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ABSTRACT
We call a semigroup S right noetherian if it satisfies the ascending chain condition on right ideals, and we say that S satisfies ACCPR if it satisfies the ascending chain condition on principal right ideals. We investigate the behavior of these two conditions with respect to ideals and ideal extensions, with a particular focus on minimal and 0-minimal one-sided ideals. In particular, we show that the property of satisfying ACCPR is inherited by right and left ideals. On the other hand, we exhibit an example of a right noetherian semigroup with a minimal ideal that is not right noetherian.
1 Introduction
A finiteness condition for a class of universal algebras is a property that is satisfied by at least all finite members of that class. Ascending chain conditions are classic examples. A poset P satisfies the ascending chain condition if every ascending chaineventually stabilizes. Ascending chain conditions on ideals of rings, introduced by Noether in the landmark paper [Citation19], have played a crucial role in the development of ring theory, appearing in major results such as Hilbert’s basis theorem, Krull’s height theorem and the Noether-Lasker theorem. Analagous conditions naturally arise in semigroup theory. A right ideal of a semigroup S is a subset
such that
We call S right noetherian if its poset of right ideals (under containment) satisfies the ascending chain condition, and we say that S satisfies ACCPR if its poset of principal right ideals satisfies the ascending chain condition. Right noetherian semigroups have received a significant amount of attention; see for instance [Citation1, Citation6, Citation9, Citation18, Citation20]. Semigroups satisfying ACCPR have been considered in [Citation15, Citation16, Citation21].
A related semigroup finiteness condition arises from the notion of a right congruence, that is, an equivalence relation on a semigroup S such that
implies
for all
. We call a semigroup strongly right noetherian if its poset of right congruences satisfies the ascending chain condition. 1 The study of such semigroups was initiated by Hotzel in [Citation8], and further developed in [Citation11, Citation12, Citation17]. As the name suggests, strongly right noetherian semigroups are right noetherian [17, Lemma 2.7]. The converse, however, does not hold. Indeed, unlike the situation for rings, the lattice of right ideals of a semigroup is not in general isomorphic to the lattice of right congruences. For example, the lattice of right ideals of a group is trivial, whereas its lattice of right congruences is isomorphic to its lattice of subgroups. Consequently, every group is trivially right noetherian, but a group is strongly right noetherian if and only if it satisfies the ascending chain condition on subgroups.
For any finiteness condition, it is natural to investigate the behavior of the condition with respect to substructures, quotients and extensions. In particular, for a semigroup finiteness condition the following questions arise. For a semigroup S and an ideal I of S:
does I satisfy
if S satisfies
?
does the Rees quotient
satisfy
if S satisfies
?
does S satisfy
if both I and
satisfy
?
The purpose of this article is to study the finiteness conditions of satisfying ACCPR and of being right noetherian, with (1)–(3) as our guiding questions.
The paper is organized as follows. In Section 2 we provide the necessary preliminary material. In particular, we collect some known results regarding the properties of satisfying ACCPR and being right noetherian, including some equivalent formulations of these conditions. The main results of the paper are contained in Sections 3 and 4. In Section 3 we consider the property of satisfying ACCPR, while Section 4 is concerned with the property of being right noetherian. These two sections follow the same format: they split into two subsections, the first concerning ideals in general and the second focusing on (0-)minimal ideals.
2 Preliminaries
2.1 Ideals and related concepts
Let S be a semigroup. We denote by the monoid obtained from S by adjoining an identity if necessary (if S is already a monoid, then
). Similarly, we denote by
the monoid obtained from S by adjoining a zero if necessary.
Recall that a right ideal of S is a subset such that
and the principal right ideals of S are those of the form
Dually, we have the notion of (principal) left ideals. An ideal of S is a set that is both a right ideal and left ideal of S, and the principal ideals of S are the sets
. Principal (one-sided) ideals determine the five Green’s relations on a semigroup. Green’s relation
on S is given by
Green’s relations and
are defined similarly, in terms of principal left ideals and principal ideals, respectively. Green’s relation
is defined as
and finally we have
. It is clear from the definitions that Green’s relations are equivalence relations on S. Moreover,
is a right congruence on S and
is a left congruence on S.
It is easy to see that the following inclusions between Green’s relations hold:
It can be easily shown that every right (resp. left, two-sided) ideal is a union of -classes (resp.
-classes,
-classes). A semigroup with no proper right (resp. left) ideals is called right (resp. left) simple. A semigroup is simple if it has no proper ideals. Clearly any right/left simple semigroup is simple.
A right (resp. left, two-sided) ideal I of S is said to be minimal if there is no right (resp. left, two-sided) ideal of S properly contained in I. It turns out that, considered as semigroups, minimal right (resp. left) ideals are right (resp. left) simple [3, Theorem 2.4], and minimal ideals are simple [3, Theorem 1.1]. A semigroup contains at most one minimal ideal but may possess multiple minimal right/left ideals. The minimal ideal of a semigroup S is also known as the kernel, and we denote it by If S has a minimal right (resp. left) ideal, then
exists and is equal to the union of all the minimal right (resp. left) ideals [3, Theorem 2.1]. A completely simple semigroup is a simple semigroup that possesses both minimal right ideals and minimal left ideals. A semigroup has both minimal right ideals and minimal left ideals if and only if it has a completely simple kernel [3, Theorem 3.2].
For semigroups with zero, the theory of minimal ideals becomes trivial, so we require the notion of 0-minimality. Suppose that S has a zero element 0. For convenience, we will usually just write the set as 0. We say that S is right (resp. left) 0-simple if
and S contains no proper right (resp. left) ideals except 0, and S is called 0-simple if
and 0 is its only proper ideal. A right (resp. left, two-sided) ideal I of S is said to be 0-minimal if 0 is the only proper right (resp. left, two-sided) ideal of S contained in I. A 0-minimal ideal I of S is either null or 0-simple [4, Theorem 2.29] (a semigroup T is null if
). If I is a 0-minimal ideal of S containing a 0-minimal right ideal of S, then I is the union of all the 0-minimal right ideals of S contained in I [4, Theorem 2.33]. A completely 0-simple semigroup is a 0-simple semigroup that possesses both 0-minimal right ideals and 0-minimal left ideals.
