nLab band

Definition

A band is a semigroup in which every element is idempotent.

Order structures on bands

There are two distinct order structures on bands, referred to in the literature by the “natural poset” structure and the “natural preorder” structure.

The natural preorder structure is defined by declaring $x \preceq y$ iff $x = x y x$. This is indeed a preorder by the following result.

Theorem

The relation $\preceq$ is reflexive and transitive.

Proof

(This proof appears in a relevant MathStackExchange discussion.) Reflexivity is obvious from $x = x x = x x x$. Transitivity follows from the following argument: let $A$ denote an instance of associativity and $I$ an instance of idempotency. From $x y x = x$ (condition 1) and $y z y = y$ (condition 2), we have

$$x =_1 x y x =_2 x y z y x =_I x y(z y x)(z y x) =_A x(y z y)x z y x =_2 x y x z y x =_1 x z y x\qquad(P).$$

Then

$$x z x =_P x z(x z y x) =_A (x z)(x z)y x =_I x z y x =_P x$$

which completes the proof.

As is the case with any preorder structure, there is an associated equivalence relation $\sim$, where $x \sim y$ iff the two conditions $x \preceq y$ and $y \preceq x$ hold.

Proposition

Every $\sim$-equivalence class is a rectangular band.

Proof

Let $[x]$ denote the equivalence class of $x$. If $y, z \in [x]$, then $x y x = x$, and also $y z y = y$ (since $y \sim z$), so by equation $P$ above, $x z y x = x$. By the same token, since $x, y, z$ are all equivalent, we can equally well say $z y x z = z$, whence $(z y)x(z y) = z y$. From these two conclusions, it follows that $z y$ is also in $[x]$, so that each equivalence class $[x]$ is closed under multiplication, and therefore is a band.

The rectangularity condition is immediate by definition: for any $\sim$-equivalent elements $x, y$, we have $x = x y x$.

Examples of bands

• A semilattice (in this article, we mean semilattices without identity) is the same thing as a commutative band.

• An idempotent monoid is a unital band.

• A rectangular band is a band (see there).

• A skew lattice $(L, \wedge, \vee)$ is a band, using either $\wedge$ or $\vee$ as the semigroup operation.

• A Boolean ring is a ring whose multiplication operation is a band.

• Similarly, a multiplicatively idempotent semiring is a semiring whose multiplication operation is a band.

• An idempotent semiring is a semiring whose addition operation is a band.

To be added: varieties of bands, and particularly of regular bands.

• A split band is defined in definition 1.1 of McAlister & Blyth 1978.

Relation to semilattices

The category of semilattices is a full subcategory of the category of bands; by general categorical considerations, the full inclusion has a left adjoint, called the reflector from bands to semilattices.

The following result, due to Clifford and Preston, is fundamental.

Theorem

Given a band $B$, the equivalence relation induced from the natural preorder, where $x \sim y$ iff $(x \preceq y) \wedge (y \preceq x)$, is a band congruence. The quotient $B/\sim$ is a semilattice, and $B \to B/\sim$ is universal among all maps from $B$ to semilattices, thus defining the reflector $B \mapsto B/\sim$ from bands to semilattices.

Properties of bands

By a theorem of McLean, finitely generated bands are finite; see Howie 76, Section IV.4.

Relation to rectangular bands

• Every rectangular band is a band since the defining identity implies $x y z = x z$ for all $x,y,z$ whence by taking $y=z=x$ one gets $x x x = x x$ and from the defining identity $x x x=x$ hence $x x = x$. In order to get the first equation expand $x y z$ by substituting $x z x$ for $x$: $x y z = (x z x) y z = x (z (xy) z) = x z \; .$

• Every band $S$ has a decomposition as a disjoint union $\coprod_{x\in L} R_x$ where $L$ is a semilattice, each $R_x$ is a sub-semigroup that is a rectangular band, and $R_x R_y \subseteq R_{x y}$ for every $x$ and $y$. This is a bit weaker than saying we have a functor from the poset $L$ to the category of rectangular bands, because we lack connecting morphisms $R_x \to R_y$.

Regular bands

A band $S$ satisfying the graphic identity $x y x = x y$ for all $x$ and $y$ is said to be left-regular. Left-regular bands can arise from hyperplane arrangements and there has been work studying random walks? on these hyperplane arrangements by analysing the semigroup algebras of the associated bands: see Brown 00 and Margolis-Saliola-Steinberg 15. Left-regular band monoids are also called graphic monoids which are examples of 1-object graphic categories.

Skeletons and split bands

For the definition of a skeleton of a band, see definition 1.2 of McAlister & Blyth 1978 for the time being…

But according to lemma 1.3 of McAlister & Blyth 1978, a band is split if and only if it has a skeleton.

Related concepts

• semigroup

• semilattice

• idempotent monoid

• Boolean ring

References

• David McLean, Idempotent Semigroups, The American Mathematical Monthly, Vol. 61, No. 2 (February 1954), pp. 110-113.

• J. Howie, An introduction to semigroup theory, Academic Press 1976.

• K. S. Brown, Semigroups, Semirings, and Markov Chains, J. Theor. Prob. 13 no.3 (2000) pp.871-938. (arXiv:math/0006145)

• Stuart Margolis, Franco Saliola, Benjamin Steinberg, Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry (arXiv:1508.05446)

Split bands and the skeleton of a band are defined in:

• Donald McAlister, Tom Blyth, Split orthodox semigroups, Journal of Algebra, Volume 51, Issue 2, April 1978, Pages 491-525, [doi:10.1016/0021-8693(78)90118-7]]

• Kaiqing Huang, Yizhi Chen, Aiping Gan, Structure of split additively orthodox semirings, AIMS Mathematics, 2022, 7(6): 11345-11361. [doi: 10.3934/math.2022633, pdf]

See also

• Wikipedia, Band (algebra)