In mathematics, a band is a semigroup in which every element is idempotent (in other words equal to its own square). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation The lattice of varieties of bands was described independently by Birjukov, Fennemore and Gerhard. In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities 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. Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such

Semilattices

Semilattices are exactly commutative bands. A semilattice is a mathematical concept with two definitions one as a type of Ordered set, the other as an Algebraic structure. In Mathematics, commutativity is the ability to change the order of something without changing the end result

Rectangular and zero bands

A rectangular band is a band S which satisfies

• xyx = x for all , equivalently xyz = xz.

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

The resulting semigroup is a rectangular band because

1. for any pair (i,j) we have
2. for any two pairs we have

In fact, any rectangular band is isomorphic to one of the above form. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

A left-zero band is a band satisfying xy = y, whence its Cayley table has constant columns. A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a Symmetrically, a right-zero band is one satisfying xy = x, constant rows. In particular right-zero and left-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, whence all rectangular bands of prime order are zero bands, either left or right.

Regular bands

A regular band is a band S satisfying

• xyxzx = xyzx for all

Lattice of varieties of bands

Figure 1. Lattice of varieties of regular bands.

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. In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' It can be shown that this lattice is countable because each variety of bands can be defined by a finite set of defining identities. In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities The lattice of the 13 varieties of regular bands are shown in Figure 1. The varieties of left-zero bands, semilattices, and right-zero bands are the three atoms (non-trivial minimal elements) of this lattice.

References

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

• Nagy, Attila (2001). Special Classes of Semigroups. Dordrecht: Kluwer Academic Publishers. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic ISBN 0792368908.