Bands

Background

** ** A semigroup $(S, \cdot)$ is a set $S$ together with an associative binary operation $\cdot$ on $S$ . If a semigroup has an identity element, then it is called a monoid. Throughtout this page, we let $S^1$ denote the monoid coming from $S$ by adding an identity if one doesn't already exist.

How can we verify if a pair $\left( S, \cdot \right)$ is a semigroup? There are two things we need to check for:

Closure (so if $a, b \in S$ then $a\cdot b \in S$ .)
Associativity (i.e. $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ ).

To test for associativity, we can use Light's associativity test. Light's associativity test might seem complicated, but in essence it's trying to do associativity using matrices to make life "easier". Since we're trying to see if $(a\cdot b) \cdot c = a \cdot (b \cdot c)$ for every $a, b, c$ , what we can do is look at the middle element and see what happens. We start off by creating a matrix to record a new multiplication table for our middle element $b$ . We look at $a \cdot b$ , which turns out to just be the $b$ -column in our original multiplication table. This column we record on the left-hand column of our new multiplication table (step 1 below). To record $(a \cdot b) \cdot c$ , we copy down each row from our original multiplication table to our new one, with respect to the label on the left-hand column (step 2 below). The entries in our new multiplication table are now $(a \cdot b) \cdot c$ . To see whether this is equal to $a \cdot (b \cdot c)$ , we must do the "opposite" direction. Since $b$ appears first, we look at the row of $b$ in our original table and copy it to the top of our new multplication table (step 3 below). Finally, for each column in our new table we must compare each entry to make sure they line up, in other words that $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ (step 4 below).

Light's associativity test Let $\left( T \right)$ be the multiplication table of $(S, \cdot)$ . Let $a\in S$ be a generator. Construct a new table as follows:

Write the column of $a$ on the left.
For each entry $t = x \cdot a$ in the column of $a$ , the row of $t$ in the new table is the row of $t$ in $(T)$ .
Write the row of $a$ at the top.
For each entry $u = ay$ in the row of $a$ , the column of $u$ in the new table should equal the column of $u$ in $(T)$ .

If the above is true for every generator, then $(S, \cdot)$ is associative.

This is a little confusing, so let's look at some examples.

Let $S$ be the semigroup with multiplication table as below: $\text{[math]}$ Note that $d$ is our only generator since: $\text{[math]}$

Following each step, we have:

Write the column of $d$ on the left: $\text{[math]}$
For each entry $t = x \cdot d$ in the column of $d$ , the row of $t$ in the new table is the row of $t$ in the original table: $\text{[math]}$
Write the row of $d$ (from the original table) at the top. $\text{[math]}$
Verify for each entry $u = d \cdot y$ in the row of $d$ , the column of $u$ in the new table should eqaul the column of $u$ in the original table. Looking column by column, they all match up!

Let's look at a non-example to see where this might fail. Let $S = \left\{ a, b \right\}$ be our set with generator $a$ and multiplication table: $\text{[math]}$

When we construct our new table we end up getting the following: $\text{[math]}$

But notice that the $b$ column is wrong! What this basically tells us is that our operation is not associative. In fact, we can verify this: $(a \cdot b) \cdot a = a \cdot a = b \neq a = a \cdot b = a \cdot (b \cdot a)$ .

Relations and Orders

Definition: Green's Preorders are given by: $\text{[math]}$

Definition: Green's Relations (introduced by Green in [Gre51]) on a semigroup $S$ are the equivalence relations below.

$\text{[math]}$

Definition: We say that a finite monoid $S$ is weakly ordered if there is a finite join-semilattice $\left( L, \leq \right)$ together with the maps $C, D: S \to L$ such that:

$C$ is a monoid morphism, i.e. $C(uv) = C(u) \join C(v) \quad \forall u,v \in S$ .
$C$ is a surjection.
If $u,v \in S$ are such that $u \leq_{\msR} uv$ then $C(v) \leq D(u)$ .
If $u,v \in S$ are such that $C(v) \leq D(u)$ then $uv = u$ .

This definition was first defined in [Sch08].

The idea of weakly ordered is coming from the $0$ -Hecke monoid with (simple) reflections $T_i$ . In this case, the map $C$ is nothing more than the content/support map, i.e. $C(T_{i_1}\cdots T_{i_\ell}) = \left\{ i_1, \ldots, i_\ell \right\}$ . The map $D$ is the (right) descent set. The lattice $L$ is the lattice of subsets ordered by inclusion and $\leq_{\msR}$ is the (poset) dual of the (right) weak order.

Definition: A monoid is $R$ -trivial if for all $x, y \in S$ then $xS = yS$ implies $x = y$ .

