In mathematics, a rectangular band is a particular type of semigroup. A semigroup S is a rectangular band if

1. each of its elements is an idempotent (meaning that S is a band);
2. xyz = xz for all $x,\; y,\; z\; \backslash in\; S$ (a property sometimes known as the rectangular property).

For example, given arbitrary non-empty sets I and J one can define a semigroup operation on $I\; \backslash times\; J$ by setting

$(i,\; j)\; \backslash cdot\; (k,\; l)\; =\; (i,\; l)$

The resulting semigroup is a rectangular band because

1. for any pair (i,j) we have $(i,\; j)\; \backslash cdot\; (i,\; j)\; =\; (i,j)$
2. for any three pairs $\backslash big\; (i\_x,\; j\_x),\; (i\_y,\; j\_y),\; (i\_z,\; j\_z)$ we have

In fact, any rectangular band is isomorphic to one of the above form.

Every rectangular band is also isomorphic to the direct product of a left-zero band and a right-zero band.

Abstract algebra | Semigroup theory