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 are specific subclasses of bands that lie near the bottom of this lattice and which are of particular interest; they are briefly described below.
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Varieties of bands
A class of bands forms a variety if it is closed under formation of subsemigroups, homomorphic images and direct products. Each variety of bands can be defined by a single defining identity.
Semilattices
Semilattices are exactly commutative bands; that is, they are the bands satisfying the equation
xy = yx for all x and y.
Bands induce a preorder that may be defined as
x
≤
y
{\displaystyle x\leq y}
if
x
y
=
x
{\displaystyle xy=x}
. Requiring commutativity implies that this preorder becomes a (semilattice) partial order.
Zero bands
A left-zero band is a band satisfying the equation
xy = x,
whence its Cayley table has constant rows.
Symmetrically, a right-zero band is one satisfying
xy = y,
so that the Cayley table has constant columns.
Rectangular bands
A rectangular band is a band S that satisfies
xyx = x for all x, y ∈ S, or equivalently,
xyz = xz for all x, y, z ∈ S,
In any semigroup, the first identity is sufficient to characterize a nowhere commutative semigroup, the proof of this follows.
Let a semigroup be nowhere commutative. In any flexible magma
(
a
a
)
a
=
a
(
a
a
)
{\displaystyle (aa)a=a(aa)}
so every element commutes with its square. So in any nowhere commutative semigroup every element is idempotent which means it is a band. Thus in any nowhere commutative semigroup
x
(
x
y
x
)
=
(
x
x
Normal bands
A normal band is a band S satisfying
zxyz = zyxz for all x, y, and z ∈ S.
We can also say a normal band is a band S satisfying
axyb = ayxb for all a, b, x, and y ∈ S.
This is the same equation used to define medial magmas, 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 S satisfying
xyx = xy for all x, y ∈ S
If we take a semigroup and define a ≤ b if ab = b, we obtain a partial ordering 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 S satisfying
xyx = yx for all x, y ∈ S
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 S satisfying
zxzyz = zxyz for all x, y, z ∈ S
Lattice of varieties
When partially ordered by inclusion, varieties of bands naturally form a lattice, 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, 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.