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.

A class of bands forms a variety if it is closed under formation of subsemigroups, homomorphic images and direct products.

Requiring commutativity implies that this preorder becomes a (semilattice) partial order.

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 In any semigroup, the first identity is sufficient to characterize a nowhere commutative semigroup, the proof of this follows.

So in any nowhere commutative semigroup every element is idempotent which means it is a band.

Now assume that the first identity holds in a semigroup.

In a band the second identity obviously implies the first but that requires idempotence.

There is a complete classification of rectangular bands.

Given arbitrary sets I and J one can define a magma operation on I × J by setting This operation is associative because for any three pairs (ix, jx), (iy, jy), (iz, jz) we have These two magma identities (xy)z = xz and x(yz) = xz are together equivalent to the second characteristic identity above.

The two together also imply associativity (xy)z =x(yz).

So any magma that satisfies both the characteristic identities (four separate magma identities) is a band and therefore a rectangular band.

The magma operation defined above is a rectangular band because for any pair (i, j) we have (i, j) · (i, j) = (i, j) so every element is idempotent and the first characteristic identity follows from the second together with idempotence.

[3] In categorical language, one can say that the category of nonempty rectangular bands is equivalent to

is the category with nonempty sets as objects and functions as morphisms.

This implies not only that every nonempty rectangular band is isomorphic to one coming from a pair of sets, but also these sets are uniquely determined up to a canonical isomorphism, and all homomorphisms between bands come from pairs of functions between sets.

[4] If the set I is empty in the above result, the rectangular band I × J is independent of J, and vice versa.

Rectangular bands are also the T-algebras, where T is the monad on Set with T(X)=X×X, T(f)=f×f,

Left-regular bands thus show up naturally in the study of posets.

Indeed, every variety of bands has an 'opposite' version; this gives rise to the reflection symmetry in the figure below.

[1] The sublattice consisting of the 13 varieties of regular bands is shown in the figure.

The varieties of left-zero bands, semilattices, and right-zero bands are the three atoms (non-trivial minimal elements) of this lattice.

Each variety of bands shown in the figure is defined by just one identity.

This is not a coincidence: in fact, every variety of bands can be defined by a single identity.