Band (algebra)

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In mathematics, a band is a semigroup in which every element is idempotent (in other words equal to its own square). The lattice of varieties of bands was described independently by Birjukov, Fennemore and Gerhard. Semilattices, left-zero bands, right-zero bands, rectangular 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. Bands have found applications in various branches of mathematics, notably in theoretical computer science.

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Semilattices are exactly commutative bands.

A rectangular band is a band S which satisfies

  • xyz = xz for all x, y, z \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 \times J by setting

(i, j) \cdot (k, l) = (i, l)

The resulting semigroup is a rectangular band because

  1. for any pair (i,j) we have (i, j) \cdot (i, j) = (i,j)
  2. for any three pairs \big (i_x, j_x), (i_y, j_y), (i_z, j_z) we have

(i_x, j_x) \cdot (i_y, j_y) \cdot (i_z, j_z) = (i_x, j_z) = (i_x, j_x) \cdot (i_z, j_z)

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

A left-zero band is a band satisfying xy = y. Symmetrically, a right-zero band is one satisfying xy = x. In particular right-zero and left-zero bands are rectangular bands and in fact every rectangular band is isomorphic to the direct product of a left-zero band and a right-zero band.

A regular band is a band S satisfying

  • xyxzx = xyzx for all x, y, z \in S

A class of bands forms a variety if it is closed under formation of subsemigroups, homomorphic images and direct product, and varieties of bands naturally form a lattice. It can be shown that this lattice is countable because each variety of bands can be defined by a finite set of defining identities. The varieties of semilattices, right-zero and left-zero bands are the three non-trivial minimal elements of this lattice.

  • Clifford, Alfred Hoblitzelle; Preston, Gordon Bamford (1972). The Algebraic Theory of Semigroups, Russian translation, Moskva: Mir. 
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