In
mathematics, a band (also called idempotent semigroup) is a
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
in which every element is
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
(in other words equal to its own square). Bands were first studied and named by ; the
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
of
varieties
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
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 that lie near the bottom of this lattice, are of particular interest and are briefly described below.
Varieties of bands
A class of bands forms a
variety
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
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
Semilattices are exactly
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
bands; that is, they are the bands satisfying the equation
* for all and .
Bands induce a preorder that may be defined as
if and only if
. Requiring commutativity implies that this preorder becomes a (semilattice) partial order.
Zero bands
A left-zero band is a band satisfying the equation
* ,
whence its
Cayley table Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplic ...
has constant rows.
Symmetrically, a right-zero band is one satisfying
* ,
so that the Cayley table has constant columns.
Rectangular bands
A rectangular band is a band ' that satisfies
# for all , or equivalently,
# for all ,
In any semigroup the first identity is sufficient to characterize a
Nowhere commutative semigroup.
Nowhere commutative semigroup implies the first identity.
In any
flexible
Flexible may refer to:
Science and technology
* Power cord, a flexible electrical cable.
** Flexible cable, an Electrical cable as used on electrical appliances
* Flexible electronics
* Flexible response
* Flexible-fuel vehicle
* Flexible rake re ...
magma
Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural sa ...
so every element commutes with its square. So in any Nowhere commutative semigroup every element is idempotent thus any Nowhere commutative semigroup is in fact a Nowhere commutative band.
Thus in any Nowhere commutative semigroup
::
So
commutes with
and thus
- the first characteristic identity.
In a any semigroup the first identity implies idempotence since
so
so idempotent (a band). Then
nowhere commutative since a band
So in a band
::
In any semigroup the first identity also implies the second because .
The idempotents of a rectangular semigroup form a sub band that is a rectangular band but a rectangular semigroup may have elements that are not idempotent. In a band the second identity obviously implies the first but that requires idempotence. There exist semigroups that satisfy the second identity but are not bands and do not satisfy the first.
There is a complete classification of rectangular bands. Given arbitrary sets ' and ' one can define a magma operation on by setting
:
This operation is associative because for any three pairs , , we have
::
and likewise
::
These two magma identities
:: and
:: are together equivalent to the second characteristic identity above.
The two together also imply associativity . Any magma that satisfies these two rectangular identities and idempotence is therefore a rectangular band. 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 we have so every element is idempotent and the first characeristic identity follows from the second together with idempotence.
But a magma that satisfies only the identities for the first characteristic and idempotence need not be associative so the second characteristic only follows from the first in a semigroup.
Any rectangular band is
isomorphic to one of the above form (either
is empty, or pick any element
, and then (
) defines an isomorphism
). 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-zero or right-zero band.
[.]
In
categorical language, one can say that the category of nonempty rectangular bands is
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
to
, where
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. If the set ' is empty in the above result, the rectangular band is independent of , and vice versa. This is why the above result only gives an equivalence between nonempty rectangular bands and pairs of nonempty sets.
Rectangular bands are also the -algebras, where is the
monad
Monad may refer to:
Philosophy
* Monad (philosophy), a term meaning "unit"
**Monism, the concept of "one essence" in the metaphysical and theological theory
** Monad (Gnosticism), the most primal aspect of God in Gnosticism
* ''Great Monad'', a ...
on Set with , ,
being the diagonal map
, and
.
Normal bands
A normal band is a band satisfying
* for all , , and .
We can also say a normal band is a band satisfying
* for all , , , and .
This is the same equation used to define
medial magma
In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that is, a set with a binary operation) which satisfies the identity
:(x \cdot y) \cdot (u \cdot v) = (x \cdot u) \cdot (y \cdot v), or more simply xy\cdot uv = xu\ ...
s, and so a normal band may also be called a medial band, and normal bands are examples of medial magmas.
Left-regular bands
A left-regular band is a band satisfying
* for all ',
If we take a semigroup and define if and only if , we obtain a
partial ordering
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
if and only if this semigroup is a left-regular band. Left-regular bands thus show up naturally in the study of
posets.
Right-regular bands
A right-regular band is a band satisfying
* for all
Any right-regular band becomes a left-regular band using the opposite product. Indeed, every variety of bands has an 'opposite' version; this gives rise to the reflection symmetry in the figure below.
Regular bands
A regular band is a band satisfying
* for all
Lattice of varieties
![Bands](https://upload.wikimedia.org/wikipedia/commons/f/f7/Bands.svg)
When
partially ordered
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
by inclusion, varieties of bands naturally form a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
, in which the meet of two varieties is their intersection and the join of two varieties is the smallest variety that contains both of them. The complete structure of this lattice is known; in particular, it is
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
,
complete, and
distributive.
[; ; ; .] 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.
See also
*
Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2.
Every Boolean ring gives rise to a Boolean al ...
, a ring in which every element is (multiplicatively) idempotent
*
Nowhere commutative semigroup
*
Special classes of semigroups
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consis ...
*
Orthodox semigroup In mathematics, an orthodox semigroup is a regular semigroup whose set of idempotents forms a subsemigroup. In more recent terminology, an orthodox semigroup is a regular ''E''-semigroup. The term ''orthodox semigroup'' was coined by T. E. Hall and ...
*
Notes
References
*.
*.
*.
*.
*.
*.
*.
*.
*.
*{{citation
, last = Yamada , first = Miyuki
, doi = 10.1007/BF02572956
, issue = 1
, journal =
Semigroup Forum
Semigroup Forum (print , electronic ) is a mathematics research journal published by Springer. The journal serves as a platform for the speedy and efficient transmission of information on current research in semigroup theory. Coverage in the jour ...
, pages = 160–167
, title = Note on exclusive semigroups
, volume = 3
, year = 1971.
Algebraic structures
Semigroup theory