== | ||
| Line 41: | Line 43: | ||
| (Recall notation in [[First-Order Logic Semantics]].) | (Recall notation in [[First-Order Logic Semantics]].) | ||
| - | Let $I = (D,\alpha)$ be an interpretation of language ${\cal L}$ with $eq \in {\cal L}$ for which [[Axioms for Equality]] hold, that is, $\alpha(eq)$ is a congruence relation for $I$. | + | Let $I = (D,\alpha)$ be an interpretation of language ${\cal L}$ with $eq \in {\cal L}$ for which [[Axioms for Equality]] hold, that is, $\alpha(eq)$ is a congruence relation for $I$. We will construct a new model $I_Q$. |
| For each element $x \in D$, define | For each element $x \in D$, define | ||