Band (algebra)

In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square). Bands were first studied and named by A. H. Clifford (1954); the lattice of varieties of bands was described independently in the early 1970s by Biryukov, Fennemore and Gerhard.Semilattices, left-zero bands, right-zero bands, rectangular bands, normal bands, left-regular bands, right-regular bands and regular bands, specific subclasses of bands which lie near the bottom of this lattice, are of particular interest and are briefly described below.

A class of bands forms a variety if it is closed under formation of subsemigroups, homomorphic images and direct product. Each variety of bands can be defined by a single defining identity.

Semilattices are exactly commutative bands; that is, they are the bands satisfying the equation

A left zero band is a band satisfying the equation

whence its Cayley table has constant rows.

Symmetrically, a right zero band is one satisfying

so that the Cayley table has constant columns.

A rectangular band is a band S that satisfies

or equivalently,

The second characterization clearly implies the first, and conversely the first implies xyz = xy(zxz) = (x(yz)x)z = xz.

There is a complete classification of rectangular bands. Given arbitrary sets I and J one can define a semigroup operation on I × J by setting

The resulting semigroup is a rectangular band because

In fact, any rectangular band is isomorphic to one of the above form. Left zero and right zero bands are rectangular bands, and in fact every rectangular band is isomorphic to a direct product of a left zero band and a right zero band. All rectangular bands of prime order are zero bands, either left or right. A rectangular band is said to be purely rectangular if it is not a left or right zero band.

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