For any 0-minimal right ideal R of S, since is a right ideal of S contained in R, it follows by 0-minimality that either R is null or
Similarly, for any
we have either
or
If R is a 0-minimal right ideal such that
we say that R is globally idempotent. In contrast to the situation for 0-minimal two-sided ideals, globally idempotent 0-minimal right ideals need not be right 0-simple; see the remark immediately after Lemma 2.31 in [Citation4].
Let R be a globally idempotent 0-minimal right ideal of S. For any the set sR is either 0 or a 0-minimal right ideal of S [4, Lemma 2.32]. Thus, the set SR, the (two-sided) ideal of S generated by R, is a union of 0-minimal right ideals of S. Let
denote the union of
and all the null 0-minimal right ideals of S contained in SR, and let
denote the union of all the globally idempotent 0-minimal right ideals of S contained in SR. We call
the null part of SR, and
the globally idempotent part of SR. We note that
may equal 0. We provide a structure theorem describing SR in terms of
and
; in order to do so, we first recall a couple of definitions.
Let S be a semigroup with 0 that is the union of subsemigroups
If
for all
we say that S is the 0-disjoint union of
(
). If, additionally,
for all
we say that S is the 0-direct union of
(
).
Theorem 2.1.
[5, Theorem 6.19] Let S be a semigroup with a globally idempotent 0-minimal right ideal R. Then:
SR is a 0-disjoint union of
and
;
2.
is a null semigroup and an ideal of S;
3.
is a 0-simple semigroup and a right ideal of S;
a subset of
is a (0-minimal) right ideal of
if and only if it is a (0-minimal) right ideal of S.
Let S be a semigroup with 0. The right socle of S is the union of 0 and all the 0-minimal right ideals of S. We denote the right socle by or just
when there is no ambiguity. It turns out that
is a (two-sided) ideal of S [5, Theorem 6.22].
Let denote the union of 0 and all the null 0-minimal right ideals of S, and let
denote the union of 0 and all the globally idempotent 0-minimal right ideals of S. We call
the null part of
and
the globally idempotent part of
Of course, if S has no 0-minimal right ideals then
Theorem 2.2.
[5, Theorem 6.23] Let S be a semigroup with 0. Then:
is a 0-disjoint union of
and
;
is a null semigroup and an ideal of S;
is a right ideal of S;
either
or there exists a set
of globally idempotent 0-minimal right ideals of S such that
is the 0-direct union of the 0-simple semigroups
The above definitions and results regarding 0-minimal right ideals have obvious duals for 0-minimal left ideals, and we use analogous notation (
etc.).
Given an ideal I of S, the Rees quotient of S by I, denoted by is the set
with multiplication given by
Let J be a -class of S. The principal factor of J is defined as follows. If
then its principal factor is itself. Otherwise, the principal factor of J is the Rees quotient of the principal ideal
where x is any element of J, by the ideal
The principal factors of S are the principal factors of its
-classes. As mentioned above, if
exists then it is simple; all other principal factors are either 0-simple or null.
2.2 Acts
Semigroup acts play the analagous role in semigroup theory as that of modules in the theory of rings. We provide some basic definitions about acts; one should consult [Citation10] for more information.
A (right) S -act is a non-empty set A together with a mapsuch that
for all
and
A subset B of an S-act A is a subact of A if
for all
and
Note that S itself is an S-act via right multiplication, and its subacts are precisely its right ideals. For clarity, a right ideal I of S will be written as
when we are viewing it as a subact (including the case
).
Given an S-act A and a subact B of A, the Rees quotient of A by B, denoted by is the S-act with universe
and action given by: for all
and
A subset X of an S-act A is a generating set for A if and A is said to be finitely generated (resp. principal) if it has a finite (resp. one-element) generating set. Thus, the principal right ideals of S are precisely the principal subacts of
.
Note that when we speak of a right ideal I of a semigroup S being generated by a set X, we mean that X generates I as an S-act, i.e.
We call an S-act A is noetherian if the poset of subacts of A (under containment) satisfies the ascending chain condition, and we say that A satisfies ACCP if the poset of principal subacts satisfies the ascending chain condition. In particular, the S-act is noetherian (resp. satisfies ACCP) if and only if S is right noetherian (resp. satisfies ACCPR).
Given an S-act A, we define an equivalence relation on A by
Notice that on the S-act
coincides with Green’s relation
on S. We denote the
-class of an element
by
. There is a natural partial order
on the set of
-classes of A given by
It is easy to see that the poset of -classes is isomorphic to the poset of principal subacts of A via the isomorphism
We call an S-act A simple if it contains no proper subact. If an S-act A has a zero 0 (that is, for all
), we say that A is 0-simple if
is its only proper subact. Notice that the simple subacts of
are precisely the minimal right ideals of S, and, if S has a zero 0, the 0-simple subacts of
are precisely the 0-minimal right ideals of S.
2.3 Foundational results
In this subsection we establish some foundational results, many of which will be required later in the paper. Some of these results are folklore but we provide proofs for completeness. We begin by presenting some equivalent characterizations of the property of satisfying ACCP.
Proposition 2.3.
Let S be a semigroup and let A be an S-act. Then the following are equivalent:
A satisfies ACCP;
the poset of
-classes of A satisfies the ascending chain condition;
every non-empty set of principal subacts of A contains a maximal element.
Proof.
(1) (2) follows from the fact, established above, that the poset of
-classes of A is isomorphic to the poset of principal subacts of A.
(1) (3). Suppose for a contradiction that there exists a non-empty set
of principal subacts of A with no maximal element. Pick any
Since
is not maximal, there exists
such that
Continuing in this way, we obtain an infinite ascending chain
of principal subacts of A, contradicting the fact that A satisfies ACCP.
(3) (1). Consider an ascending chain
where
By assumption, the set
contains a maximal element, say
Then we must have that
for all
Thus A satisfies ACCP. □
Corollary 2.4 .
The following are equivalent for a semigroup S:
S satisfies ACCPR;
every non-empty set of principal right ideals of S contains a maximal element;
the poset of
-classes of S satisfies the ascending chain condition.
We now provide several equivalent formulations of the property of being noetherian for acts. For this result, recall that an antichain of a poset is a subset consisting of pairwise incomparable elements.
Theorem 2.5.
Let S be a semigroup and let A be an S-act. Then the following are equivalent:
A is noetherian;
every subact of A is finitely generated;
every non-empty set of subacts of A contains a maximal element;
A satisfies ACCP and contains no infinite antichain of principal subacts;
the poset of
-classes of A satisfies the ascending chain condition and contains no infinite antichain.
Proof.
The proof that (1), (2), and (3) are equivalent is essentially the same as that of the analogue for modules over rings; see [13, Section 10.1]. follows from the fact that the poset of
-classes of A is isomorphic to the poset of principal subacts of A.
Clearly A satisfies ACCP. Suppose for a contradiction that there exists an infinite antichain
of principal subacts of A. For each
let
be the subact
Clearly
. We cannot have
, for otherwise we would have
for some
and hence
contradicting the fact that
and
are incomparable. Thus, we have an infinite strictly ascending chain
of right ideals of S, contradicting the assumption that A is noetherian.
Suppose that A is not noetherian but does satisfy ACCP. We need to construct an infinite antichain of principal subacts of A. Since A is not noetherian, there exists an infinite strictly ascending chain
of subacts of A. Choose elements
and
for
Then certainly
is not contained in any
since
and
Consider the infinite set of principal subacts of A. Since A satisfies ACCP,
contains a maximal element, say
by Proposition 2.3. Now consider the infinite set
Again,
contains a maximal element, say
Then
is not contained in
since
and
is not contained in
since
is maximal in
. Similarly, the infinite set
contains a maximal element, say
and
and
are pairwise incomparable. Continuing this process ad infinitum, we obtain an infinite antichain
of principal subacts of A, as required. □
From Theorem 2.5 we deduce a number of corollaries.
Corollary 2.6 .
[18, Proposition 3.1 and Theorem 3.2] The following are equivalent for a semigroup S:
S is right noetherian;
every right ideal of S is finitely generated;
every non-empty set of right ideals of S contains a maximal element;
S satisfies ACCPR and contains no infinite antichain of principal right ideals;
the poset of
-classes of S satisfies the ascending chain condition and contains no infinite antichain.
Corollary 2.7 .
Let S be a semigroup. Any S-act A with finitely many -classes is noetherian.
Corollary 2.8 .
Any semigroup with finitely many -classes is right noetherian.
Corollary 2.9 .
Let S be a semigroup, and let A be an S-act (with 0) that is the union of (0-)simple subacts Then A satisfies ACCP. Furthermore, A is noetherian if and only if I is finite.
Proof.
It is clear that A satisfies ACCPR. The (0-)simple subacts of A are clearly principal and form an antichain (under containment), so the second statement follows from Corollary 2.6. □
Corollary 2.10 .
Let S be a semigroup (with 0) that is the union of (0-)minimal right ideals of S. Then S satisfies ACCPR. Furthermore, S is right noetherian if and only if I is finite.
The next result states, for both the properties of being noetherian and satisfying ACCP, an act has the property if and only if both a subact and the associated Rees quotient do.
Proposition 2.11 .
Let S be a semigroup, let A be an S-act, and let B be a subact of A. Then A is noetherian (resp. satisfies ACCP) if and only if both B and are noetherian (resp. satisfy ACCP).
Proof.
Suppose that A is noetherian (resp. satisfies ACCP). Since any ascending chain of (principal) subacts of B is also an ascending chain of (principal) subacts of A, it follows that B is noetherian (resp. satisfies ACCP). Now consider an ascending chainof (principal) subacts of
Let
be the quotient map, and set
for all
Then we have an ascending chain
of (principal) subacts of A. Since A is noetherian, there exists
such that
for all
Then
for all
Hence,
is noetherian (resp. satisfies ACCP).
Conversely, suppose that both B and are noetherian (resp. satisfy ACCP). Consider an ascending chain
of (principal) subacts of A. If
for all
then each
is a subact of
and hence the above chain must eventually stabilize since
is noetherian. Assume then that there exists
such that
Setting
and
for all
we obtain ascending chains
of B and
respectively. Since B and
are noetherian, these chains eventually stabilize, and thus there exists
such that
and
for all
Then we have that
for all
Hence, A is noetherian. □
We now focus on the semigroup conditions of being right noetherian and of satisfying ACCPR. Every free semigroup satisfies ACCPR, but a free semigroup is right noetherian if and only if it is monogenic:
Proposition 2.12.
[18, Proposition 3.5] Let X be a non-empty set. The free semigroup on X satisfies ACCPR, but
is right noetherian if and only if
Since every free semigroup satisfies ACCPR, this property is certainly not closed under quotients. On the other hand, the property of being right noetherian is closed under quotients:
Lemma 2.13.
[18, Lemma 4.1] Let S be a semigroup and let be a congruence on S. If S is right noetherian, then so is
The property of being right noetherian is not in general inherited by ideals; see [18, Remark 6.10]. Going in the other direction, if both an ideal and the associated Rees quotient are right noetherian, then so is the ideal extension:
Proposition 2.14.
[18, Corollary 4.5] Let S be a semigroup and let I be an ideal of S. If both I and are right noetherian, then so is S.
Recall that an element a of a semigroup S is regular if there exists such that
and S is regular if all its elements are regular. The property of being right noetherian is inherited by regular subsemigroups:
Proposition 2.15.
[18, Corollary 5.7] Let S be a semigroup with a regular subsemigroup T. If S is right noetherian then so is T.
The corresponding statement for the property of satisfying ACCPR also holds:
Proposition 2.16.
Let S be a semigroup with a regular subsemigroup T. If S satisfies ACCPR then so does T.
Proof.
Consider an ascending chain
of principal right ideals of T. Then clearly we have an ascending chainof principal right ideals of S. Since S is right noetherian, there exists
such that
for all
Therefore, for any
there exists
such that
Since T is regular, there exists
such that
Then we have that
and hence
Thus T satisfies ACCPR. □
3 Semigroups satisfying ACCPR
In this section we consider the relationship between semigroups and their (one-sided) ideals with respect to the property of satisfying ACCPR. We first consider ideals in general, and we then focus on minimal and 0-minimal ideals.
3.1 General ideals
It turns out that, unlike the property of being right noetherian, the property of satisfying ACCPR is closed under ideals. In fact, we show that this property is closed under the more general class of (m, n)-ideals, introduced by Lajos in [Citation14].
Let An (m, n)-ideal of a semigroup S is a subsemigroup A of S such that
Notice that any one-sided ideal is an (m, n)-ideal. (1,1)-ideals are also known as bi-ideals, which were introduced by Good and Hughes in [Citation7].
Theorem 3.1.
Let S be a semigroup, and let A be an (m, n)-ideal of S for some If S satisfies ACCPR, then so does A.
Proof.
Assume for a contradiction that there exists an infinite strictly ascending chain
of principal right ideals of A. Then clearly we have an ascending chainof principal right ideals of S. Since S satisfies ACCPR, there exists
such that
for all
Now, we have
for each
and
for each
Thus, we have
where the final containment follows from the fact that A is an (m, n)-ideal of S. But then
contradicting the assumption. □
Corollary 3.2 .
Let S be a semigroup and let I be a right/left/two-sided ideal of S. If S satisfies ACCPR, then so does I.
It was noted in Section 2 that the property of satisfying ACCPR is not closed under quotients. However, we shall see that this property is closed under Rees quotients. First note that, given an ideal I of S, we have both the semigroup Rees quotient and the S-act Rees quotient
(with the same universe).
Lemma 3.3.
Let S be a semigroup and let I be an ideal of S. Then satisfies ACCPR if and only if
satisfies ACCP.
Proof.
Since I is an ideal of S, for any we have that
if and only if
From this fact the result readily follows. □
Corollary 3.4 .
Let S be a semigroup and let I be an ideal of S. If S satisfies ACCPR, then both I and satisfy ACCPR.
Proof.
We have that I satisfies ACCPR by Corollary 3.2. Since satisfies ACCP, the quotient
satisfies ACCP by Proposition 2.11, and hence
satisfies ACCPR by Lemma 3.3.□
Corollary 3.5 .
Let S be a semigroup and let I be an ideal of S. Then S satisfies ACCPR if and only if satisfies ACCPR and (the S-act)
satisfies ACCP.
Proof.
If S satisfies ACCPR, then satisfies ACCPR by Corollary 3.4. Since
satisfies ACCP, the subact
satisfies ACCP by Proposition 2.11. The converse follows from Proposition 2.11 and Lemma 3.3. □
Recall that a principal factor of a semigroup is either the minimal ideal (if it exists) or the Rees quotient of a certain ideal by another ideal. Thus, Corollary 3.4 yields:
Corollary 3.6 .
If a semigroup S satisfies ACCPR, then so do all its principal factors.
We shall show that the converse of Corollary 3.4 does not hold. To this end, we introduce the following construction.
Construction 3.7 .
Let S be a semigroup and let A be an S-act. Let be a set in one-to-one correspondence with A and disjoint from S, and let 0 be an element disjoint from
Define a multiplication on
extending that on S, by
for all
and
With this multiplication, U is a semigroup, and we denote it by
Notice that
is a null semigroup and an ideal of S.
Proposition 3.8.
Let S be a semigroup, let A be an S-act, and let Then U satisfies ACCPR if and only if S satisfies ACCPR and A satisfies ACCP.
Proof.
Let By Corollary 3.5, we have that U satisfies ACCPR if and only if
satisfies ACCPR and
satisfies ACCP. Clearly
satisfies ACCPR if and only if S satisfies ACCPR. It is easy to show that, for any
we have
and
if and only if
Thus, the poset of principal subacts of
has the form
where P is isomorphic to the poset of principal subacts of A. It follows that
satisfies ACCP if and only if A satisfies ACCP. This completes the proof. □
We now show that the converse of Corollary 3.4 does not hold.
Let S be a semigroup that satisfies ACCPR with an S-act A that does not satisfy ACCP. (For example, we can take A to be any semigroup that does not satisfy ACCPR and S to be a free semigroup with a surjective homomorphism We turn A into an S-act by defining
for all
and
We have that S satisfies ACCPR by Proposition 2.12, and it is straightforward to show that A does not satisfy ACCP.) The semigroup
does not satisfy ACCPR by Proposition 3.8. On the other hand, the ideal
certainly satisfies ACCPR (indeed, any null semigroup satisfies ACCPR by Corollary 2.10), and the Rees quotient
satisfies ACCPR since S satisfies ACCPR.
We now consider conditions on an ideal I such that converse of Corollary 3.4 does hold.
Given a semigroup S, we say that an element has a local right identity (in S) if there exists
such that
; i.e.
If S is a monoid or a regular semigroup, then clearly every element has a local right identity.
Proposition 3.9.
Let S be a semigroup, let I be an ideal of S, and suppose that every element of I has a local right identity in I. Then S satisfies ACCPR if and only if both I and satisfy ACCPR.
Proof.
We show that the S-act satisfies ACCP. The result then follows from Corollary 3.5. So, consider an ascending chain
of principal subacts of
. Then for each
we have that
using the fact that has a local right identity in I. Therefore, we have an ascending chain
of principal right ideals of I. Since I satisfies ACCPR, there exists
such that
for all
Thus
for all
□
Proposition 3.10.
Let S be a semigroup, let I be an ideal of S, and suppose that there is no infinite antichain of principal right ideals of I. Then the following are equivalent:
S satisfies ACCPR;
both I and
satisfy ACCPR;
I is right noetherian and
satisfies ACCPR.
Proof.
(1) (2) is Corollary 3.4.
(2) (3). Since I satisfies ACCPR and has no infinite antichain of principal right ideals, it is right noetherian by Corollary 2.6.
(3) (1). Assume for a contradiction that S does not satisfy ACCPR. Then there exists an infinite strictly ascending chain
of principal right ideals of S. We cannot have for any
for then we would have an infinite ascending chain
of principal right ideals of . Thus
for all
Consider the set of principal right ideals of I. By assumption, this set does not contain an infinite antichain. Also, we cannot have
for any
for then we would have
Thus, there exist
with
such that
Hence
Now consider the set By a similar argument as above, there exist
with
such that
Now, we have
and hence Continuing this process ad infinitum, we obtain an infinite strictly ascending chain
of principal right ideals of I, contradicting the fact that I is right noetherian. Hence, S satisfies ACCPR. □
3.2 Minimal and 0-minimal ideals
In the remainder of this section we focus on minimal and 0-minimal (one-sided) ideals. Recall that the minimal ideal of a semigroup S, if it exists, is denoted by
Proposition 3.11.
Let S be a semigroup with at least one minimal right ideal, and let Then S satisfies ACCPR if and only if
satisfies ACCPR.
Proof.
Clearly being the union of all the minimal right ideals of S, satisfies ACCPR by Corollary 2.10. Consider
Then
for some minimal right ideal R of S. Clearly
is a right ideal of S contained in R, so
by the minimality of R, and hence
Thus every element of
has a local right identity. The result now follows from Proposition 3.9.□
We now consider semigroups satisfying ACCPR with minimal left ideals.
Theorem 3.12.
Let S be a semigroup that satisfies ACCPR. Then S has a minimal left ideal if and only if S has a completely simple kernel.
Proof.
If S has a completely simple kernel, then, as established in Section 2, S has minimal left ideals.
Now suppose that S has a minimal left ideal. Then is the union of all the minimal left ideals of S. We shall prove that
has an idempotent, and then
is completely simple by [5, Theorem 8.14].
Let L be a minimal left ideal of S, and consider the set of principal right ideals of S. This set contains a maximal element, say
Since Lx is left ideal of S contained in L, we have that
by the minimality of L. Thus
for some
and hence
Since
is maximal in the set
we conclude that
Then
for some
and hence
Thus, has an idempotent, as required. □
Corollary 3.13 .
Let S be a semigroup. Then S satisfies ACCPR and has a minimal left ideal if and only if S has a completely simple minimal ideal and
satisfies ACCPR.
Proof.
The forward implication follows from Theorem 3.12 and Corollary 3.5. Conversely, since S has a completely simple minimal ideal, it certainly has a minimal left ideal, and S satisfies ACCPR by Proposition 3.9, since every element of has a local right identity. □
The following result is an analogue of Theorem 3.12 for 0-minimal ideals.
Theorem 3.14.
Let be a semigroup that satisfies ACCPR and has a 0-minimal ideal I. Then I contains a globally idempotent 0-minimal left ideal of S if and only if I is completely 0-simple.
Proof.
() Suppose that I contains a globally idempotent 0-minimal left ideal L of S. Since
therefore
and hence I is 0-simple. We shall prove that I contains an idempotent, and then it is completely 0-simple by [5, Theorem 8.22].
Recall that for any either
or
Consider the set
of principal right ideals of S. By the 0-minimality of L, we have
for each
Since
there exist
such that
and hence
Thus P is non-empty. Since S satisfies ACCPR, P contains a maximal element, say
Then
and
Thus
for some
and hence
Since
we cannot have
so
and hence
Since
is maximal in P, we conclude that
Then, as in the proof of Theorem 3.12, we have
so I contains an idempotent, as required.
() If I is completely 0-simple, then it has a globally idempotent 0-minimal ideal L. We have that
and
since
so
by the 0-minimality of L. Thus, L is a left ideal of S. Clearly any left ideal of S contained in L also a left ideal of I, so it follows from the 0-minimality of L in I that L is 0-minimal in S. □
Corollary 3.15.
Let S be a 0-simple semigroup. Then S satisfies ACCPR and has a 0-minimal left ideal if and only if S is completely 0-simple.
Proof.
Suppose that S satisfies ACCPR and has a 0-minimal left ideal L. Since S is 0-simple, we have that by [4, Lemma 2.34], and hence L must be globally idempotent. It follows from Theorem 3.14 that S is completely 0-simple.
The converse clearly holds.□
Corollary 3.16.
Let be a semigroup with a globally idempotent 0-minimal left ideal L. If S satisfies ACCPR, then the globally idempotent part
of LS is completely 0-simple.
Proof.
By the left-right dual of Theorem 2.1, is a left ideal of S. Therefore, since S satisfies ACCPR,
satisfies ACCPR by Corollary 3.2. Also by the left-right dual of Theorem 2.1,
is 0-simple and has globally idempotent 0-minimal left ideals (of itself). Hence, by Corollary 3.15,
is completely 0-simple.□
Recall that the left socle of a semigroup S with 0 is the 0-disjoint union of
and
which are the null part and globally idempotent part of
, respectively. Note that since
is an ideal of S, we may view it as a subact of
.
Theorem 3.17.
Let be a semigroup, and let
Then the following are equivalent:
S satisfies ACCPR;
is either 0 or the 0-direct union of completely 0-simple semigroups
the S -act
satisfies ACCP, and
satisfies ACCPR.
Proof.
(1) (2). Suppose that
Then, by the left-right dual of Theorem 2.2, there exists a set
of globally idempotent 0-minimal left ideals of S such that
is the 0-direct union of the 0-simple semigroups
Each
is completely 0-simple by Corollary 3.16. The subact
of
satisfies ACCP by Proposition 2.11, and
satisfies ACCPR by Corollary 3.4.
(2) (1). Let T denote the Rees quotient
Since
satisfies ACCP, by Corollary 3.5 it suffices to prove that T satisfies ACCPR. Notice that
is (isomorphic to) an ideal of T. Since
is either 0 or the 0-direct union of completely 0-simple semigroups, it satisfies ACCPR by Corollary 2.10, and every element of
has a local right identity. Now,
(by the Third Isomorphism Theorem), so
satisfies ACCPR by assumption. Hence, by Proposition 3.9, T satisfies ACCPR, as required. □
If has no null 0-minimal ideals then
, so by Theorem 3.17 we have:
Corollary 3.18.
Let be a semigroup without null 0-minimal ideals, and let
Then the following are equivalent:
S satisfies ACCPR;
is either 0 or the 0-direct union of completely 0-simple semigroups, and
satisfies ACCPR.
We shall find some necessary and sufficient conditions for a semigroup to satisfy ACCPR, but first we provide the following lemma.
Lemma 3.19.
Let be a semigroup such that
and let
Then the following statements hold.
If
then
is a 0-minimal right ideal of S for each
and
is an ideal of S.
is the 0-direct union of
and
Proof.
By the left-right dual of Theorem 2.2, is either 0 or the 0-direct union of 0-simple semigroups
(
). Consider
. Since either
or x belongs to a 0-simple semigroup, we have
, where
denotes the
-class of x. We have
since
is an ideal, and hence
so
Thus
and hence
Since
is an ideal of S and
is a left ideal of S, it follows that
Since
and
we conclude that
Therefore, if
then
is a 0-minimal right ideal for each
Thus,
and hence
Then
so
If then it is clear that
is an ideal of S and that statements (3) and (4) hold, so we may assume that
Let
Consider
We have that
since
is 0-simple, and hence
as
is an ideal. We must have that
for otherwise we would have
(using the fact
is 0-simple and
is an ideal). It follows that
is the 0-direct union of
,
Now, is a right ideal of S by Theorem 2.2, so to prove that it is an ideal, it suffices to show that it is a left ideal. So, let
and
If
then
Suppose that
We have that
and
for some
If
then
If
then
Thus
is an ideal of S.
Since and
are both ideals of S, and
it follows that
Thus
is the 0-direct union of
and
Since and
is an ideal of S, it is certainly an ideal of
Observing that the universe of
is
it is easy to see that
□
Theorem 3.20.
Let be a semigroup such that
and let
Then the following are equivalent:
S satisfies ACCPR;
is either 0 or the 0-direct union of completely 0-simple semigroups;
is either a null semigroup or the 0-direct union of a null semigroup and completely 0-simple semigroups, and either
or
is the 0-direct union of completely 0-simple semigroups.
Proof.
(1) (2) follows immediately from Theorem 3.17.
(2) (3). By Lemma 3.19,
and
is the 0-direct union of
and
If
then
is a null semigroup. If
then
so
is the 0-direct union of completely 0-simple semigroups
(
). As in the proof of Lemma 3.19, there exists a set
such that
is the 0-direct union of
,
By Lemma 3.19, we have that
Thus, if
then
is (isomorphic to) the 0-direct union of
(3) (1). We have that
is a 0-direct union of completely 0-simple semigroups, and hence satisfies ACCPR by Corollary 2.10. Therefore, to prove that S satisfies ACCPR, by Corollary 3.5 it suffices to show that
is noetherian (as an S-act). If
then it is obviously noetherian. Otherwise, by Lemma 3.19, we have that
is the union of 0-simple subacts
(
), and hence
is noetherian by Corollary 2.9.□
4 Right noetherian semigroups
In this section we consider right noetherian semigroups. Paralleling the previous section, this section splits into two parts, the first of which deals with ideals in general, and the section concerns minimal and 0-minimal ideals.
4.1 General ideals
As mentioned in Section 3, unlike the property of satisfying ACCPR, the property of being right noetherian is not closed under ideals. The following result provides a condition under which ideals, and more generally (m, n)-ideals, inherit the property of being right noetherian. In what follows, a right ideal I of a semigroup A is decomposable (in A) if
Proposition 4.1.
Let S be a semigroup, let A be an (m, n)-ideal of S, and suppose that every right ideal of A is decomposable in A. If S is right noetherian, then so is A.
Proof.
Let I be a right ideal of A. Then by assumption. This implies that
Since S is right noetherian and
is a right ideal of S, there exists a finite set
such that
For each
choose
such that
and let
We claim that
Clearly
Now consider
Then
for some
and
and
for some
and
Now,
for some
Therefore, we have that
using the fact that A is an (m, n)-ideal of S. Thus
and hence
as desired. □
Corollary 4.2.
Let S be a semigroup, and suppose that A is a left ideal of S such that every element of A is regular in S. If S is right noetherian, then so is A.
Proof.
Let I be a right ideal of A. For any there exists
such that
Since A is a left ideal, we have
so
Thus
is decomposable. Hence, by Proposition 4.1, A is right noetherian.□
By Propositions 2.14 and 4.1 we have:
Corollary 4.3.
Let S be a semigroup, let I be an ideal of S, and suppose that every right ideal of I is decomposable. Then S is right noetherian if and only if both I and are right noetherian.
Recall that a semigroup is strongly right noetherian if its poset of right congruences satisfies the ascending chain condition. The following result, due to Kozhukhov, describes the non-null principal factors of a strongly right noetherian semigroup.
Proposition 4.4.
[11, Lemma 1.3] Any (0-)simple principal factor of a strongly right noetherian semigroup is completely (0-)simple and has only finitely many -classes.
From Proposition 4.4 and Corollary 2.10 we immediately deduce:
Corollary 4.5.
Every non-null principal factor of a strongly right noetherian semigroup is right noetherian.
Corollary 4.6.
Let S be a semigroup with an ideal I such that for every -class
the principal factor of J is either simple or 0-simple. If S is strongly right noetherian, then I is regular and hence right noetherian.
Proof.
The ideal I is a union of -classes. For every
-class
its principal factor is either completely simple or completely 0-simple by Proposition 4.4. It follows that element of I is regular (in I), so I is a regular semigroup. Hence, by Proposition 2.15, I is right noetherian.□
A semigroup is said to be semisimple if each of its principal factors is simple or 0-simple. If a semigroup has a null principal factor, then the non-zero elements of that principal factor are not regular. Thus regular semigroups are semisimple. This fact, together with Corollary 4.6, yields:
Corollary 4.7.
Let S be a strongly right noetherian semigroup. Then S is semisimple if and only if it is regular, in which case every ideal of S is right noetherian.
Remark 4.8.
Ideals, indeed kernels, of strongly right noetherian (regular) semigroups need not be strongly right noetherian; see [17, Example 6.5 and Proposition 6.6].
We end this subsection with some results that will be useful in the next subsection.
Lemma 4.9.
Let S be a semigroup and let I be an ideal of S. Then is right noetherian if and only if the S-act
is noetherian.
Proof.
It can easily seen that a subset of is a right ideal of
if and only if it is a subact of
, and that finite generation is preserved in both directions.□
Corollary 4.10.
Let S be a semigroup and let I be an ideal of S. Then S is right noetherian if and only if is right noetherian and (the S-act)
is noetherian.
Proof.
If S is right noetherian, then so is by Lemma 2.13. Since
is noetherian, the subact
is noetherian by Proposition 2.11. The converse follows from Lemma 4.9 and Proposition 2.14.□
Recalling Construction 3.7, an argument similar to the proof of Proposition 3.8 yields:
Proposition 4.11.
Let S be a semigroup, let A be an S-act, and let Then U is right noetherian if and only if S is right noetherian and A is noetherian.
4.2 Minimal and 0-minimal ideals
From now on we focus on minimal and 0-minimal ideals. We begin by exhibiting an example of a right noetherian semigroup with a kernel that is not right noetherian.
Example 4.12.
Let S be the semigroup defined by the presentation
Corresponding to the above presentation, we have a rewriting system on consisting of the rules
and
It is straightforward to check that this rewriting is complete (i.e. noetherian and confluent) and hence yields the following set of normal forms for S:
that is, the set of all the words over
that do not contain
or aba as a subword. For more information about rewriting systems, one may consult [Citation2] for instance.
Let We have that
and hence
Thus,
is the
-class of b. Since
is an ideal of S, we conclude that it is the kernel
(1) S is right noetherian.
Since is right noetherian, by Corollary 4.10 it suffices to prove that
is noetherian. So, let
be a subact of
We shall prove that
is finitely generated. Let
be minimal such that
If there exist
such that
let
be the minimal such j and set
; otherwise, let
If there exist
such that
let
be the minimal such k and set
; otherwise, let
We claim that
is generated by
So, let
There are two cases to consider.
Case 1:
for some
and
If then
Suppose then that
Now
so
and hence
It follows that
and hence
Case 2:
for some
and
If then
so assume that
We have that
so
and
Thus
(2) is not right noetherian.
We claim that the infinite set is an antichain of principal right ideals of
and hence
is not right noetherian by Corollary 2.6. Indeed, consider
where
Suppose first that for some
and
If
then
and
If
then
Now suppose that for some
and
If
then
If
then
Finally, if then
In any case, in view of the normal form for we conclude that
for any
It follows that
and
are incomparable whenever
The next two results show that in a right noetherian semigroup with minimal one-sided ideals, the kernel is also right noetherian.
Proposition 4.13.
Let S be a semigroup with at least one minimal right ideal, and let If S is right noetherian, then
has finitely many
-classes (of itself), and hence
is right noetherian.
Proof.
The kernel is the union of all the minimal right ideals of S. By [3, Theorem 2.4], each of these minimal right ideals is a minimal right ideal of
Moreover, due to their minimality, they form an antichain of principal right ideals of S. Hence, by Corollary 2.6,
is the union of finitely many minimal right ideals. Hence, by Corollary 2.10,
is right noetherian.□
Proposition 4.14.
Let S be a semigroup with at least one minimal left ideal, and let If S is right noetherian, then
is completely simple and right noetherian (and hence has finitely many
-classes).
Proof.
Since S satisfies ACCPR, the kernel is completely simple by Theorem 3.12. Since
is a regular subsemigroup of S, it is right noetherian by Proposition 2.15, and hence
has finitely many
-classes by Corollary 2.10. □
Corollary 4.15.
Let S be a semigroup with a minimal one-sided ideal, and let Then S is right noetherian if and only if both
and
are right noetherian.
Proof.
The forward direction follows from Lemma 2.13 and Propositions 4.13 and 4.14, and the reverse implication follows from Proposition 2.14.□
The remainder of this section concerns semigroups with zero. The following example demonstrates that a right noetherian semigroup can have a right/left socle that is not right noetherian.
Example 4.16.
(1) Let S be any right noetherian semigroup, let A be a noetherian S-act (such as ), and let
Then U is right noetherian by Proposition 4.11. For each
the set
is a null 0-minimal left ideal of U, and
If A is infinite, then
is not right noetherian; indeed, any infinite null semigroup is not right noetherian by Corollary 2.10.
(2) Let S be the free commutative semigroup on two generators y and z. Let be the S-act with action given by
It is easy to see that A has no proper subacts, and hence A is noetherian. Since S is right noetherian, we have that
is right noetherian by Proposition 4.11. We have that
is a null 0-minimal right ideal of U, and
is not right noetherian.
The following result provides a necessary and sufficient condition for a 0-minimal right ideal to be right noetherian.
Theorem 4.17.
Let R be a 0-minimal right ideal of a semigroup S. Then R is right noetherian if and only if the set is finite.
Proof.
For each we have either
or
Thus, if
with
then
It follows that R satisfies ACCPR. Thus, by Corollary 2.6, R is right noetherian if and only if it has no infinite antichain of principal right ideals. Now, for any
with
the principal right ideals
and
are incomparable if and only if
and
if and only if
and
if and only if
The result now follows.□
Completely 0-simple semigroups have the following well-known representation, due to Rees. Let G be a group, let I and J be non-empty sets, and let be a
matrix over
in which every row and column contains at least one element of G. The Rees matrix semigroup with zero over G with respect to P is the semigroup
with universe
and multiplication given by
The 0-minimal right ideals of are the sets
(
). From Theorem 4.17 we deduce:
Corollary 4.18.
Let be a completely 0-simple semigroup. Then a 0-minimal right ideal
of S is right noetherian if and only if the set
is finite.
Corollary 4.19 .
Let be a completely 0-simple semigroup where G is infinite. Then a 0-minimal right ideal
of S is right noetherian if and only if
for all
Remark 4.20.
Let where
and
. Then S is strongly right noetherian by [12, Corollary 2.2], but neither of its two 0-minimal right ideals are right noetherian by Corollary 4.19.
Although the right socle of a right noetherian semigroup need not be right noetherian itself, it is necessary that the globally idempotent part of the right socle be right noetherian.
Proposition 4.21.
Let be a right noetherian semigroup. Then S has finitely many 0-minimal right ideals. Moreover, if S has a globally idempotent 0-minimal right ideal, then the globally idempotent part
of
is a union of finitely many 0-minimal right ideals of itself, and hence
is right noetherian.
Proof.
