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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 14:23, 21 April 2012
, 14:23, 21 April 2012→6.6. Basic Notions of Group Theory: markup
: <math>h(x) *_2 h(y) = h(z).\!</math>
: <math>h(x) *_2 h(y) = h(z).\!</math>
From these two equations one derives, by substituting <math>x *_1 y\!</math> for <math>z\!</math> in <math>h(z),\!</math> a succinct formulation of the condition for a mapping <math>h : X_1 \to X_2\!</math> to be a relation homomorphism from a system <math>(X_1, *_1)\!</math> to a system <math>X_2, *_2,\!</math> expressed in the form of a ''distributive law'' or ''linearity condition'':
From these two equations one derives, by substituting x *1 y for z in h(z), a succinct formulation of the condition for a mapping h : X1 >X2 to be a relation homomorphism from a system <X1, *1> to a system <X2, *2>, expressed in the form of a "distributive law" or "linearity condition":
h(x *1 y) = h(x) *2 h(y).
: <math>h(x *_1 y) ~=~ h(x) *_2 h(y).\!</math>
<pre>
To sum up the development so far in a general way: A "homomorphism" is a mapping from a system to a system that preserves an aspect of systematic structure, usually one that is relevant to an understood purpose or context. When the pertinent aspect of structure for both the source and the target system is a binary operation or a LOC, then the condition that the LOCs be preserved in passing from the pre image to the image of the mapping is frequently expressed by stating that "the image of the product is the product of the images". That is, if h : X1 >X2 is a homomorphism from X1 = <X1, *1> to X2 = <X2, *2>, then for every x, y C X1 the following condition holds:
To sum up the development so far in a general way: A "homomorphism" is a mapping from a system to a system that preserves an aspect of systematic structure, usually one that is relevant to an understood purpose or context. When the pertinent aspect of structure for both the source and the target system is a binary operation or a LOC, then the condition that the LOCs be preserved in passing from the pre image to the image of the mapping is frequently expressed by stating that "the image of the product is the product of the images". That is, if h : X1 >X2 is a homomorphism from X1 = <X1, *1> to X2 = <X2, *2>, then for every x, y C X1 the following condition holds: