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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 15:15, 21 April 2012→6.6. Basic Notions of Group Theory: markup
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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 <math>h : X_1 \to X_2\!</math> is a homomorphism from <math>\underline{X}_1 = (X_1, *_1)\!</math> to <math>\underline{X}_2 = (X_2, *_2),\!</math> then for every <math>x, y \in X_1\!</math> 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:
h(x *1 y) = h(x) *2 h(y).
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| <math>h(x *_1 y) ~=~ h(x) *_2 h(y).\!</math>
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Next, the concept of a homomorphism or "structure preserving map" is specialized to the different kinds of structure of interest here.
Next, the concept of a homomorphism or a ''structure-preserving map'' is specialized to the different kinds of structure of interest here.
<pre>
A "semigroup homomorphism" from a semigroup X1 = <X1, *1> to a semigroup X2 = <X2, *2> is a mapping between the underlying sets that preserves the structure appropriate to semigroups, namely, the LOCs. This makes it a map h : X1 >X2 whose induced action on the LOCs is such that it takes every element of *1 to an element of *2. That is:
A "semigroup homomorphism" from a semigroup X1 = <X1, *1> to a semigroup X2 = <X2, *2> is a mapping between the underlying sets that preserves the structure appropriate to semigroups, namely, the LOCs. This makes it a map h : X1 >X2 whose induced action on the LOCs is such that it takes every element of *1 to an element of *2. That is: