Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6

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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 "nth multiple" of x in a monoid X = <X, +, 0>, for integer n > 0, is defined the same way for n > 0, letting 0x = 0 when n = 0.

The "nth multiple" of x in a group X = <X, +, 0>, for any integer n, is defined the same way for n > 0, letting nx = ( n)( x) for n < 0.

A group X = <X, +, 0> is "cyclic" if and only if there is an element g C X such that every x C X can be written as x = ng for some n C Z. In this case, an element such as g is called a "generator" of the group.

Version 2

In a semigroup \(\underline{X} = (X, +, 0),\!\) 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 \(\underline{X} = (X, +, 0),\!\) \(nx\!\) is defined for every non-negative integer \(n\!\) by letting \(0x = 0\!\) and proceeding the same way for \(n > 0.\!\)

In a group \(\underline{X} = (X, +, 0),\!\) \(nx\!\) is defined for every integer \(n\!\) by letting \(nx = (-n)(-x)\!\) for \(n < 0\!\) and proceeding the same way for \(n \ge 0.\!\)

A group \(\underline{X} = (X, +, 0),\!\) is cyclic if and only if there is an element \(g \in X\!\) such that every \(x \in X\!\) can be written as \(x = ng\!\) for some \(n \in \mathbb{Z}.\!\) In this case, an element such as \(g\!\) is called a generator of the group.