# On Maximal Ideals of Compact Connected Topological Semigroups

Authors: Phoebe McLaughlin; Shing S. So; Haohao Wang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

Several results concerning ideals of a compact topological semigroupS with S2=S can be found in the literature.
In this paper, we further investigate in a compact connected topological semigroup S how the conditions S2=S and S2≠S affect
the structure of ideals of S, especially the maximal ideals.


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## Body

## 1. Introduction

First, we list some standard definitions which can be found in [1–3].Definition 1.1.
Atopological semigroup is a topological space S together with a continuous function m:S×S→S such that S is Hausdorff and m is associative.
Asubsemigroup of a semigroup S is a nonvoid set A⊂S such that A2⊂A, and A is called a subgroup of S if it is a group with respect to m.
An elemente of a topological semigroup S is called an idempotent if e2=e. Similarly, an element e of S is called a left identity (right identity) if ea=a (ae=a) for all a∈S. An element of S is called an identity of S if it is both a left and a right identity of S.
The set of all idempotents ofS will be denoted by E throughout this paper. For each e∈E, let H(e) be the union of all subgroups of S containing e. It is shown in [3] that H(e) is the maximal subgroup of S containing e.Definition 1.2.
A nonempty subsetA of a semigroup S is called a left ideal (right ideal) of S if SA⊂A (AS⊂A) and an ideal if it is both a left and a right ideal. A left ideal (right ideal, ideal) is said to be proper if it is not S itself.
An (left, right) idealM of a semigroup S is called minimal if it does not properly contain any (left, right) ideal of S. It follows that there can be at most one minimal ideal of S. If S has a minimal ideal K, then K is called the kernel of S.
Amaximal (left, right) ideal of a semigroup S is a proper (left, right) ideal of S that is not properly contained in any other (left, right) ideal.Definition 1.3.
LetA be a subset of a topological semigroup S, then J0(A) is defined as follows:
(1.1)J0(A)={∅ifAcontainsnoidealofS,∪{I:IisanidealofSandI⊂A}.Theorem 1.4.
LetS be a compact connected topological semigroup without zero, and let  K be the kernel of S. Then, either K∩E is infinite or K is a topological subgroup of S.Proof.
SinceS is a compact topological semigroup, K=∪{H(e):e∈K∩E}, and H(e)=eSe by [3, Theorem1.2.6]. Suppose that  K∩E is finite and K is not a topological subgroup of S. Let eK∈K∩E. Then,  K∖H(eK)≠∅. Otherwise, K=H(eK) is both the kernel and a maximal subgroup of S by [3, Theorem 1.3.14], and hence  K is topological subgroup of S with the relative topology, which contradicts our assumption.
Furthermore, sinceK∩E is finite and K∖H(eK)=∪{H(e):e∈K∩E,e≠eK}, it follows that K∖H(eK) and H(eK) form a separation of K. Hence,  K is disconnected, which contradicts [1, Theorem 1.28]. Therefore, we can deduce that either K∩E is infinite or K is a maximal subgroup of S.


