View source for Free semigroup generated by

{{Stub page|grade=A|msg=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]]
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==Definition==
Given a [[set]], {{M|X}}, there is a ''free [[semigroup]]'', {{M|(F,*)}}, 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]]){{rAAPAG}}, defined as follows:
* The elements of {{M|F}} are ''non-empty'' [[tuple|tuples]] of elements of {{M|X}}, {{M|(x_1,\ldots,x_n)\in F}} for {{M|n\ge 1}}
* The operation {{M|*::F\times F\rightarrow F}} is concatenation of the tuples:
** {{M|*:((x_1,\ldots,x_m),(y_1,\ldots,y_n))\mapsto(x_1,\ldots,x_m,y_1,\ldots,y_n)}}
==Terminology==
* Be sure to mention word terminology here, this page should be pretty close to [[free monoid generated by]] with the obvious differences.
==References==
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{{Definition|Abstract Algebra||Semigroup theory}}