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In this setting the mapping <math>f : X \to Y</math> is said to be ''linear'', and to ''preserve'' the structure of <math>L_X\!</math> in the structure of <math>L_Y,\!</math> if and only if <math>f(a ~\#~ b) = f(a) + f(b),</math> for all pairs <math>a, b\!</math> in <math>X.\!</math> In other words, <math>f\!</math> ''distributes'' over the additions <math>\#</math> to <math>+,\!</math> just as if it were a form of multiplication, analogous to <math>m(a + b) = ma + mb.\!</math>
In this setting the mapping <math>f : X \to Y</math> is said to be ''linear'', and to ''preserve'' the structure of <math>L_X\!</math> in the structure of <math>L_Y,\!</math> if and only if <math>f(a ~\#~ b) = f(a) + f(b),</math> for all pairs <math>a, b\!</math> in <math>X.\!</math> In other words, <math>f\!</math> ''distributes'' over the additions <math>\#</math> to <math>+,\!</math> just as if it were a form of multiplication, analogous to <math>m(a + b) = ma + mb.\!</math>
Writing this more directly in terms of the 3-adic relations ''L''<sub>''X''</sub> and ''L''<sub>''Y''</sub> instead of via their operation symbols, we would say that ''f'' : ''X'' → ''Y'' is linear with regard to ''L''<sub>''X''</sub> and ''L''<sub>''Y''</sub> if and only if <''a'', ''b'', ''c''> being in the relation ''L''<sub>''X''</sub> determines that its map image <''f''(''a''), ''f''(''b''), ''f''(''c'')> be in ''L''<sub>''Y''</sub>. To see this, observe that <''a'', ''b'', ''c''> being in ''L''<sub>''X''</sub> implies that ''c'' = ''a'' # ''b'', and <''f''(''a''), ''f''(''b''), ''f''(''c'')> being in ''L''<sub>''Y''</sub> implies that ''f''(''c'') = ''f''(''a'') + ''f''(''b''), so we have that ''f''(''a'' # ''b'') = ''f''(''c'') = ''f''(''a'') + ''f''(''b''), and the two notions are one.
Writing this more directly in terms of the 3-adic relations <math>L_X\!</math> and <math>L_Y\!</math> instead of via their operation symbols, we would say that <math>f : X \to Y</math> is linear with regard to <math>L_X\!</math> and <math>L_Y\!</math> if and only if <math>(a, b, c)\!</math> being in the relation <math>L_X\!</math> determines that its map image <math>(f(a), f(b), f(c))\!</math> be in <math>L_Y.\!</math> To see this, observe that <math>(a, b, c)\!</math> being in <math>L_X\!</math> implies that <math>c = a ~\#~ b,</math> and <math>(f(a), f(b), f(c))\!</math> being in <math>L_Y\!</math> implies that <math>f(c) = f(a) + f(b),\!</math> so we have that <math>f(a ~\#~ b) = f(c) = f(a) + f(b),</math> and the two notions are one.
The idea of mappings that preserve 3-adic relations should ring a few bells here.
The idea of mappings that preserve 3-adic relations should ring a few bells here.
