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Revision as of 16:16, 8 February 2009
, 16:16, 8 February 2009→Digression on Derived Relations: mathematical markup
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
| <math>\upharpoonleft \operatorname{Der}(L) \upharpoonright (x, y) = \underset{o ~\in~ O}{\operatorname{Conj}} ~\underline{((}~ \upharpoonleft L_{SO} \upharpoonright (x, o) ~,~ \upharpoonleft L_{OS} \upharpoonright (o, y) ~\underline{))}~.</math>
| <math>\upharpoonleft \operatorname{Der}(L) \upharpoonright (x, y) = \underset{o \in O}{\operatorname{Conj}} ~\underline{((}~ \upharpoonleft L_{SO} \upharpoonright (x, o) ~,~ \upharpoonleft L_{OS} \upharpoonright (o, y) ~\underline{))}~.</math>
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From this may be abstracted a form of composition, <math>P \circeq Q,</math> where <math>P ~\subseteq~ X \times M</math> and <math>Q ~\subseteq~ M \times Y</math> are otherwise arbitrary dyadic relations, and where <math>P \circeq Q ~\subseteq~ X \times Y</math> is defined as follows:
From this abstract a form of composition, temporarily notated as "P#Q", where P c XxM and Q c MxY are otherwise arbitrary dyadic relations, and where P#Q c XxY is defined as follows:
{P#Q}(x, y) = Conj(m C M) (( {P}(x, m) , {Q}(m, y) )).
{| align="center" cellpadding="8" width="90%"
| <math>\upharpoonleft P \circeq Q \upharpoonright (x, y) = \underset{m \in M}{\operatorname{Conj}} ~\underline{((}~ \upharpoonleft P \upharpoonright (x, m) ~,~ \upharpoonleft Q \upharpoonright (m, y) ~\underline{))}~.</math>
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<pre>
Compare this with the usual form of composition, typically notated as "P.Q" and defined as follows:
Compare this with the usual form of composition, typically notated as "P.Q" and defined as follows:
