2.1. Basic Results on Semigroups
Let S be a semigroup. For a subset X of S, we use for the subset of all idempotents of S contained in X and is the subsemigroup generated by X; and will denote the sets obtained from by adjoining an identity element 1 and a zero element 0, respectively. If Y is a subset of G, we write for all . If or , we simply write and respectively.
A subset I of a semigroup S is said to be an ideal of S if . In this case, the set is a semigroup in which products falling in I are zero.
The notion of ideal leads naturally to the consideration of an equivalence relation
introduced by Green which is defined by the rule if, and only if, . We will use to denote the -class of an element . In particular, if S is a semigroup with a zero element, then .
Since
is defined in terms of ideals, the inclusion order among these ideals induce a preorder among the elements of S: if, and only if, . If and , we write .
The following well-known lemma will be very useful when dealing with
-classes of products.
Lemma 1. Let . Then and .
Note that induces a partial order among the
-classes of S: if are
-classes of S for some , then if, and only if, . If and , we write .
In general, we write just
to denote the relation
on a semigroup S; however, when
is considered in different semigroups S and T, we write and to denote, respectively, the relation error on each of such semigroups.
Let S be a semigroup with zero. A
-class J of S is called 0-minimal, if for every , implies . In particular, for every , . By Lemma 1, for all . Hence, , for every with . Consequently, is a subsemigroup of S.
If
, then
is the set of all elements of
S for which
e is a two-sided identity. The group of units of
is denoted by
. Clearly
contains all subgroups of
S having
e as their identity element. Hence
is called the
maximal subgroup of
S with
e as the identity element. This subgroup of
S is denoted by
in [
8,
9] and called the
maximal subgroup of
S at
e. Our notation is more convenient when maximal subgroups in different subsemigroups containing the same idempotent
e are considered.
Two ideas of great importance in semigroup theory are those of regular element and an inverse of an element. An element x of a semigroup S is called regular if , i.e., , for some . If s is an element of S, we say that t is an inverse of s, if and .
If x is regular, there exists such that , and then , ; i.e., is an inverse of x. Moreover, .
A
-class J of a semigroup S is said to be regular if for some regular element , or equivalently for some . A semigroup S is called regular if every element of S is regular, or equivalently, if every element of S has an inverse element.
A regular semigroup S is said to be an inverse semigroup if every element has a unique inverse, say. If X is a subset of an inverse semigroup S we denote .
The following lemma is very useful (see [
8]).
Lemma 2. - 1.
A subsemigroup T of an inverse semigroup S is an inverse subsemigroup if implies .
- 2.
If are elements of an inverse semigroup S, we have that and .
- 3.
S is an inverse semigroup if, and only if, S is regular and its idempotents commute. As a consequence, if S is an inverse semigroup, is a subsemigroup of S.
A semigroup S with zero is called 0-simple if and 0 is the only proper ideal of S.
An inverse 0-simple semigroup is called a Brandt semigroup.
Let S be an inverse semigroup and J a 0-minimal
-class of S. Then, is an inverse subsemigroup of S, since if, and only if, . Moreover, 0 is the only proper ideal of , because , for every . On the other hand, because J is a regular
-class. Hence, is a Brandt subsemigroup of S. Note that if , then , for every . In particular, , for every .
Let be non-empty finite sets and let G be a group. A Rees matrixC is a map . We say that C is regular if every row and every column of C has a non-zero entry.
The
Rees matrix semigroup with sandwich matrix
C is the semigroup
with underlying set
and the operation:
, and
for all
,
,
.
It is known that the Rees matrix semigroup with sandwich matrix C is a regular semigroup if, and only if, C is a regular matrix, and in this case is a 0-simple semigroup.
The following theorem due to Rees [
10] is fundamental.
Theorem 1. Let S be a 0-simple semigroup. Then, there exists a group G such that is isomorphic to G, for all , and S is isomorphic to a regular Rees matrix semigroup . Conversely, every regular Rees matrix semigroup is a 0-simple semigroup.
In particular, inverse 0-simple semigroups are called Brandt semigroups and are isomorphic to Rees matrix semigroups of the form , where G is a group and is a set of indices.
We bring the section to a close with the following:
Construction: Let be a
-class of a semigroup S with zero. Set . By Lemma 1, is an ideal of S. Therefore is a semigroup with zero. Note that if S is inverse, then is inverse as well.