It turns out that the last two definitions are equivalent.

Theorem [Theorem 2.18 BBBS 2011] A finite monoid $S$ is a weakly ordered monoid if and only if it is an $R$ -trivial monoid.

Definition: A band is a semigroup $S$ such that every element is idempotent.

The term band is first used in English in [Cli54]. Clifford uses this terminology since a semigroup $S$ contains only idempotent elements if and only if the decomposition of relative inverses has only groups of order one, or as he states, it is a "band of groups of order one". The idea of a band is much older and was likely defined first by Fritz Klein-Barmen in 1940 (see [K-B40]. In essence, Klein-Barmen was looking at semilattices (Halbverbänden) and wanted to take away commutativity. He called these non-commutative semilattices skew-semilattices (Schief-Halbverbänden).

If we take a band and we reinstate commutativity, then we get a join-semilattice (or equivalently, a meet-semilattice). This was first noticed by Clifford which showed the two structures are the same.

Theorem [p. 1041 Cli41] A band is commutative if and only if it is a (meet or join)-semilattice.

It's easy to see that a (meet or join)-semilattice is a commutative band where the multiplication is the meet or join operation. Converseley, if we have a commutative band, we can define $a \leq b$ to mean $ab = a$ giving us a partially ordered set where $ab$ is nothing more than the meet.

Definitions: Given a semigroup $S$ , an element $a$ is said to have an inverse $b$ if $aba = a$ and $bab = b$ . The element $a$ is said to be regular if it has an inverse or (equivalently) if $axa = a$ for some $x \in S$ . (see [Lemma 2.2] for equivalence)

A left regular band is a band $S$ such that for every $x, y \in S$ then $xyx = xy$ .

The first person to work on left regular bands was Maurice-Paul Schützeberger in 1947. Note that Schützenberger adopted Klein-Barmen's terminology and called these non-commutative lattices (trellis non-commutatives). In his article on [page 777], Schützenberger defines axiom $\overline{CJ}$ to be the relation $xyx = xy$ and refers to left regular bands as non-commutative lattices that satisfy the axiom $\overline{CJ}$ (trellis non-comutatives avec l'axiome $\overline{CJ}$ ).

Why work with bands? (and, in particular, left regular ones)

It turns out that working with bands makes Green's preorders much nicer to state. For example, if we have $a \leq_{\msR} b$ then $a = bv$ for some $v \in S^1$. Multiplying both sides by $b$ , we have $ba = bbv = bv$ . In other words, for bands $a \leq_{\msR} b \iff a = ba$ .

The reason we restrict to left regular bands is due to the fact that Green's $\msR$ -preorder is an order if and only if our semi groups is a left regular band.

Theorem [Proposition 7 Brown 2000] Green's $\msR$ -preorder is a partial order if and only if the semigroup is a left regular band.

Relative Inverses

Definitions: An element $a$ has a relative inverse $b$ if there exists an element $e$ such that:

$ea = ae = a$
$ab = ba = e$

In this case we say that $a$ belongs to $e$ . We let $S_e$ denote the set of all elements belonging to $e$ .

Lemma [Lemma 1.1-1.3 Clifford 1941] If $a$ belongs to $e_1$ and $e_2$ then $e_1 = e_2$ and $e_1^2 = e_1$ . Furthermore, the set $S_e$ of all elements of a semigroup $S$ belonging to $e$ is a groupw ith identity $e$ .

Theorem [Theorem 1 Clifford 1941] A semigroup $S$ admits relative inverses if and only if the groups $S_e$ are mutually disjoint groups.

The above theorem is the basis of where the term band comes from. A band is precisely a semigroup where the groups $S_e$ are mutually disjoing groups of order $1$ .

Note: The term "relative inverses" are not commonly used in the literature outside of Clifford's 1941 article. We use the term here in order to match their article.

Face Poset

Definition: Given a band $S$ , let $\left( S, \leq_F \right)$ denote the face poset where $\text{[math]}$ For a left regular band, as $xyx = xy$ , this is equivalent to saying $\text{[math]}$

The face poset was first introduced by D. Rees on page 393 in [Ree40]. Rees said that $e$ is under $f$ if $ef = fe = e$ . In terms of the face poset above, $e$ is under $f$ if and only if $f \leq_F e$ . Although Rees defined this partial order, it wasn't until Clifford in [Cli41] that this was stated to be a partial order.

Support Map

Definition: Given a semigroup $S$ , let $\varphi: S \to T$ be a (semigroup) homomorphism to a join-semilattice $(T, \leq_T)$ . Then $y = yx$ if and only $\varphi(x) \leq_T \varphi(y)$ and $y = yx$ , $x = xy$ if and only if $\varphi(x) = \varphi(y)$ . The map $\varphi$ is known as the support map.

