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sav08:axioms_for_equality [2008/04/02 14:28] vkuncak |
sav08:axioms_for_equality [2008/04/02 14:46] vkuncak |
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| * Congruence for function symbols: for $f \in {\cal L}$ function symbol with $ar(f)=n$, ++ | | * Congruence for function symbols: for $f \in {\cal L}$ function symbol with $ar(f)=n$, ++ | | ||
| \[ | \[ | ||
| - | \forall x_1,\ldots,x_n, y_1,\ldots,y_n.\ (\bigwedge_{i=1}^n x_i = y_i) \rightarrow f(x_1,\ldots,x_n) = f(y_1,\ldots,y_n) | + | \forall x_1,\ldots,x_n, y_1,\ldots,y_n.\ (\bigwedge_{i=1}^n eq(x_i,y_i)) \rightarrow eq(f(x_1,\ldots,x_n),f(y_1,\ldots,y_n)) |
| \] | \] | ||
| ++ | ++ | ||
| * Congruence for relation symbols: for $R \in {\cal L}$ relation symbol with $ar(R)=n$, ++ | | * Congruence for relation symbols: for $R \in {\cal L}$ relation symbol with $ar(R)=n$, ++ | | ||
| \[ | \[ | ||
| - | \forall x_1,\ldots,x_n, y_1,\ldots,y_n.\ (\bigwedge_{i=1}^n x_i = y_i) \rightarrow (R(x_1,\ldots,x_n) \leftrightarrow R(y_1,\ldots,y_n)) | + | \forall x_1,\ldots,x_n, y_1,\ldots,y_n.\ (\bigwedge_{i=1}^n eq(x_i,y_i)) \rightarrow (R(x_1,\ldots,x_n) \leftrightarrow R(y_1,\ldots,y_n)) |
| \] | \] | ||
| ++ | ++ | ||
| - | Note: if an interpretation $(D,I)$ satisfies $AxEq$, then we call $e_F(I)(eq)$ (the interpretation of eq) a //congruence// relation for $(D,I)$. | + | Note: if an interpretation $(D,I)$ satisfies $AxEq$, then we call $I(eq)$ (the interpretation of eq) a //congruence// relation for $(D,I)$. |
| === Example: quotient on pairs of natural numbers === | === Example: quotient on pairs of natural numbers === | ||