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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 8th 2021

creating a stub entry, following the discussion here

• CommentRowNumber2.
• CommentAuthorGuest
• CommentTimeJan 8th 2021
This is Yemon Choi here; I've forgotten my password, or perhaps never created one after OpenID stopped working. I have tried to add some basic information from Howie's Introduction To Semigroup Theory, but the spam detector keeps blocking me.
• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeJan 9th 2021

@Yemon, can you paste the relevant text here, just so that it’s not lost? Experts can try to sort out the spam detector issue in your absence and maybe even insert your edit.

• CommentRowNumber4.
• CommentAuthorGuest
• CommentTimeJan 9th 2021
OK David - here goes. I didn't intend to write anything polished, just wanted to get the ball rolling and correct some terminology.

The reference here is J. Howie's book "An introduction to semigroup theory", Academic Press 1976.

A band is a semigroup in which every element is idempotent. Commutative bands are usually known as semilattices. This is the semigroup theoretic definition, but there is also an order theoretic definition: given a semilattice L in this semigroup-theoretic sense, it has a canonical partial order given by $e\precreq f$ when $ef=e$. So semilattices are also posets.

Finitely generated bands are finite: see Section IV.4 of Howie's book.

A rectangular band may be described as a semigroup satisfying the identity $aba=a$ for all elements $a$ and $b$. If $S$ is a rectangular band, then there exist non-empty sets $I$ and $J$ such that $S$ is semigroup-isomorphic to $I\times J$ equipped with the multiplication
$(i, j)(p,q) = (i,q)$ for $i,p\in I$ and $j,q\in J$.

Every band S has a decomposition as a disjoint union $\coprod_{x\in L} R_x$ where $L$ is a semilattice, each $R_x$ is a subsemigroup that is a rectangular band, and $R_x R_y \subseteq R_{xy}$ for every $x$ and $y$. This is a bit weaker than saying we have a functor from the poset L to the category of rectangular bands, because we lack connecing morphisms $R_x \to R_y$.

A band $S$ satisfying the identity $xyx=xy$ for all $x$ and $y$ is said to be left-regular. Left-regular bands can arise from hyperplane arrangements and there has been work studying random walks on these hyperplane arrangements by analysing the semigroup algebras of the associated bands: see https://arxiv.org/abs/math/0006145 and https://arxiv.org/abs/1508.05446
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 9th 2021
• (edited Jan 10th 2021)

I have pasted Yemon’s material into the entry, touching it slightly for formatting and hyperlinks.

(Apparently the spam filter gets triggered by lenghty additions of users without a history of making lenghty additions. If it happens again to you, you can try to add the material incrementally in smaller chunks.)

For the moment I made band a redirect to rectangular band. If you want to change that, just remove the redirect and create an entry for “band”.

• CommentRowNumber6.
• CommentAuthorThomas Holder
• CommentTimeJan 9th 2021

• CommentRowNumber7.
• CommentAuthorThomas Holder
• CommentTimeJan 9th 2021

Added some details and general properties of the variety.

• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeJan 9th 2021
• (edited Jan 10th 2021)

deleted