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| − | ==Discussion==
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| − | | |
| − | ==Work Area==
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| − | | |
| − | ===Alternate Text===
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| − | 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 <math>X = (X, *),\!</math> interpreted to mean that a semigroup <math>X\!</math> is specified by giving two pieces of data, a nonempty set that conventionally, if somewhat ambiguously, goes under the same name <math>{}^{\backprime\backprime} X {}^{\prime\prime},\!</math> plus an associative binary operation denoted by <math>{}^{\backprime\backprime} * {}^{\prime\prime}.\!</math> 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 <math>X_i = (X, *_i)\!</math> to distinguish them.
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| − | | |
| − | ===Additive Presentation===
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| − | | |
| − | ====Version 1====
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| − | | |
| − | : The <math>n^\text{th}\!</math> '''multiple''' of an element <math>x\!</math> in a semigroup <math>\underline{X} = (X, +, 0),\!</math> for integer <math>n > 0,\!</math> is notated as <math>nx\!</math> and defined as follows. Proceeding recursively, for <math>n = 1,\!</math> let <math>1x = x,\!</math> and for <math>n > 1,\!</math> let <math>nx = (n-1)x + x.\!</math>
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| − | | |
| − | : The <math>n^\text{th}\!</math> '''multiple''' of <math>x\!</math> in a monoid <math>\underline{X} = (X, +, 0),\!</math> for integer <math>n \ge 0,\!</math> is defined the same way for <math>n > 0,\!</math> letting <math>0x = 0\!</math> when <math>n = 0.\!</math>
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| − | | |
| − | : The <math>n^\text{th}\!</math> '''multiple''' of <math>x\!</math> in a group <math>\underline{X} = (X, +, 0),\!</math> for any integer <math>n,\!</math> is defined the same way for <math>n \ge 0,\!</math> letting <math>nx = (-n)(-x)\!</math> for <math>n < 0.\!</math>
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| − | | |
| − | ====Version 2====
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| − | | |
| − | : In a semigroup written additively, the <math>n^\text{th}\!</math> '''multiple''' of an element <math>x\!</math> is notated as <math>nx\!</math> and defined for every positive integer <math>n\!</math> in the following manner. Proceeding recursively, let <math>1x = x\!</math> and let <math>nx = (n-1)x + x\!</math> for all <math>n > 1.\!</math>
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| − | | |
| − | : In a monoid written additively, the multiple <math>nx\!</math> is defined for every non-negative integer <math>n\!</math> by letting <math>0x = 0\!</math> and proceeding the same way for <math>n > 0.\!</math>
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| − | | |
| − | : In a group written additively, the multiple <math>nx\!</math> is defined for every integer <math>n\!</math> by letting <math>nx = (-n)(-x)\!</math> for <math>n < 0\!</math> and proceeding the same way for <math>n \ge 0.\!</math>
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| − | | |
| − | ==Table Work==
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| − | | |
| − | <br>
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| − | | |
| − | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
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| − | |+ <math>\text{Table 32.1}~~\text{Scheme of a Group Operation Table}</math>
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| − | |- style="height:50px"
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| − | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math>
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| − | | style="border-bottom:1px solid black" | <math>x_0\!</math>
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| − | | style="border-bottom:1px solid black" | <math>\cdots\!</math>
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| − | | style="border-bottom:1px solid black" | <math>x_j\!</math>
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| − | | style="border-bottom:1px solid black" | <math>\cdots\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>x_0\!</math>
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| − | | <math>x_0 * x_0\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>x_0 * x_j\!</math>
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| − | | <math>\cdots\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\cdots\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\cdots\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>x_i\!</math>
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| − | | <math>x_i * x_0\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>x_i * x_j\!</math>
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| − | | <math>\cdots\!</math>
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| − | |- style="height:50px"
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| − | | width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
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| − | | width="22%" | <math>\cdots\!</math>
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| − | | width="22%" | <math>\cdots\!</math>
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| − | | width="22%" | <math>\cdots\!</math>
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| − | | width="22%" | <math>\cdots\!</math>
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| − | |}
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| − | | |
| − | <br>
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| − | | |
| − | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
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| − | |+ <math>\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}</math>
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| − | |- style="height:50px"
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| − | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
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| − | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>x_0\!</math>
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| − | | <math>\{\!</math>
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| − | | <math>(x_0 ~,~ x_0 * x_0),\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>(x_j ~,~ x_0 * x_j),\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\cdots\!</math>
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| − | | <math>\{\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>x_i\!</math>
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| − | | <math>\{\!</math>
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| − | | <math>(x_0 ~,~ x_i * x_0),\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>(x_j ~,~ x_i * x_j),\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
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| − | | width="4%" | <math>\{\!</math>
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| − | | width="18%" | <math>\cdots\!</math>
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| − | | width="22%" | <math>\cdots\!</math>
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| − | | width="22%" | <math>\cdots\!</math>
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| − | | width="18%" | <math>\cdots\!</math>
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| − | | width="4%" | <math>\}\!</math>
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| − | |}
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| − | | |
| − | <br>
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| − | | |
| − | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
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| − | |+ <math>\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}</math>
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| − | |- style="height:50px"
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| − | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
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| − | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>x_0\!</math>
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| − | | <math>\{\!</math>
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| − | | <math>(x_0 ~,~ x_0 * x_0),\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>(x_j ~,~ x_j * x_0),\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\cdots\!</math>
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| − | | <math>\{\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>x_i\!</math>
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| − | | <math>\{\!</math>
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| − | | <math>(x_0 ~,~ x_0 * x_i),\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>(x_j ~,~ x_j * x_i),\!</math>
