Changes

For example, let's start with the archetype of all morphisms, namely, a ''linear function'' or a ''linear mapping'' <math>f : X \to Y.</math>

For example, let's start with the archetype of all morphisms, namely, a ''linear function'' or a ''linear mapping'' <math>f : X \to Y.</math>

To say that the function ''f'' is ''linear'' is to say that we have already got in mind a couple of relations on ''X'' and ''Y'' that have forms roughly analogous to "addition tables", so let's signify their operation by means of the symbols "#" for "addition in ''X''" and "+" for "addition in ''Y''".

To say that the function <math>f\!</math> is ''linear'' is to say that we have already got in mind a couple of relations on <math>X\!</math> and <math>Y\!</math> that have forms roughly analogous to "addition tables", so let's signify their operation by means of the symbols <math>{}^{\backprime\backprime} \# {}^{\prime\prime}</math> for addition in <math>X\!</math> and <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> for addition in <math>Y.\!</math>

More exactly, the use of "#" refers to a 3-adic relation ''L''_''X'' &sube; ''X'' &times; ''X'' &times; ''X'' that licenses the formula "''a'' # ''b'' = ''c''" just when <''a'', ''b'', ''c''> is in ''L''_''X'', and the use of "+" refers to a 3-adic relation ''L''_''Y'' &sube; ''Y'' &times; ''Y'' &times; ''Y'' that licenses the formula "''p'' + ''q'' = ''r''" just when <''p'', ''q'', ''r''> is in ''L''_''Y''.

More exactly, the use of <math>{}^{\backprime\backprime} \# {}^{\prime\prime}</math> refers to a 3-adic relation <math>L_X \subseteq X \times X \times X</math> that licenses the formula <math>a ~\#~ b = c</math> just when <math>(a, b, c)\!</math> is in <math>L_X\!</math> and the use of <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> refers to a 3-adic relation <math>L_Y \subseteq Y \times Y \times Y</math> that licenses the formula <math>p + q = r\!</math> just when <math>(p, q, r)\!</math> is in <math>L_Y.\!</math>

In this setting, the mapping ''f'' : ''X'' &rarr; ''Y'' is said to be ''linear'', and to ''preserve'' the structure of ''L''_''X'' in the structure of ''L''_''Y'', if and only if ''f''(''a'' # ''b'') = ''f''(''a'') + ''f''(''b''), for all pairs ''a'', ''b'' in ''X''.  In other words, ''f'' ''distributes'' over the additions # to +, just as if it were a form of multiplication, like ''m''(''a'' + ''b'') = ''ma'' + ''mb''.

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''_''X'' and ''L''_''Y'' instead of via their operation symbols, we would say that ''f'' : ''X'' &rarr; ''Y'' is linear with regard to ''L''_''X'' and ''L''_''Y'' if and only if <''a'', ''b'', ''c''> being in the relation ''L''_''X'' determines that its map image <''f''(''a''), ''f''(''b''), ''f''(''c'')> be in ''L''_''Y''.  To see this, observe that <''a'', ''b'', ''c''> being in ''L''_''X'' implies that ''c'' = ''a'' # ''b'', and <''f''(''a''), ''f''(''b''), ''f''(''c'')> being in ''L''_''Y'' 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 ''L''_''X'' and ''L''_''Y'' instead of via their operation symbols, we would say that ''f'' : ''X'' &rarr; ''Y'' is linear with regard to ''L''_''X'' and ''L''_''Y'' if and only if <''a'', ''b'', ''c''> being in the relation ''L''_''X'' determines that its map image <''f''(''a''), ''f''(''b''), ''f''(''c'')> be in ''L''_''Y''.  To see this, observe that <''a'', ''b'', ''c''> being in ''L''_''X'' implies that ''c'' = ''a'' # ''b'', and <''f''(''a''), ''f''(''b''), ''f''(''c'')> being in ''L''_''Y'' 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.