Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6
Discussion
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 …
Work Area
Alternate Text
A semigroup consists of a nonempty set with an associative LOC on it. On formal occasions, a semigroup is introduced by means a formula like \(X = (X, *),\!\) interpreted to mean that a semigroup \(X\!\) is specified by giving two pieces of data, a nonempty set that conventionally, if somewhat ambiguously, goes under the same name \({}^{\backprime\backprime} X {}^{\prime\prime},\!\) plus an associative binary operation denoted by \({}^{\backprime\backprime} * {}^{\prime\prime}.\!\) In contexts where there is only one semigroup being discussed, or where the additional structure is otherwise understood, it is common practice to call the semigroup by the name of the underlying set. In contexts where more than one semigroup is formed on the same set, one may use notations like \(X_i = (X, *_i)\!\) to distinguish them.
Additive Presentation
Version 1
- The \(n^\text{th}\!\) multiple of an element \(x\!\) in a semigroup \(\underline{X} = (X, +, 0),\!\) for integer \(n > 0,\!\) is notated as \(nx\!\) and defined as follows. Proceeding recursively, for \(n = 1,\!\) let \(1x = x,\!\) and for \(n > 1,\!\) let \(nx = (n-1)x + x.\!\)
- The \(n^\text{th}\!\) multiple of \(x\!\) in a monoid \(\underline{X} = (X, +, 0),\!\) for integer \(n \ge 0,\!\) is defined the same way for \(n > 0,\!\) letting \(0x = 0\!\) when \(n = 0.\!\)
- The \(n^\text{th}\!\) multiple of \(x\!\) in a group \(\underline{X} = (X, +, 0),\!\) for any integer \(n,\!\) is defined the same way for \(n \ge 0,\!\) letting \(nx = (-n)(-x)\!\) for \(n < 0.\!\)
Version 2
- In a semigroup written additively, the \(n^\text{th}\!\) multiple of an element \(x\!\) is notated as \(nx\!\) and defined for every positive integer \(n\!\) in the following manner. Proceeding recursively, let \(1x = x\!\) and let \(nx = (n-1)x + x\!\) for all \(n > 1.\!\)
- In a monoid written additively, the multiple \(nx\!\) is defined for every non-negative integer \(n\!\) by letting \(0x = 0\!\) and proceeding the same way for \(n > 0.\!\)
- In a group written additively, the multiple \(nx\!\) is defined for every integer \(n\!\) by letting \(nx = (-n)(-x)\!\) for \(n < 0\!\) and proceeding the same way for \(n \ge 0.\!\)
Table Work
| \(*\!\) | \(x_0\!\) | \(\cdots\!\) | \(x_j\!\) | \(\cdots\!\) |
| \(x_0\!\) | \(x_0 * x_0\!\) | \(\cdots\!\) | \(x_0 * x_j\!\) | \(\cdots\!\) |
| \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) |
| \(x_i\!\) | \(x_i * x_0\!\) | \(\cdots\!\) | \(x_i * x_j\!\) | \(\cdots\!\) |
| \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) |
| \(\text{Element}\!\) | \(\text{Function as Set of Ordered Pairs of Elements}\!\) | |||||
| \(x_0\!\) | \(\{\!\) | \((x_0 ~,~ x_0 * x_0),\!\) | \(\cdots\!\) | \((x_j ~,~ x_0 * x_j),\!\) | \(\cdots\!\) | \(\}\!\) |
| \(\cdots\!\) | \(\{\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\}\!\) |
| \(x_i\!\) | \(\{\!\) | \((x_0 ~,~ x_i * x_0),\!\) | \(\cdots\!\) | \((x_j ~,~ x_i * x_j),\!\) | \(\cdots\!\) | \(\}\!\) |
| \(\cdots\!\) | \(\{\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\}\!\) |
| \(\text{Element}\!\) | \(\text{Function as Set of Ordered Pairs of Elements}\!\) | |||||
| \(x_0\!\) | \(\{\!\) | \((x_0 ~,~ x_0 * x_0),\!\) | \(\cdots\!\) | \((x_j ~,~ x_j * x_0),\!\) | \(\cdots\!\) | \(\}\!\) |
| \(\cdots\!\) | \(\{\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\}\!\) |
| \(x_i\!\) | \(\{\!\) | \((x_0 ~,~ x_0 * x_i),\!\) | \(\cdots\!\) | \((x_j ~,~ x_j * x_i),\!\) | \(\cdots\!\) | \(\}\!\) |
| \(\cdots\!\) | \(\{\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\cdots\!\) | \(\}\!\) |
| \(\cdot\!\) | \(\operatorname{e}\) | \(\operatorname{f}\) | \(\operatorname{g}\) | \(\operatorname{h}\) |
| \(\operatorname{e}\) | \(\operatorname{e}\) | \(\operatorname{f}\) | \(\operatorname{g}\) | \(\operatorname{h}\) |
| \(\operatorname{f}\) | \(\operatorname{f}\) | \(\operatorname{e}\) | \(\operatorname{h}\) | \(\operatorname{g}\) |
| \(\operatorname{g}\) | \(\operatorname{g}\) | \(\operatorname{h}\) | \(\operatorname{e}\) | \(\operatorname{f}\) |
