nLab rectangular band

Contents

Definition

A rectangular band or nowhere commutative semigroup is a semigroup which satisfies any of these equivalent conditions

it is nowhere commutative in the sense that $a b = b a$ implies that $a = b$ for all $a$ and $b$ .
$a b a = a$ for all $a$ and $b$
$a a a = a$ and $a b c = a c$ for all $a$ , $b$ , and $c$

However, despite its name, there exist commutative nowhere commutative semigroups, such as the empty semigroup and the trivial group.

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 $R_x \to R_y$ .

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).

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