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sav08:notes_on_congruences [2009/05/06 10:00] vkuncak |
sav08:notes_on_congruences [2009/05/06 11:56] vkuncak |
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| We next fix $D$ as well as functions and relations and consider the set of all congruences on the set $D$ with respect to these functions and relations. | We next fix $D$ as well as functions and relations and consider the set of all congruences on the set $D$ with respect to these functions and relations. | ||
| - | We assume no relation symbols other than congruence itself. We can represent predicate $p(x_1,\ldots,x_n)$ as $f_p(x_1,\ldots,x_n)=true$. | + | We assume no relation symbols other than congruence itself. (We represent a predicate $p(x_1,\ldots,x_n)$ as $f_p(x_1,\ldots,x_n)=true$.) |
| ===== Intersection of Congruences ===== | ===== Intersection of Congruences ===== | ||
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| \[\begin{array}{rcl} | \[\begin{array}{rcl} | ||
| \bigwedge_{i=0}^n (x_i,y_1) \in \bigcap S & \rightarrow & \bigwedge_{i=0}^n (x_i,y_1) \in r_1,r_2 \\ | \bigwedge_{i=0}^n (x_i,y_1) \in \bigcap S & \rightarrow & \bigwedge_{i=0}^n (x_i,y_1) \in r_1,r_2 \\ | ||
| - | r_1,r_2 ~~ \text{congruence relations} & \rightarrow & f(x_1,\ldots, x_n) = f(y_1,\ldots, y_n) \in r_1,r_2 \\ | + | r_1,r_2 ~~ \text{congruence relations} & \rightarrow & (f(x_1,\ldots, x_n), f(y_1,\ldots, y_n)) \in r_1,r_2 \\ |
| & \rightarrow & f(x_1,\ldots, x_n) = f(y_1,\ldots, y_n) \in \bigcap S | & \rightarrow & f(x_1,\ldots, x_n) = f(y_1,\ldots, y_n) \in \bigcap S | ||
| \end{array} \] | \end{array} \] | ||