| \(\operatorname{h}\) | \(\operatorname{h}\) | \(\operatorname{g}\) | \(\operatorname{f}\) | \(\operatorname{e}\) |
| \(\text{Element}\!\) | \(\text{Function as Set of Ordered Pairs of Elements}\!\) | |||||
| \(\operatorname{e}\) | \(\{\!\) | \((\operatorname{e}, \operatorname{e}),\) | \((\operatorname{f}, \operatorname{f}),\) | \((\operatorname{g}, \operatorname{g}),\) | \((\operatorname{h}, \operatorname{h})\) | \(\}\!\) |
| \(\operatorname{f}\) | \(\{\!\) | \((\operatorname{e}, \operatorname{f}),\) | \((\operatorname{f}, \operatorname{e}),\) | \((\operatorname{g}, \operatorname{h}),\) | \((\operatorname{h}, \operatorname{g})\) | \(\}\!\) |
| \(\operatorname{g}\) | \(\{\!\) | \((\operatorname{e}, \operatorname{g}),\) | \((\operatorname{f}, \operatorname{h}),\) | \((\operatorname{g}, \operatorname{e}),\) | \((\operatorname{h}, \operatorname{f})\) | \(\}\!\) |
| \(\operatorname{h}\) | \(\{\!\) | \((\operatorname{e}, \operatorname{h}),\) | \((\operatorname{f}, \operatorname{g}),\) | \((\operatorname{g}, \operatorname{f}),\) | \((\operatorname{h}, \operatorname{e})\) | \(\}\!\) |
| \(\text{Element}\!\) | \(\text{Function as Set of Ordered Pairs of Symbols}\!\) | |||||
| \(\operatorname{e}\) | \(\{\!\) | \(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})\) | \(\}\!\) |
| \(\operatorname{f}\) | \(\{\!\) | \(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})\) | \(\}\!\) |
| \(\operatorname{g}\) | \(\{\!\) | \(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})\) | \(\}\!\) |
| \(\operatorname{h}\) | \(\{\!\) | \(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\) | \(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})\) | \(\}\!\) |
| \(\cdot\!\) | \(\operatorname{1}\) | \(\operatorname{a}\) | \(\operatorname{b}\) | \(\operatorname{c}\) |
| \(\operatorname{1}\) | \(\operatorname{1}\) | \(\operatorname{a}\) | \(\operatorname{b}\) | \(\operatorname{c}\) |
| \(\operatorname{a}\) | \(\operatorname{a}\) | \(\operatorname{b}\) | \(\operatorname{c}\) | \(\operatorname{1}\) |
| \(\operatorname{b}\) | \(\operatorname{b}\) | \(\operatorname{c}\) | \(\operatorname{1}\) | \(\operatorname{a}\) |
| \(\operatorname{c}\) | \(\operatorname{c}\) | \(\operatorname{1}\) | \(\operatorname{a}\) | \(\operatorname{b}\) |
| \(\text{Element}\!\) | \(\text{Function as Set of Ordered Pairs of Elements}\!\) | |||||
| \(\operatorname{1}\) | \(\{\!\) | \((\operatorname{1}, \operatorname{1}),\) | \((\operatorname{a}, \operatorname{a}),\) | \((\operatorname{b}, \operatorname{b}),\) | \((\operatorname{c}, \operatorname{c})\) | \(\}\!\) |
| \(\operatorname{a}\) | \(\{\!\) | \((\operatorname{1}, \operatorname{a}),\) | \((\operatorname{a}, \operatorname{b}),\) | \((\operatorname{b}, \operatorname{c}),\) | \((\operatorname{c}, \operatorname{1})\) | \(\}\!\) |
| \(\operatorname{b}\) | \(\{\!\) | \((\operatorname{1}, \operatorname{b}),\) | \((\operatorname{a}, \operatorname{c}),\) | \((\operatorname{b}, \operatorname{1}),\) | \((\operatorname{c}, \operatorname{a})\) | \(\}\!\) |
| \(\operatorname{c}\) | \(\{\!\) | \((\operatorname{1}, \operatorname{c}),\) | \((\operatorname{a}, \operatorname{1}),\) | \((\operatorname{b}, \operatorname{a}),\) | \((\operatorname{c}, \operatorname{b})\) | \(\}\!\) |
| \(+\!\) | \(\operatorname{0}\) | \(\operatorname{1}\) | \(\operatorname{2}\) | \(\operatorname{3}\) |
| \(\operatorname{0}\) | \(\operatorname{0}\) | \(\operatorname{1}\) | \(\operatorname{2}\) | \(\operatorname{3}\) |
| \(\operatorname{1}\) | \(\operatorname{1}\) | \(\operatorname{2}\) | \(\operatorname{3}\) | \(\operatorname{0}\) |
| \(\operatorname{2}\) | \(\operatorname{2}\) | \(\operatorname{3}\) | \(\operatorname{0}\) | \(\operatorname{1}\) |
| \(\operatorname{3}\) | \(\operatorname{3}\) | \(\operatorname{0}\) | \(\operatorname{1}\) | \(\operatorname{2}\) |
Table 35.2 Regular Representation of the Group Z4(+)
Element Function as Set of Ordered Pairs of Elements
0 { <0, 0>, <1, 1>, <2, 2>, <3, 3> }
1 { <0, 1>, <1, 2>, <2, 3>, <3, 0> }
2 { <0, 2>, <1, 3>, <2, 0>, <3, 1> }
3 { <0, 3>, <1, 0>, <2, 1>, <3, 2> }