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--- sav08:axioms_for_equality [2008/04/02 21:25]
vkuncak
+++ sav08:axioms_for_equality [2008/04/02 21:27]
vkuncak
@@ Line 18: / Line 18: @@
 **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 $I$.
-**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 $\alpha$.
+**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 ${\can N}$, with one binary operation whose interpretation is $+$ and one binary relation whose interpretation is $\leq$ can be written as $({\cal N},+,\leq)$.  This way we avoid writing $\alpha$ all the time, but it becomes more cumbersome to describe correspondence between structures.
 === Example: quotient on pairs of natural numbers ===