Recall that the product in
is defined by:
Let . Assume that . Then, there exist such that . By Lemma 1, and so . This shows that relations
coincide in both semigroups, so that J is the unique 0-minimal
-class of .
2.2. Relational Morphisms and Kernels
A
relational morphism S
T between two semigroups
S and
T is a map from
S into
, the set of subsets of
T, such that
and
, for all
. If
is a relational morphism between monoids a third condition is required:
. In particular, homomorphisms of semigroups are relational morphisms.
Given
S
T and
T
U two relational morphisms, we can define:
Then
S
U is again a relational morphism between
S and
U.
If
S
G is a relational morphism between a semigroup
S and a group
G, the set
is a subsemigroup of
S called the
kernel of τ. If
,
, because
is a subgroup of
G.
Suppose, in addition, that S is an inverse semigroup and let . Then we have that . In fact, take and . Since G is finite, there exists such that . Therefore, and . Hence, we have . Analogously, and the equality holds.
For the sake of clarity, given an element of a Rees matrix semigroup , we denote its image under a relational morphism instead of .
The following proposition analyses the behavior of the images of a unique 0-minimal
-class of an inverse semigroup S under relational morphisms.
Proposition 1. Let S be an inverse semigroup with a unique 0-minimal
-class J, such that . Let H be a group and S
H be a relational morphism. Then, the following properties hold: - 1.
For every ,
- 2.
Given , for every , .
- 3.
For every , and , in case that .
Proof. 1. It follows from the fact that relational morphisms between a semigroup and a group send idempotents to subgroups.
2. Let . Since S is an inverse semigroup we have .
Let
. Then
and therefore:
Let . Then, , thus . We can conclude that .
Analogously, since , we can also prove that .
3. By Statement 2, , for every . Therefore, , for every .
By 2, , for all . Moreover, if , we have that and then , i.e., . □
From now on, we shall be interested here in relational morphisms between semigroups and groups in a variety.
Recall that a formation is a class of groups which is closed under taking epimorphic images and subdirect products. A variety (or pseudovariety) is a formation which is closed under taking subgroups.
Let
be a formation. Each group
G has a smallest normal subgroup with quotient in
; this subgroup is called the
-
residual of
G and it is denoted by
. Clearly
is a characteristic subgroup of
G and it is the intersection of all normal subgroups
N of
G such that
(see [
11],
Section 2.2 for further details).
Let
be a variety of groups. The intersection
of the kernels of all relational morphisms
S
G between
S and every group
is a subsemigroup of
S called the
-kernel of
S. Since
S is finite, there exists
S
G with
such that
. Moreover, if
W is a subsemigroup of
S, then
is a subsemigroup of
. If, in addition,
W is a subgroup of
S, then
is contained in
.
If
is a variety, the
-kernel
of a semigroup
S is computable if, and only if,
is computable, for every
-class
J of
S. Hence, in the sequel, we shall be concerned about the computability of
. In this context, the following theorem of Steinberg [
12] is the most optimal result so far.
Theorem 2. Let be a variety of groups. We can compute the regular elements of for every semigroup S if, and only if, is computable for every inverse semigroup and every
-class J of .
The following proposition allows us to conclude that it is enough to consider inverse semigroups with zero with a unique 0-minimal
-class in order to compute the
-kernel. It was proved by Rhodes and Tilson in [
1] for the variety of all groups, and it still holds for a general variety of groups
.
Proposition 2 ([
1], Fact 2.17).
Let I be an ideal of S. Then Lemma 3. Let S be a semigroup with zero and let J be a
-class of S. Then Proof. Consider the ideal
, where
and
. Then
. By Proposition 2,
In particular, since , we have that . □
According to the construction at the end of
Section 2.1,
is a semigroup with zero and
J is the unique 0-minimal
-class of
. Hence we have:
Corollary 1. Let be a variety of groups. Then, is computable for every inverse semigroup S and every
-class J of S if, and only if, is computable for every inverse semigroup with zero having a unique 0-minimal
-class .
Proposition 3. Let S be a semigroup and let be a variety of groups. Then, for every , is a subgroup of .
Proof. Let . Since , we have that . Since is a subgroup of and e is the identity element of , it follows that . □