nLab rectangular band

Idea

… idempotent semigroup …

Definition

Recall that 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 $e f = e$. So semilattices are also posets.

Finitely generated bands are finite: see Howie 76, Section IV.4.

Now a rectangular band may be described as a semigroup satisfying the identity $a b a = a$ for all elements $a$ and $b$.

Properties

• A rectangular band is indeed a band since the defining identity implies $x y z = x z$ for all $x,y,z$ whence by taking $y=z=x$ one gets $x x x = x x$ and from the defining identity $x x x=x$ hence $x x = x$. In order to get the first equation expand $x y z$ by substituting $x z x$ for $x$: $x y z = (x z x) y z = x (z (xy) z) = x z \; .$

• If $S$ is a rectangular band, then there exist non-empty sets $I$ and $J$ such that $S$ is isomorphic as a semigroup 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 sub-semigroup that is a rectangular band, and $R_x R_y \subseteq R_{x y}$ 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 connecting morphisms $R_x \to R_y$.

The category of rectangular bands

Let $Rect$ be the category of rectangular bands with semigroup homomorphisms as morphisms.

• Since rectangular bands are an equationally defined subclass of the class of all semigroups, $Rect$ is a subvariety of the variety $SGr$ of semigroups and hence enjoys all the usual (co)completeness properties of a variety.

• Since $Rect$ is the 2-valued collapse? of the topos $Set\times Set$ it is even a cartesian closed variety. Since the distributive law holds for (finite) coproducts in cartesian closed categories, $Rect$ is a distributive category. Since it is not locally cartesian closed it is neither a topos nor even an extensive category. For more on this see Johnstone (1990).

Some ramifications

• A band $S$ satisfying the graphic identity $x y x = x y$ 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 Brown 00 and Margolis-Saliola-Steinberg 15. Left-regular band monoids are also called graphic monoids which are examples of 1-object graphic categories.

Related concepts

• distributive category

• reader monad

• graphic monoid

• collapsed topos?

References

• J. Howie, An introduction to semigroup theory, Academic Press 1976.

• K. S. Brown, Semigroups, Semirings, and Markov Chains, J. Theor. Prob. 13 no.3 (2000) pp.871-938. (arXiv:math/0006145)

• Peter Johnstone, Collapsed toposes and cartesian closed varieties , JPAA 129 (1990) pp.446-480.

• N. Kimura, The structure of idempotent semigroups I , Pacific Journal of Mathematics 8 no.2 (1958) pp.257-275. (pdf)

• Stuart Margolis, Franco Saliola, Benjamin Steinberg, Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry (arXiv:1508.05446)