Are there any interesting and natural examples of semigroups that

petrusrexcs

petrusrexcs

Answered question

2021-12-30

Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)?

Answer & Explanation

Kayla Kline

Kayla Kline

Beginner2021-12-31Added 37 answers

Step 1
Consider the following definition of a group:
Definition A semigroup S is said to be a group if the following hold:
1. There is an eS such that ea=a for all aS
2. For each aS there is an element a1S with a1a=e
At one point in my life, it seemed natural to ask what happens if we replace axiom 2 with the very similar axiom
2'. For each aS there is an element a1S with aa1=e.
It is a fun exercise to work out some of the consequences that result from this. Here are a few facts about a semigroup S which satisfies 1 and 2':
If e is the unique element of S satisfying axiom 1, then S is a group
If S has an identity (in the usual sense) then S is a group
The principal left ideal Sa={sasS} is a group for all aS, and in fact all principal left ideals of S are isomorphic as groups.
It is not difficult to find examples of such semigroups that are not groups. For example, consider the following set of 2×2 matrices (with matrix multiplication as the operation):
{(ab00)|a, bR, a0}
Or, an example that appears as exercise 30 in section 4 of Fraleigh's abstract algebra text: the nonzero real numbers under the operation × defined by a×b=|a|b.
Certainly it is debatable whether or not semigroups satisfying axioms 1 and 2′ are "interesting" or "natural". But I guess I think they are. And, I am not the only one (or the first one, by a long shot!) to think this. See Mann, On certain systems which are almost groups (MR).
lovagwb

lovagwb

Beginner2022-01-01Added 50 answers

Step 1
Let (M, , e) be a monoid and let M be the set of finite words constructed from M. For two words let's define operation × as point-wise application of , truncating according to the shorter word:
(u1,,um)×(v1,,vn)=(u1v1,,umin(m, n)vmin(m, n))
Then (M, ×) is a semigroup, but it's not a monoid. Any candidate y for the unit element would have only a finite length, so for any x that is longer we'd have y×xx. The unit element would have to be an infinite sequence (e, e,), but M
This example is not arbitrary, it is closely related to zipping lists and convolution.
Step 2
Update: A very simple example of a semigroup that is not a monoid is (Z, min). While min is clearly associative, there is no single element in Z that would serve as the identity. (It is actually a homomorphic image of the previous example, mapping words to their length.)
karton

karton

Expert2022-01-09Added 613 answers

I don't know if this counts as interesting, but a simple example is what C programmers know as the comma operator: Ignore the first argument and return the second.
Or written as multiplication rule: ab=b for all a, bS.
Or written as multiplication rule: ab=b for all a, bS.. This is easily shown to be a semigroup, but as long as there are at least two elements in S, this is not a monoid, as with neutral element e you'd have for all aS the identity a=e.

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