Here's a weirdo semigroup:

0 1 2 3 4 5
0| 5 5 5 2 0 3
1| 5 5 5 2 2 3
2| 5 5 5 2 2 3
3| 2 2 2 3 3 5
4| 2 1 2 3 4 5
5| 3 3 3 5 5 2

Besides associativity, it has the properties that:
* There is no left-identity or right-identity
* Neither a*_ nor _*a is ever a constant function (i.e., no row nor column is constant)
* The operator is surjective (no element fails to appear in the table)

I wrote a constraint solver to find such semigroups, and the curious thing is that if we want to limit the number of entries where a*a = a, then it appears that n-2 such constraints are satisfiable, but n-1 of them are not.

Relevance: https://blog.computationalcomplexity.org/2025/04/i-want-applicaion-of-this-application.html where there is a proof that at least one element must satisfy a*a = a, but Bill Gasarch asks for a "non-obvious" finite example. Is it true that all non-obvious examples must have _two_ such elements?

I realized about halfway through that this is probably the wrong way to go about it -- it would be better to generate rules instead of tables, and test those rules for associativity and non-obviousness. Then we could find if there's some "compactly described" semigroup with the desired properties.

#semigroup

🚀 just uploaded to https://fpilluminated.com

Folding Cheat Sheet #8 - Folding with Monoids.

39 slides - Twelve examples.

Direct link: https://fpilluminated.com/deck/240

#haskell #scala #monoid #semigroup #fold #foldMap #combineAll #cats #fp #functional_programming

The more I learn about quasigroups, the more I think they deserve the name in a way that semigroups don't.

Semigroups that faithfully embed into a group maybe are ok b/c they're "half the group" (e.g. positive reals inside the reals, under addition).

But not general semigroups. They need a new name.

#algebra #quasigroup #semigroup

We just received another publication from 2 other Ukrainian Research Fellows in the research area #algebra. 🔣 Click at the link below!

#Congruence #semigroup #FreeKNilpotent #NTuple
@univienna

https://www.tandfonline.com/doi/figure/10.1080/00927872.2023.2195000?scroll=top&needAccess=true&role=tab

New maintenance release for GAP v4.12.2

https://www.gap-system.org/

#computer #algebra #grouptheory #semigroup