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| − | | <math>\cdots\!</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
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| − | | width="4%" | <math>\{\!</math>
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| − | | width="18%" | <math>\cdots\!</math>
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| − | | width="22%" | <math>\cdots\!</math>
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| − | | width="22%" | <math>\cdots\!</math>
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| − | | width="18%" | <math>\cdots\!</math>
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| − | | width="4%" | <math>\}\!</math>
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| − | |}
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| − | | |
| − | <br>
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| − | | |
| − | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
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| − | |+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
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| − | |- style="height:50px"
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| − | | width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
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| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>
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| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>
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| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>
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| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{e}</math>
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| − | | <math>\operatorname{e}</math>
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| − | | <math>\operatorname{f}</math>
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| − | | <math>\operatorname{g}</math>
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| − | | <math>\operatorname{h}</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{f}</math>
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| − | | <math>\operatorname{f}</math>
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| − | | <math>\operatorname{e}</math>
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| − | | <math>\operatorname{h}</math>
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| − | | <math>\operatorname{g}</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{g}</math>
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| − | | <math>\operatorname{g}</math>
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| − | | <math>\operatorname{h}</math>
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| − | | <math>\operatorname{e}</math>
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| − | | <math>\operatorname{f}</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{h}</math>
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| − | | <math>\operatorname{h}</math>
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| − | | <math>\operatorname{g}</math>
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| − | | <math>\operatorname{f}</math>
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| − | | <math>\operatorname{e}</math>
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| − | |}
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| − | | |
| − | <br>
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| − | | |
| − | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
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| − | |+ <math>\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4</math>
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| − | |- style="height:50px"
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| − | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
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| − | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
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| − | |- style="height:50px"
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| − | | width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
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| − | | width="4%" | <math>\{\!</math>
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| − | | width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math>
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| − | | width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math>
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| − | | width="20%" | <math>(\operatorname{g}, \operatorname{g}),</math>
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| − | | width="16%" | <math>(\operatorname{h}, \operatorname{h})</math>
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| − | | width="4%" | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{f}</math>
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| − | | <math>\{\!</math>
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| − | | <math>(\operatorname{e}, \operatorname{f}),</math>
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| − | | <math>(\operatorname{f}, \operatorname{e}),</math>
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| − | | <math>(\operatorname{g}, \operatorname{h}),</math>
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| − | | <math>(\operatorname{h}, \operatorname{g})</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{g}</math>
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| − | | <math>\{\!</math>
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| − | | <math>(\operatorname{e}, \operatorname{g}),</math>
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| − | | <math>(\operatorname{f}, \operatorname{h}),</math>
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| − | | <math>(\operatorname{g}, \operatorname{e}),</math>
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| − | | <math>(\operatorname{h}, \operatorname{f})</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{h}</math>
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| − | | <math>\{\!</math>
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| − | | <math>(\operatorname{e}, \operatorname{h}),</math>
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| − | | <math>(\operatorname{f}, \operatorname{g}),</math>
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| − | | <math>(\operatorname{g}, \operatorname{f}),</math>
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| − | | <math>(\operatorname{h}, \operatorname{e})</math>
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| − | | <math>\}\!</math>
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| − | |}
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| − | | |
| − | <br>
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| − | | |
| − | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
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| − | |+ <math>\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4</math>
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| − | |- style="height:50px"
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| − | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
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| − | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>
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| − | |- style="height:50px"
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| − | | width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
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| − | | width="4%" | <math>\{\!</math>
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| − | | width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
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| − | | width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
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| − | | width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
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| − | | width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})</math>
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| − | | width="4%" | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{f}</math>
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| − | | <math>\{\!</math>
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| − | | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
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| − | | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
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| − | | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
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| − | | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{g}</math>
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| − | | <math>\{\!</math>
