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I'm wondering if I can "make up" an identity element, like so:

I can define an element I such that any element x + I is equal to x, i.e.: I can redefine my set as [the old set] union with {I}, and redefine my binary operation such that if one of the operands is I, then the result is the other operand. Then I will have an identity element.

For example, imagine the set of integers greater than 100 and addition; this forms a semigroup but has no identity. I can define a new element I (it doesn't have any "numerical value" but it doesn't need to). Then I say 100 + I = 100, 101 + I = 101, etc. Of course, I wouldn't be able to do multiplication or take the square root of I, but those operations aren't a part of my semigroup anyways.

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  • $\begingroup$ Note that the fact this is true is part of the formalized math library that is Lean's mathlib. The result in question is with_zero.add_monoid, so named because the adjoined element "I" acts like a 0. $\endgroup$
    – Eric
    May 9 at 13:05
  • $\begingroup$ @Eric: Zero is different from identity element. $\endgroup$
    – spin
    May 9 at 13:57
  • $\begingroup$ If you're writing your semigroup operation as $+$ as this question does, then all I'm saying is it can be convenient to write your monoid identity as $0$. $\endgroup$
    – Eric
    May 9 at 14:15

2 Answers 2

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Yes, this construction works, and is the standard way to embed a semigroup without identity into a monoid.

Given a semigroup $S$ without identity, let $u \not\in S$. Then you can define a binary operation on $S \cup \{u\}$ by extending the binary operation on $S$: define $ux = xu = x$ for all $x \in S \cup \{u\}$.

To show that this makes $S \cup \{u\}$ into a monoid with identity $u$, you just need to prove that the binary operation on $S \cup \{u\}$ is associative. That should be an easy exercise.

It might also be instructive to think about what happens if you do this construction in the case where $S$ already has an identity.

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Yes, you can always do it!

Suppose $(S, .)$ is a semigroup (i.e., $.$ an associative binary operation). Then consider the set $S'=S\cup\{*\}$ together with the binary operation $.'$ satisfying

  • $a.'b=a.b$ for all $a, b\in S$
  • $a.'*=*.'a=a$ for all $a\in S$
  • $*.'*=*$

Now, it is an easy exercise to prove $(S', .')$ is a monoid, and the obvious inclusion mapping $\iota : (S, .)\to (S', .')$ with $\iota(a.b)=a.'b$ is a semigroup homomorphism.

Also, you can go a bit further and notice this semigroup homomorphism is universal in the sense that any semigroup homomorphism $\varphi: S\to M$ to a monoid $M$, there is a unique monoid homomorphism $\varphi': S'\to M$ such that $\varphi=\varphi'\iota.$ Pictorially,

enter image description here

The natural mathematical domain that discusses this type of construction is "category theory", but I will not go into details.

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