San José State University |
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
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Concerning Monoids |
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A monoid is an associative binary function defined over a set which has a left identity element. Semigroups are the only simpler algebraic structure, They have no identity element.
If f(x, y) is associative then
Lemma: Function composition is associative.
Definition. A function monoid is a set of functions defined over a set which is closed under function composition and includes the identity function,
Proof:
Let M be a monoid over the set S and f:S×S→S its binary associative function with e its left identity element.
For each element a of S create the function g_{a}(x)=f(a, x). The set G of such functions is at least a semigroup with respect to function composition. The left identity element of G is the function g_{e}(x)=f(e, x). Therefore G is a monoid.
It is isomorphic to M under the two-way mapping
because if f(a, b)=c then it also holds that g_{a}°g_{b}=g_{c} and vice versa. The composition g_{a}°g_{b} is, in terms of their definitions,
This theorem can be considered as Cayley's Theorem for monoids.
Any semigroup may be converted into a monoid by adjoining a left identity element to its set. Let S be any semigroup and Z be its set. Let the binary function for S be denoted as f( , ); i.e., f:Z×Z→Z. Then let e be an element not in Z. Create Z* as Z∪{e} and define f(e, z)=z for all z in Z*.
(To be continued.)
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