The set of 0-minimal right ideals of S, if non-empty, is an antichain of principal right ideals of S. Therefore, since S is right noetherian, it has finitely many 0-minimal right ideals by Corollary 2.6.
Now suppose that S has a globally idempotent 0-minimal right ideal. By Theorem 2.2, there exists a set of globally idempotent 0-minimal right ideals of S such that
is the 0-direct union of the
Then I is finite, and it follows from Theorem 2.1(4) that each
is a union of 0-minimal right ideals of itself. It then clearly follows that
is a union of finitely many 0-minimal right ideals of itself. Hence, by Corollary 2.10,
is right noetherian.□
Corollary 4.22.
Let be a semigroup, and let
Then the following are equivalent:
S is right noetherian;
S has finitely many 0-minimal right ideals and
is right noetherian.
Proof.
(1) (2) follows immediately from Proposition 4.21 and Lemma 2.13.
(2) (1). The right socle
contains only finitely many right ideals of S; equivalently,
contains only finitely many subacts of
Thus
is noetherian. Since
is right noetherian, we have that S is right noetherian by Corollary 4.10.□
Corollary 4.23.
Let be a semigroup without null 0-minimal ideals, and let
Then the following are equivalent:
S is right noetherian;
is a union of finitely many 0-minimal right ideals of itself, and
is right noetherian;
both
and
are right noetherian.
Proof.
(1) (2). We have
, so
is a union of finitely many 0-minimal right ideals of itself by Proposition 4.21. By Lemma 2.13,
is right noetherian.
(2) (3) follows from Corollary 2.10, and (3)
(1) follows from Proposition 2.14.□
The following result is an analogue of Proposition 4.14 for 0-minimal left ideals.
Proposition 4.24.
Let be a semigroup with a globally idempotent 0-minimal left ideal L. If S is right noetherian, then the globally idempotent part
of LS is completely 0-simple and right noetherian (and hence has finitely many
-classes). Moreover, L is right noetherian.
Proof.
Since S satisfies ACCPR, is completely 0-simple by Corollary 3.16. Therefore,
is right noetherian by Proposition 2.15, and hence
has finitely many
-classes by Corollary 2.10. Since L is contained in
, which is regular, L is right noetherian by Corollary 4.2.□
We now characterize the property of being right noetherian in terms of the left socle.
Theorem 4.25.
Let be a semigroup, and let
Then the following are equivalent:
S is right noetherian;
is either 0 or the 0-direct union of finitely many completely 0-simple semigroups that each have finitely many
-classes, the S -act
is noetherian, and
is right noetherian.
both
and
are right noetherian.
Proof.
(1) (2). By Corollary 4.10,
is noetherian and
is right noetherian. Suppose that
Then, by the left-right dual of Theorem 2.2, there exists a set
of globally idempotent 0-minimal left ideals such that
is a 0-direct union of
where
By Proposition 4.24, each
is completely 0-simple and has finitely many
-classes. For each
let
be a non-zero idempotent in
We cannot have
for any
for that would imply that
contradicting the fact that
Thus,
is an antichain of principal right ideals of S, and hence I is finite by Corollary 2.6.
(2) (3). We have that
is right noetherian by Corollary 2.10. Therefore, since
is noetherian,
is right noetherian by Corollary 4.10.
(3) (1) follows from Proposition 2.14.□
Corollary 4.26.
Let be a semigroup without null 0-minimal ideals, and let
Then the following are equivalent:
S is right noetherian;
2.
is either 0 or the 0-direct union of finitely many completely 0-simple semigroups that each have finitely many
-classes, and
satisfies ACCPR.
We now find several equivalent characterizations for a semigroup to be right noetherian.
Theorem 4.27.
Let be a semigroup such that
and let
Then the following are equivalent:
S is right noetherian;
2.
is finite, and
is either 0 or the 0-direct union of finitely many completely 0-simple semigroups that each have finitely many
-classes;
3.
is either a finite null semigroup or the 0-direct union of a finite null semigroup and finitely many completely 0-simple semigroups that each have finitely many
-classes, and either
or
is the 0-direct union of finitely many completely 0-simple semigroups that each have finitely many
-classes;
S has finitely many
-classes.
Proof.
(1) (2). Given Theorem 4.25, we only need to prove that
is finite. By Lemma 3.19, either
or
is a 0-minimal right ideal of
for each
Since
is right noetherian, it has only finitely many 0-minimal right ideals by Proposition 4.21, so
is finite.
The proof of (2) (3) is essentially the same as that of (2)
(3) of Theorem 3.20. (3)
(4) is obvious, and (4)
(1) follows from Corollary 2.8.□
Corollary 4.28.
Let be a semigroup such that
Then S is right noetherian if and only if it is either a finite null semigroup or the 0-direct union of a finite null semigroup and finitely many completely 0-simple semigroups that each have finitely many
-classes.
We now present an example to illustrate Theorem 4.27, and to demonstrate that a right noetherian semigroup can be the union, but not 0-direct union, of its 0-minimal left ideals.
Example 4.29.
Let V be the 0-disjoint union of two completely 0-simple semigroups S and T, each with finitely many -classes, and let x be an element disjoint from V. Let
and define a multiplication on U, extending that on V, as follows:
for all
and
It is straightforward to show that U is a semigroup under this multiplication. It is easy to see that the 0-minimal left ideals of U are
and the 0-minimal left ideals of S and T. Thus
where
and
and U is right noetherian by Theorem 4.27. The 0-minimal right ideals of U are
and the 0-minimal right ideals of T, and
is the 0-direct union of
and T. On the other hand, U is not the 0-direct union of its 0-minimal left ideals (since
for all
).
Remark 4.30.
Let S be any right simple semigroup, and let Then S is right noetherian by Proposition 4.11. It is easy to see that U is the union of its two 0-minimal right ideals,
and
Thus
where
and
Since
for all
the semigroup U is not the 0-direct union of I and
Note
1 Strongly right noetherian semigroups are also known in the literature as ‘right noetherian’, and the term ‘weakly right noetherian’ has been used to denote semigroups that satisfy the ascending chain condition on right ideals.
Acknowledgments
This work was supported by the Engineering and Physical Sciences Research Council [EP/V002953/1]. The author thanks the referee for a number of helpful suggestions.
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