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@slava@mathstodon.xyz
Post #1827202
2026-05-01 01:28 UTC
A funny little monoid: ⟨a, b | ba=ab, aaa=bb⟩
It is commutative, because ba=ab. It is cancellative, because ax=ay or bx=by implies x=y, for all x, y.
A cancellative commutative monoid embeds in a group with the same presentation, and the group you get is Z, so this is a submonoid of the integers.
But which submonoid? One possible embedding sends ‘a’ to 2 and ‘b’ to 3, so this monoid is the set of all non-negative linear combinations of 2 and 3.
What numbers can you get this way? Well, our submonoid must include the identity, and indeed 0 = 0*2 + 0*3. Also, every positive even integer can be written as a sum of 2’s, while every odd integer greater than or equal to 3 can be written as the sum of 3 and some non-negative even integer. However, you can’t reach any negative integers in this way at all.
Our monoid is isomorphic to the set of natural numbers but without 1, with the operation of addition. This is called a _numerical semigroup_ generated by 2 and 3. Quite simple in this case, but numerical semigroups can be rather complex, even though they’re all “just” submonoids of N.
Check out the Cayley graph of ⟨a, b | ba=ab, aaa=bb⟩: https://monoids.net/2,2/2262.html
Replies (3)
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@slava@mathstodon.xyz @joe@f.duriansoftware.com I like the "Words To Reduce" form:
abababa ⇒ a2b2aba ⇒ a2bab2a ⇒ a3b3a ⇒ b5a ⇒ b4ab ⇒ b3ab2 ⇒ b2ab3 ⇒ bab4 ⇒ ab5
Can't say I understand what a monoid is, but seeing the term-rewriting steps are fun.
Open ##1831685
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@slava In "Monoids of natural Numbers" Steven Finch says:
"A numerical monoid \(S\) is a subset of \(\mathbf{N}\) that is closed under addition, contains 0, and has a finite complement in \(\mathbf{N}\). (Most authors use the phrase ``numerical semigroup''' but semigroups by defintion need not contain 0, hence the usage is puzzling.)"
https://studylib.net/doc/7964892/monoids-of-natural-numbers-steven-finch-march-17--2009-let-n
Open ##1844254
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@slava This is also related to the braid group on three strands, B₃ = .
Open ##1844255