2.9 Future work
3.1.3 Completely (0-)simple semigroups: linked triples
There is a special way of describing a congruence on a completely simple or completely 0-simple semigroup: using a linked triple. We will start by explaining the termscompletely simple and
completely 0-simple, then we will define a semigroup’s linked triples and explain how they are related to its congruences.
Definition 3.3. A semigroupS is:
• simpleif its only ideal isS;
• 0-simpleif it contains a zero, and has precisely two ideals.
Simple and 0-simple semigroups are closely related. Note that if S is a simple semigroup, then S0, the semigroup created by appending a zero element to S, is 0-simple. A 0-simple
semigroup’s ideals are {0} and S. Note also that the trivial semigroup is simple but not 0- simple.
Next, we consider a slightly stronger condition, after a preliminary definition relating to idempotents.
Definition 3.4. An idempotent p ∈ S is primitive if it is non-zero and there is no other non-zero idempotenti∈S such thatip=pi=i.
Definition 3.5. A semigroup is:
• completely simpleif it is simple and contains a primitive idempotent;
Definitions 3.3 and 3.5 are equivalent for finite semigroups – that is to say, a finite semigroup is completely simple if and only if it is simple, and it is completely 0-simple if and only if it is 0-simple. Some of the conversions described in this chapter will be applicable only to finite semigroups, and in those circumstances we will refer to finite simple or finite 0-simple
semigroups, knowing that these are completely simple or completely 0-simple, respectively. Note that a finite semigroup is simple if and only if it isJ-trivial.
Completely simple and completely 0-simple semigroups have a strong and useful isomor- phism property, which allows us to say a great deal about their structure and, in particular, their congruences. We will consider first the more complicated case, that of completely 0-simple semigroups, and then at the end of this section we will explain how this theory can be adapted for the much less complicated case, that of completely simple semigroups.
Definition 3.6 ([How95,§3.2]). ARees 0-matrix semigroupM0[T;I,Λ;P] is the set
(I×T×Λ)∪ {0}
with multiplication given by
(i, a, λ)·(j, b, µ) = (
(i, apλjb, µ) ifpλj 6= 0,
0 otherwise,
for (i, a, λ),(j, b, µ)∈I×T×Λ, and 0x=x0 = 0 for all x∈ M0[T;I,Λ;P], where
• T is a semigroup,
• Iand Λ are non-empty index sets,
• P is a |Λ| × |I|matrix with entries (pλi)λ∈Λ,i∈I taken fromT0,
• 0 is an element not inI×T×Λ.
We will require a certain property of the matrixP, which we should define first: we call a matrixregularif it contains at least one non-zero entry in each row and each column.
The following theorem shows how we can use Rees 0-matrix semigroups to classify completely 0-simple semigroups.
Theorem 3.7 (Rees). Every completely 0-simple semigroup is isomorphic to a Rees 0-matrix semigroup M0[G;I,Λ;P], where Gis a group and P is regular. Conversely, every such Rees
0-matrix semigroup is completely 0-simple.
Proof. Theorem 3.2.3 in [How95].
Now we can replace any completely 0-simple semigroup with its isomorphic Rees 0-matrix semigroup when we wish to perform any isomorphism-invariant calculations – hence we can restrict our further investigations just to this type of semigroup. Note that methods exist in the Semigroupspackage for performing this replacement: in a session, we can decide whether a finite semigroupSis completely 0-simple usingIsZeroSimpleSemigroup, and if the result is positive we can useIsomorphismReesZeroMatrixSemigroup to obtain a Rees 0-matrix semi- group isomorphic toS, as well as a map between the elements of the two semigroups [M+19].
IfM0[G;I,Λ;P] is finite, thenG, I, Λ andP must all be finite, so all the components of the
semigroup that we work with will also be finite.
Next we consider the congruences of a finite 0-simple semigroup.
Definition 3.8([How95,§3.5]). LetS be a finite Rees 0-matrix semigroupM0[G;I,Λ;P] over
the groupGwith regular matrixP. Alinked tripleonS is a triple (N,S,T)
consisting of a normal subgroup N E G, an equivalence relationS on I and an equivalence relationT on Λ, such that the following are satisfied:
(i) S ⊆εI, where εI ={(i, j)∈I×I| ∀λ∈Λ :pλi= 0 ⇐⇒ pλj = 0},
(ii) T ⊆εΛ, where εΛ={(λ, µ)∈Λ×Λ| ∀i∈I:pλi= 0 ⇐⇒ pµi= 0},
(iii) For all i, j ∈ I and λ, µ ∈ Λ such that pλi, pλj, pµi, pµj 6= 0 and either (i, j) ∈ S or
(λ, µ)∈ T, we haveqλµij∈N, where
qλµij=pλip−µi1pµjp−λj1.
