Free semigroup generated by

Stub grade: A

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This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Needs fleshing out, it's like page 6 of the first reference, demote to grade D once satisfactory.

Note: the free monoid generated by a set is also a semigroup (as all monoids are semgroups), however there is a difference, see Discussion of the free monoid and free semigroup generated by a set

Definition

Given a set, [ilmath]X[/ilmath], there is a free semigroup, [ilmath](F,*)[/ilmath], generated by that set (that is distinct from the free monoid generated by (which of course is also a semigroup) - see discussion of the free monoid and free semigroup generated by a set)^[1], defined as follows:

The elements of [ilmath]F[/ilmath] are non-empty tuples of elements of [ilmath]X[/ilmath], [ilmath](x_1,\ldots,x_n)\in F[/ilmath] for [ilmath]n\ge 1[/ilmath]
The operation [ilmath]*::F\times F\rightarrow F[/ilmath] is concatenation of the tuples:
- [ilmath]*:((x_1,\ldots,x_m),(y_1,\ldots,y_n))\mapsto(x_1,\ldots,x_m,y_1,\ldots,y_n)[/ilmath]

Be sure to mention word terminology here, this page should be pretty close to free monoid generated by with the obvious differences.