rectangular band




idempotent semigroup


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 LL in this semigroup-theoretic sense, it has a canonical partial order given by eprecreqfe\precreq f when ef=ee 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 aba=aa b a = a for all elements aa and bb.


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

  • If SS is a rectangular band, then there exist non-empty sets II and JJ such that SS is isomorphic as a semigroup to I×JI\times J equipped with the multiplication (i,j)(p,q)=(i,q)(i, j)(p,q) = (i,q) for i,pIi,p\in I and j,qJj,q\in J.

  • Every band SS has a decomposition as a disjoint union xLR x\coprod_{x\in L} R_x where LL is a semilattice, each R xR_x is a sub-semigroup that is a rectangular band, and R xR yR xyR_x R_y \subseteq R_{x y} for every xx and yy. This is a bit weaker than saying we have a functor from the poset LL to the category of rectangular bands, because we lack connecting morphisms R xR yR_x \to R_y.

The category of rectangular bands

Let RectRect 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, RectRect is a subvariety of the variety SGrSGr of semigroups and hence enjoys all the usual (co)completeness properties of a variety.

  • Since RectRect is the 2-valued collapse? of the topos Set×SetSet\times Set it is even a cartesian closed variety. Since the distributive law holds for (finite) coproducts in cartesian closed categories, RectRect 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 SS satisfying the graphic identity xyx=xyx y x = x y for all xx and yy 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.


  • 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)

Last revised on January 9, 2021 at 10:44:54. See the history of this page for a list of all contributions to it.