We can see that the terms commute since: $\text{[math]}$

The support map was first introduced by D. McLean in [Mcl54].

Ideals

Definitions: Let $S$ be a semigroup and let $T$ be a subset of elements of $S$ . Then $T$ is a left ideal of $S$ if $ST \subseteq T$ where $ST = \set{st}{s \in S,\, t \in T}$ . An ideal $T$ of $S$ is a left and right ideal of $S$ . Given an element $a \in S$ , then $S^1 a$ is called the principal left ideal generated by $a$ and $S^1aS^1$ is called the principal ideal generated by $a$ .

A semigroup is left simple if it has no proper left ideals. A semigroup is simple if it has no proper ideals and if it is not the zero semigroup of order $2$ .

It's not hard to see that a semigroup $S$ is left simple if and only if $Sa = S$ for every $a \in S$ . When are semigroups simple?

Theorem (due to Wedderburn) A semigroup $S$ is simple if and only if $SxS = S$ for all $0 \neq x \in S$ .

Theorem [Ree40] If a semigroup $S$ is simple and $e$ is a non-zero idempotent of $S$ , then $eSe$ is simple.

Definitions: An idempotent $e$ is primitive in a semigroup $S$ if there does not exist an idempotent $f$ such that $ef = fe = f$ .

A semigroup $S$ is completely simple if:

$S$ is simple.
For $x \in S$ , there exists idempotents $e, f \in S$ such that $xe = x,\, fx = x$ .
For every non-primitive idempotent $e$ , there exists a primitive idempotent $f$ such that $ef = fe = f$ .
(equivalent) Every idempotent of $S$ is primitive.

Theorem [Ree40]

Let $S$ be a completely simple semigroup. Then every idempotent is primitive.
Let $S$ be a simple semigroup whose elements are of finite order. Then $S$ is completel simple.
Let $S$ be a completely simple monoid. Then $S$ is a group.

Theorem [Lemma 2.7 Cli41] A simple semigroup $S$ without zero is completely simple if and only if it admits relative inverses.

Theorem [Combo of lemmas Cli41] Let $P$ be the set of all principal ideals $\left\{ SaS \right\}$ . Since $SaS \cdot SbS = SabS$ ([Lemma 2.3]) and $SabS = SaS \cap SbS$ ([Lemma 2.2]), then $P$ is a join-semilattice with $Sa S \leq SbS$ if and only if $SaS \subseteq SbS$ . Let $S_{SaS} = \set{b \in S}{SbS = SaS}$ be the set of generators of $SaS$ . Then $S_{SaS}$ is a subsemigroups of $S$ ([Lemma 2.5]).

References

[BBBS11] Chris Berg, Nantel Bergeron, Sandeep Bhargava, Franco Saliola Primitive orthogonal idempotents for $R$ -trivial monoids, Jour. Alg. 348 (2011), no. 1, 446-461. DOI, arXiv
[Bro00] Kenneth S. Brown Semigroups, Rings, and Markov Chains, Journal of Theoretical Probability. [13] (2000), no. 3, 871-938. DOI, arXiv
[Cli41] A. H. Clifford Semigroups admitting relative inverses, Ann. of Math. 42 (1941), no. 4, 1037-1048. DOI
[Cli54] A.H. Clifford Band of Semigroups, Proc. Amer. Math. Soc. 5 (1954), 499-504. DOI
[Gre51] J.A. Green On the Structure of Semigroups, Annals of Mathematics, [54] (Jul. 1951), no. 1, 163-172.
[Gri95] P.A. Grillet Semigroups: An Introduction to Structure Theory, CRC Press 1995, 1st edition.
[K-B40] Fritz Klein-Barmen Über eine weitere Verallgemeinerung des Verbandsbegriffes, Mathematische Zeitschrift 46 (1940), 472-480. DOI
[Mcl54] David McLean Idempotent Semigroups, Amer. Math. Monthly 61 (1954), no. 2, 110-113. DOI
[Ree40] D. Rees On Semi-groups, Math. Proc. Camb. Phil. Soc. 36 (1940), no. 4, 387-400. DOI
[Sch08] Manfred Schocker Radical of weakly ordered semigroup algebras, Jour. Alg. Comb. 28 (2008), 231-234. DOI
[Sch47] Maurice-Paul Schützenberger Sur certain treillis gauches, C. R. Acad. Sci. 224 (1947), 776-778.

The Cali Garmo

Bands

Table of Contents

Background

Relations and Orders

Bands

Why work with bands? (and, in particular, left regular ones)

Relative Inverses

Face Poset

Support Map

Ideals

References