Background
A semigroup is a set together with an associative binary operation on . If a semigroup has an identity element, then it is called a monoid. Throughtout this page, we let denote the monoid coming from by adding an identity if one doesn't already exist.
How can we verify if a pair is a semigroup? There are two things we need to check for:
 Closure (so if then .)
 Associativity (i.e. ).
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 for every , 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 . We look at , which turns out to just be the column in our original multiplication table. This column we record on the lefthand column of our new multiplication table (step 1 below). To record , we copy down each row from our original multiplication table to our new one, with respect to the label on the lefthand column (step 2 below). The entries in our new multiplication table are now . To see whether this is equal to , we must do the "opposite" direction. Since appears first, we look at the row of 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 (step 4 below).
Light's associativity test Let be the multiplication table of . Let be a generator. Construct a new table as follows:
 Write the column of on the left.
 For each entry in the column of , the row of in the new table is the row of in .
 Write the row of at the top.
 For each entry in the row of , the column of in the new table should equal the column of in .
If the above is true for every generator, then is associative.
This is a little confusing, so let's look at some examples.
Let be the semigroup with multiplication table as below: Note that is our only generator since:
Following each step, we have:

Write the column of on the left:

For each entry in the column of , the row of in the new table is the row of in the original table:

Write the row of (from the original table) at the top.

