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sav08:interpretation_quotient_under_congruence [2008/04/16 09:22] maysam |
sav08:interpretation_quotient_under_congruence [2009/05/14 14:06] vkuncak |
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| 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? | 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? | ||
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| ===== Definition of Quotient of an Interpretation ===== | ===== Definition of Quotient of an Interpretation ===== | ||
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| (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 | ||