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--- sav08:axioms_for_equality [2008/04/02 14:51]
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+++ sav08:axioms_for_equality [2008/04/02 21:27]
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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 $I$.
+**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 ===