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Binary Operations : Equivalence Classes

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Note. I don't how to make a letter with a line overtop of it so the equivalent notation here is *. ex) a* = a bar (a with a line overtop of it)

Let M be a commutative monoid. Define a relation ~ on M by a ~ b if a = bu for some unit u.
(a) Show that ~ is an equivalence on M and if a* deontes the equivalence class of a, let M* = {a*| a belongs to M} denote the set of all equivalence classes. Show that a*b* = (ab)* is a well-defined operation on M* deontes.
(b) If M* is as in (a), show that M* is a commutative monoid in which the identity 1* is the only unit.

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Proof.

(a) We want to show that a*b*=(ab)*. By the definition, a* is the equivalence class of a, and b* is the equivalence class of b. So,
for any element a' in a*, it can be written as a'=au for some unit u. Similarly, for any element b' in b*, ...

Solution Summary

Equivalence of a monoid is investigated. The commutative monoid identified is provided.

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