Verify for each entry in the row of , the column of in the new table should eqaul the column of in the original table. Looking column by column, they all match up!
Let's look at a nonexample to see where this might fail. Let be our set with generator and multiplication table:
When we construct our new table we end up getting the following:
But notice that the column is wrong! What this basically tells us is that our operation is not associative. In fact, we can verify this: .
Relations and Orders
Definition: Green's Preorders are given by:
Definition: Green's Relations (introduced by Green in [Gre51]) on a semigroup are the equivalence relations below.
Definition: We say that a finite monoid is weakly ordered if there is a finite joinsemilattice together with the maps such that:
 is a monoid morphism, i.e. .
 is a surjection.
 If are such that then .
 If are such that then .
This definition was first defined in [Sch08].
The idea of weakly ordered is coming from the Hecke monoid with (simple) reflections . In this case, the map is nothing more than the content/support map, i.e. . The map is the (right) descent set. The lattice is the lattice of subsets ordered by inclusion and is the (poset) dual of the (right) weak order.
Definition: A monoid is trivial if for all then implies .
It turns out that the last two definitions are equivalent.
Theorem [Theorem 2.18 BBBS 2011] A finite monoid is a weakly ordered monoid if and only if it is an trivial monoid.
Bands
Definition: A band is a semigroup such that every element is idempotent.
The term band is first used in English in [Cli54]. Clifford uses this terminology since a semigroup 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 KleinBarmen in 1940 (see [KB40]. In essence, KleinBarmen was looking at semilattices (Halbverbänden) and wanted to take away commutativity. He called these noncommutative semilattices skewsemilattices (SchiefHalbverbänden).
If we take a band and we reinstate commutativity, then we get a joinsemilattice (or equivalently, a meetsemilattice). 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 to mean giving us a partially ordered set where is nothing more than the meet.
Definitions: Given a semigroup , an element is said to have an inverse if and . The element is said to be regular if it has an inverse or (equivalently) if for some . (see [Lemma 2.2] for equivalence)
A left regular band is a band such that for every then .
The first person to work on left regular bands was MauricePaul Schützeberger in 1947. Note that Schützenberger adopted KleinBarmen's terminology and called these noncommutative lattices (trellis noncommutatives). In his article on [page 777], Schützenberger defines axiom to be the relation and refers to left regular bands as noncommutative lattices that satisfy the axiom (trellis noncomutatives avec l'axiome ).
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 then for some $v \in S^1$. Multiplying both sides by , we have . In other words, for bands .
The reason we restrict to left regular bands is due to the fact that Green's preorder is an order if and only if our semi groups is a left regular band.
Theorem [Proposition 7 Brown 2000] Green's preorder is a partial order if and only if the semigroup is a left regular band.
Relative Inverses
Definitions: An element has a relative inverse if there exists an element such that:
In this case we say that belongs to . We let denote the set of all elements belonging to .
Lemma [Lemma 1.11.3 Clifford 1941] If belongs to and then and . Furthermore, the set of all elements of a semigroup belonging to is a groupw ith identity .
Theorem [Theorem 1 Clifford 1941] A semigroup admits relative inverses if and only if the groups 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 are mutually disjoing groups of order .
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 , let denote the face poset where For a left regular band, as , this is equivalent to saying
The face poset was first introduced by D. Rees on page 393 in [Ree40]. Rees said that is under if . In terms of the face poset above, is under if and only if . Although Rees defined this partial order, it wasn't until Clifford in [Cli41] that this was stated to be a partial order.
Support Map
We can see that the terms commute since:
The support map was first introduced by D. McLean in [Mcl54].
Ideals
Definitions: Let be a semigroup and let be a subset of elements of . Then is a left ideal of if where . An ideal of is a left and right ideal of . Given an element , then is called the principal left ideal generated by and is called the principal ideal generated by .
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 .
It's not hard to see that a semigroup is left simple if and only if for every . When are semigroups simple?
Theorem (due to Wedderburn) A semigroup is simple if and only if for all .
Theorem [Ree40] If a semigroup is simple and is a nonzero idempotent of , then is simple.
Definitions: An idempotent is primitive in a semigroup if there does not exist an idempotent such that .
A semigroup is completely simple if:
 is simple.
 For , there exists idempotents such that .
 For every nonprimitive idempotent , there exists a primitive idempotent such that .
 (equivalent) Every idempotent of is primitive.
Theorem [Ree40]
 Let be a completely simple semigroup. Then every idempotent is primitive.
 Let be a simple semigroup whose elements are of finite order. Then is completel simple.
 Let be a completely simple monoid. Then is a group.
Theorem [Lemma 2.7 Cli41] A simple semigroup without zero is completely simple if and only if it admits relative inverses.
Theorem [Combo of lemmas Cli41] Let be the set of all principal ideals . Since ([Lemma 2.3]) and ([Lemma 2.2]), then is a joinsemilattice with if and only if . Let be the set of generators of . Then is a subsemigroups of ([Lemma 2.5]).
References
 [BBBS11] Chris Berg, Nantel Bergeron, Sandeep Bhargava, Franco Saliola Primitive orthogonal idempotents for trivial monoids, Jour. Alg. 348 (2011), no. 1, 446461. DOI, arXiv
 [Bro00] Kenneth S. Brown Semigroups, Rings, and Markov Chains, Journal of Theoretical Probability. [13] (2000), no. 3, 871938. DOI, arXiv
 [Cli41] A. H. Clifford Semigroups admitting relative inverses, Ann. of Math. 42 (1941), no. 4, 10371048. DOI
 [Cli54] A.H. Clifford Band of Semigroups, Proc. Amer. Math. Soc. 5 (1954), 499504. DOI
 [Gre51] J.A. Green On the Structure of Semigroups, Annals of Mathematics, [54] (Jul. 1951), no. 1, 163172.
 [Gri95] P.A. Grillet Semigroups: An Introduction to Structure Theory, CRC Press 1995, 1st edition.
 [KB40] Fritz KleinBarmen Über eine weitere Verallgemeinerung des Verbandsbegriffes, Mathematische Zeitschrift 46 (1940), 472480. DOI
 [Mcl54] David McLean Idempotent Semigroups, Amer. Math. Monthly 61 (1954), no. 2, 110113. DOI
 [Ree40] D. Rees On Semigroups, Math. Proc. Camb. Phil. Soc. 36 (1940), no. 4, 387400. DOI
 [Sch08] Manfred Schocker Radical of weakly ordered semigroup algebras, Jour. Alg. Comb. 28 (2008), 231234. DOI
 [Sch47] MauricePaul Schützenberger Sur certain treillis gauches, C. R. Acad. Sci. 224 (1947), 776778.