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constitutes an isomorphism from <math>Z_4(\cdot)\!</math> to <math>Z_4(+).\!</math>
constitutes an isomorphism from <math>Z_4(\cdot)\!</math> to <math>Z_4(+).\!</math>
This fact can be verified in several ways: (1) by checking that the map <math>h\!</math> is bijective and that <math>h(x \cdot y) = h(x) + h(y)\!</math> for every <math>x\!</math> and <math>y\!</math> in <math>Z_4(\cdot),\!</math> (2) by noting that <math>h\!</math> transforms the whole multiplication table for <math>Z_4(\cdot)\!</math> into the whole addition table for <math>Z_4(+)\!</math> in a one-to-one and onto fashion, or (3) by finding that both systems share some collection of properties that are definitive of the abstract group, for example, being cyclic of order <math>4.\!</math>
This fact can be verified in several ways: (1) by checking that the map h is bijective and that h(x.y) = h(x) + h(y) for every x and y in Z4(.), (2) by noting that h transforms the whole multiplication table for Z4(.) into the whole addition table for Z4(+) in a one to one and onto fashion, or (3) by finding that both systems share some collection of properties that are definitive of the abstract group, for example, being cyclic of order 4.
<pre>
Table 34.1 Multiplicative Presentation of the Group Z4(.)
Table 34.1 Multiplicative Presentation of the Group Z4(.)
. 1 a b c
. 1 a b c
b b c 1 a
b b c 1 a
c c 1 a b
c c 1 a b
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<pre>
Table 34.2 Regular Representation of the Group Z4(.)
Table 34.2 Regular Representation of the Group Z4(.)
Element Function as Set of Ordered Pairs of Elements
Element Function as Set of Ordered Pairs of Elements
b { <1, b>, <a, c>, <b, 1>, <c, a> }
b { <1, b>, <a, c>, <b, 1>, <c, a> }
c { <1, c>, <a, 1>, <b, a>, <c, b> }
c { <1, c>, <a, 1>, <b, a>, <c, b> }
</pre>
<pre>
Table 35.1 Additive Presentation of the Group Z4(+)
Table 35.1 Additive Presentation of the Group Z4(+)
+ 0 1 2 3
+ 0 1 2 3
2 2 3 0 1
2 2 3 0 1
3 3 0 1 2
3 3 0 1 2
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<pre>
Table 35.2 Regular Representation of the Group Z4(+)
Table 35.2 Regular Representation of the Group Z4(+)
Element Function as Set of Ordered Pairs of Elements
Element Function as Set of Ordered Pairs of Elements
2 { <0, 2>, <1, 3>, <2, 0>, <3, 1> }
2 { <0, 2>, <1, 3>, <2, 0>, <3, 1> }
3 { <0, 3>, <1, 0>, <2, 1>, <3, 2> }
3 { <0, 3>, <1, 0>, <2, 1>, <3, 2> }
</pre>
Standard references for the above material are:
Standard references for the above material are:
Jacobson, N. Basic Algebra I.
# Jacobson, N., ''Basic Algebra I'', W.H. Freeman, San Francisco, CA, 1974.
# Lang, S., ''Algebra'', 2nd ed., Addison Wesley, Menlo Park, CA, 1984.
# Rotman, J.J., ''An Introduction to the Theory of Groups'', 3rd ed., Allyn & Bacon, Boston, MA, 1984.
Lang, S. Algebra, 2nd ed.
[buffer fragment ?] When it comes to the subject of systems theory, a particular POV is so widely propagated that it might as well be regarded as the established, received, or traditional POV. The POV in question says that there are dynamic systems and symbolic systems, and never the twain shall meet. I naturally intend to challenge this assumption, preferring to suggest that dynamic …
When it comes to the subject of systems theory, a particular POV is so widely propagated that it might as well be regarded as the established, received, or traditional POV. The POV in question says that there are dynamic systems and symbolic systems, and never the twain shall meet. I naturally intend to challenge this assumption, preferring to suggest that dynamic ... [possible buffer fragment]
===6.7. Basic Notions of Formal Language Theory===
===6.7. Basic Notions of Formal Language Theory===