We can associate the linked triples of a finite 0-simple semigroup with its non-universal congruences, as follows.
Theorem 3.9. LetS be a Rees 0-matrix semigroup defined with a group and a regular matrix. There exists a bijectionΓ between the non-universal congruences onS and the linked triples on
S.
Proof. Theorem 3.5.8 in [How95]
This theorem shows us an alternative way to look at congruences on completely 0-simple semigroups, just as normal subgroups show us an alternative way to look at congruences on groups. However, in order to use this at all in a computational setting, we must have a concrete function Γ which we can use to convert a congruence to a linked triple and back again, rather than just the knowledge that such a function exists – indeed, describing such a function is the purpose of this section. We define the function Γ as follows.
Definition 3.10 ([How95, §3.5]). Let S be a Rees 0-matrix semigroup M0[G;I,Λ;P] over
a group G and a regular matrixP. The linked triple function Γ ofS is defined, for ρ a non-universal congruence, by
Γ :ρ7→(Nρ,Sρ,Tρ),
so that it maps any non-universal congruence onto a triple whose entries are defined as follows. The relationSρ⊆I×I is defined by the rule that (i, j)∈ Sρ if and only if (i, j)∈εI and
(i, p−λi1, λ)ρ(j, p−λj1, λ)
for all λ ∈ Λ such that pλi 6= 0 (and hence pλj 6= 0). Similarly, the relation Tρ ⊆ Λ×Λ is
defined by the rule that (λ, µ)∈ Tρ if and only if (λ, µ)∈εΛ and
for all i∈ I such thatpλi 6= 0 (and hencepµi 6= 0). Finally, we define the normal subgroup
NρEGas follows. First, fix someξ∈Λ, a row of the matrixP. SinceP is regular, rowξmust
contain a non-zero entry – fix somek∈Isuch thatpξk6= 0. Now we can define
Nρ={a∈G|(k, a, ξ)ρ(k,1G, ξ)},
where 1G is the identity in the groupG.
The inverse of Γ is then such that, for a linked triple (N,S,T), the congruence (N,S,T)Γ−1 is equal to n (i, a, λ),(j, b, µ) (pξiapλk)(pξjbpµk) −1 ∈N,(i, j)∈ S,(λ, µ)∈ To∪ (0,0) , whereξ∈Λ andk∈I can be any elements such thatpξi andpλk are both non-zero, as shown
in [How95, Lemma 3.5.6]. Note thatξ andk definitely exist, sinceP is a regular matrix, and so columniand rowλmust each contain a non-zero entry.
Note that the definition ofNρdoes not depend on the choice ofξandk. Independence from
the choice ofξ is established by the following lemma, and independence from the choice of k follows by a similar argument.
Lemma 3.11. Let ξ1, ξ2∈Λ andk∈I such that pξ1k6= 0 andpξ2k6= 0. Then
(k, a, ξ1) ρ(k,1G, ξ1) if and only if (k, a, ξ2)ρ(k,1G, ξ2)
for alla∈G.
Proof. Assume that (k, a, ξ1)ρ(k,1G, ξ1). We can right-multiply both sides by (k, p−ξ11k, ξ2) to
give (k, a, ξ1)(k, p−ξ11k, ξ2)ρ(k,1G, ξ1)(k, p−ξ11k, ξ2), which simplifies to (k, apξ1kp−ξ1 1k, ξ2)ρ(k,1Gpξ1kp −1 ξ1k, ξ2),
and then to (k, a, ξ2) ρ(k,1G, ξ2), as required. The converse argument is identical, swapping
ξ1forξ2.
Our discussion so far has focused on 0-simple semigroups, but very similar structures exist for completely simple semigroups. They are isomorphic to Rees matrix semigroups, and linked triples can be defined on them in almost exactly the same way, except for the removal of complications related to the zero element. A Rees matrix semigroup follows Definition 3.6 but with the removal of the zero element, and linked triples follow Definition 3.8, where the restrictions related to placements of 0 inP are irrelevant. It should also be noted that even the universal congruence has a linked triple in this case – (G, I×I,Λ×Λ) – so the domain of Γ is not only the non-universal congruences, but all congruences onS.