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sav08:axioms_for_equality [2008/04/02 14:46] vkuncak |
sav08:axioms_for_equality [2008/04/02 21:25] 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 $I$. |
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| + | **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 === | === Example: quotient on pairs of natural numbers === | ||
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| 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 consutrction 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 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 | ||