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| − | | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
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| − | | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
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| − | | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
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| − | | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{h}</math>
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| − | | <math>\{\!</math>
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| − | | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
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| − | | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
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| − | | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
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| − | | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})</math>
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| − | | <math>\}\!</math>
| |
| − | |}
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| − | | |
| − | <br>
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| − | | |
| − | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
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| − | |+ <math>\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)</math>
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| − | |- style="height:50px"
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| − | | width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
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| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
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| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{a}</math>
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| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{b}</math>
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| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{c}</math>
| |
| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{1}</math>
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| − | | <math>\operatorname{1}</math>
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| − | | <math>\operatorname{a}</math>
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| − | | <math>\operatorname{b}</math>
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| − | | <math>\operatorname{c}</math>
| |
| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{a}</math>
| |
| − | | <math>\operatorname{a}</math>
| |
| − | | <math>\operatorname{b}</math>
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| − | | <math>\operatorname{c}</math>
| |
| − | | <math>\operatorname{1}</math>
| |
| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{b}</math>
| |
| − | | <math>\operatorname{b}</math>
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| − | | <math>\operatorname{c}</math>
| |
| − | | <math>\operatorname{1}</math>
| |
| − | | <math>\operatorname{a}</math>
| |
| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{c}</math>
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| − | | <math>\operatorname{c}</math>
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| − | | <math>\operatorname{1}</math>
| |
| − | | <math>\operatorname{a}</math>
| |
| − | | <math>\operatorname{b}</math>
| |
| − | |}
| |
| − | | |
| − | <br>
| |
| − | | |
| − | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
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| − | |+ <math>\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</math>
| |
| − | |- style="height:50px"
| |
| − | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
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| − | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
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| − | |- style="height:50px"
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| − | | width="20%" style="border-right:1px solid black" | <math>\operatorname{1}</math>
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| − | | width="4%" | <math>\{\!</math>
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| − | | width="16%" | <math>(\operatorname{1}, \operatorname{1}),</math>
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| − | | width="20%" | <math>(\operatorname{a}, \operatorname{a}),</math>
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| − | | width="20%" | <math>(\operatorname{b}, \operatorname{b}),</math>
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| − | | width="16%" | <math>(\operatorname{c}, \operatorname{c})</math>
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| − | | width="4%" | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{a}</math>
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| − | | <math>\{\!</math>
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| − | | <math>(\operatorname{1}, \operatorname{a}),</math>
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| − | | <math>(\operatorname{a}, \operatorname{b}),</math>
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| − | | <math>(\operatorname{b}, \operatorname{c}),</math>
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| − | | <math>(\operatorname{c}, \operatorname{1})</math>
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| − | | <math>\}\!</math>
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| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{b}</math>
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| − | | <math>\{\!</math>
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| − | | <math>(\operatorname{1}, \operatorname{b}),</math>
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| − | | <math>(\operatorname{a}, \operatorname{c}),</math>
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| − | | <math>(\operatorname{b}, \operatorname{1}),</math>
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| − | | <math>(\operatorname{c}, \operatorname{a})</math>
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| − | | <math>\}\!</math>
| |
| − | |- style="height:50px"
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| − | | style="border-right:1px solid black" | <math>\operatorname{c}</math>
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| − | | <math>\{\!</math>
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| − | | <math>(\operatorname{1}, \operatorname{c}),</math>
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| − | | <math>(\operatorname{a}, \operatorname{1}),</math>
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| − | | <math>(\operatorname{b}, \operatorname{a}),</math>
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| − | | <math>(\operatorname{c}, \operatorname{b})</math>
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| − | | <math>\}\!</math>
| |
| − | |}
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| − | | |
| − | <br>
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| − | | |
| − | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
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| − | |+ <math>\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math>
| |
| − | |- style="height:50px"
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| − | | width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math>
| |
| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}</math>
| |
| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
| |
| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}</math>
| |
| − | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}</math>
| |
| − | |- style="height:50px"
| |
| − | | style="border-right:1px solid black" | <math>\operatorname{0}</math>
| |
| − | | <math>\operatorname{0}</math>
| |
| − | | <math>\operatorname{1}</math>
| |
| − | | <math>\operatorname{2}</math>
| |
| − | | <math>\operatorname{3}</math>
| |
| − | |- style="height:50px"
| |
| − | | style="border-right:1px solid black" | <math>\operatorname{1}</math>
| |
| − | | <math>\operatorname{1}</math>
| |
| − | | <math>\operatorname{2}</math>
| |
| − | | <math>\operatorname{3}</math>
| |
| − | | <math>\operatorname{0}</math>
| |
| − | |- style="height:50px"
| |
| − | | style="border-right:1px solid black" | <math>\operatorname{2}</math>
| |
| − | | <math>\operatorname{2}</math>
| |
| − | | <math>\operatorname{3}</math>
| |
| − | | <math>\operatorname{0}</math>
| |
| − | | <math>\operatorname{1}</math>
| |
| − | |- style="height:50px"
| |
| − | | style="border-right:1px solid black" | <math>\operatorname{3}</math>
| |
| − | | <math>\operatorname{3}</math>
| |
| − | | <math>\operatorname{0}</math>
| |
| − | | <math>\operatorname{1}</math>
| |
| − | | <math>\operatorname{2}</math>
| |
| − | |}
| |
| − | | |
| − | <br>
| |
| − | | |
| − | <pre>
| |
| − | 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> }
| |
| − | </pre>
| |