## 2. Maximal Ideals of Compact Connected Topological Semigroups

The following theorem is a summary of the results found in [1]. It lists necessary and sufficient conditions for S2=S in a compact topological semigroup S. In this section, we characterize maximal ideals in a compact connected topological semigroup S with S2=S and S2≠S.Theorem 2.1.
LetS be a compact connected topological semigroup. The following are equivalent:  (a)
S2=S,(b)
E∩(S∖I)≠∅ for each proper ideal I of S,(c)
S=SES.
The following theorem and corollary are results from [3], which are useful for our discussion.Theorem 2.2.
LetS be a compact topological semigroup. Then, any proper (left, right) ideal of S is contained in a maximal (left, right) ideal of S, and each maximal (left, right) ideal is open.Corollary 2.3.
IfS is a compact connected topological semigroup and J a maximal ideal of S, then J is dense in S.Theorem 2.4.
Suppose thatS is a compact topological semigroup and S2≠S.  (a)
For eacha∈S∖S2, S∖{a} is a maximal ideal of S.(b)
IfS has more than one connected maximal ideal, then, S is connected.Proof.
(a) Leta∈S∖S2. For every x∈S∖{a} and y∈S, {xy,yx}⊂S2⊂S∖{a} implies that  S∖{a} is a proper ideal of S. (b) Let M1 and M2 be two distinct connected maximal ideals of S. Suppose that  S is disconnected. Then, M1∪M2=S=P∪Q such that P¯∩Q=∅=P∩Q¯. Since M1 and M2 are connected, M1⊂P and M2⊂P. It follows that M1∩M2=∅, and hence M1⊂S∖M2={a2} and M2⊂S∖M1={a1}. On the other hand, since M1 and M2 are ideals, a1a2=a2a1=a2 and a1a2=a2a1=a1, and hence M1={a2}={a1}=M2 contradicting M1 and M2 being distinct. Therefore, S is connected, and hence K is connected.
The following example shows that the conditionS having more than one connected maximal ideal is a necessary condition for Theorem 2.4(b).Example 2.52.1.
LetS=[0,1/4]∪{1/2} with the usual topology and the usual multiplication. Then, S2=[0,1/8]∪{1/4}≠S,K={0} is connected, M=[0,1/4] is the only connected maximal ideal of S, and S is disconnected.
The next theorem is Theorem2.4.3 of [3], and hence the proof is omitted.Theorem 2.6.
IfS is a connected topological semigroup and I an ideal of S, then one and only one component of I is an ideal of S.
One will call the ideal in Theorem2.6 the component ideal  of I.Theorem 2.7.
Let S be a compact connected topological semigroup andC= ⋃{MC:MCis the  ideal  component of a maximal proper ideal M}. Then either C=S or C is the maximal proper connected ideal of S. Furthermore, if C≠S, then C is the component ideal of a maximal ideal of S.Proof.
For each maximal idealM of S, let MC be its component ideal. Since K is the kernel and K⊂MC for each MC, C=⋃{MC:MC is  the  ideal  component  of  a maximal  proper  ideal M} is a connected ideal.
Suppose that there is a connected idealI such that C⊂I⊊S, then I is contained in a maximal ideal M of S. Since K⊂I∩MC, I∪MC is a connected ideal of S and is contained in M, and hence I∪MC⊂MC⊂C, a contradiction. Thus, if C≠S, then C is the maximal connected proper ideal of S. Furthermore, there exists a maximal ideal M of S such that C⊂M. Let MC be the component ideal of M. Then, MC=C.Lemma 2.8.
LetS be a compact connected topological semigroup, M a maximal ideal of S, and MC the component ideal of M. If S2≠S, then MC is not closed in S.Proof.
IfMC=M, then the result follows from Theorem 2.4(b).
IfMC⊊M, then M=MC∪KM where KM is the union of all components of M except MC. If MC were closed in S, then KM is open in S because KM=M∩(S∖MC) and M are both open. Therefore, for a∈S∖M, S=M¯=MC∪(KM∪{a}), and hence S is disconnected, which is a contradiction.The next theorem provides a necessary and sufficient condition for a compact connected topological semigroupS satisfying S2≠S by means of the component ideals of its maximal ideals.Theorem 2.9.
LetS be a compact connected topological semigroup. Then, S2≠S if and only if there exists a maximal ideal M of S with M=S∖{b}, b∈S∖S2 such that S2⊂MC where MC is a component ideal of M.Proof.
Suppose thatS2≠S. It follows from Theorem 2.1(a) that there exists a maximal ideal M of S such that E∩(S∖M)=∅. By [3, Theorem 1.3.8], S/M is either the zero semigroup of order two or else completely 0-simple.
Suppose thatS/M is the zero semigroup of order two. Then, S∖M={b} for some b∈S. If b∈S2, then b=xy with x,y∈S∖M. It is because if {x,y}∩M≠∅, then b∈M contradicting S∖M={b}. It follows that x=y=b, and hence b∈E. This contradicts E∩(S∖M)=∅. Therefore, b∈S∖S2 and S2⊂MC⊂M∖{b}. Note that the semigroup S/M is not completely 0-simple because if S/M were completely 0-simple, then S/M contains a nonzero primitive idempotent, which contradicts E∩(S∖M)=∅.
The converse is obviously true.The next example shows that the component idealMC of a maximal ideal M can be M itself.Example 2.102.10..
LetS=[0,1/2] with the usual multiplication and the usual topology. Then, S is a compact connected topological semigroup, and S2≠S. Let M=[0,1/2) and M#=S∖{5/16}. Then, M and M# are maximal ideals of S, and MC=[0,1/2)=M and MC#=[0,5/16)⊊M#.
The next theorem is Theorem1.40 of [1], and hence the proof is omitted.Theorem 2.11.
LetS be a compact connected topological semigroup. Then, S2=S if and only if each dense (left, right) ideal (containing K) is connected.
WhenS2=S, it is possible that aS=S for some a∈S. Existence of the set P={α∈S:αS=S} and its relationship to maximal ideals have been discussed in [3]. The following theorem provides a few additional properties of the set P of a compact topological semigroup S.Theorem 2.12.
Suppose thatS is a compact topological semigroup such that aS=S for some a∈S. Let P={α∈S:αS=S}. Then, the following is considered.(a)
P is a right group.(b)
IfP≠S, Then  S∖P is dense in S or S is disconnected.(c)
J0(S∖{a}) is dense in S for each a∈P if S is connected and P≠S.Proof.
(a) According to [3, Theorem 1.4.6], P=⋃e∈E∩PH(e), and P is a subtopological semigroup of S. Then, eS=S for all e∈E∩P, and hence e is a left identity of S. For each a∈P, a∈H(e) for some e∈E∩P, and hence there exists a-1∈H(e) such that aa-1=e. For any x∈P, x=(aa-1)x=a(a-1x)∈aP. It follows that P=aP for every a∈P, and hence P is right simple since S is compact and P is closed. The result follows from Theorem 1 of [4].(b) SinceP is a nonempty closed subtopological semigroup of S and the kernel K exists, S∖P is nonempty. In fact, by [3, Theorem 1.4.7], S∖P is the only maximal ideal of S because S≠P≠∅. If S∖P¯≠S, then S∖P is both open and closed by the maximality, and hence S is disconnected.(c)The result follows immediately from part (b) and the fact thatS∖P⊂J0(S∖{a}) for every a∈P.The following example shows that the conditionS≠P is necessary for Theorem 2.12(b) and (c).Example 2.13.
LetS=[0,1] with the usual topology and the multiplication xy=y for x,y∈S. Then, S=P=K.Definition 2.14.
A topological semigroupS has the left maximal property (right maximal property) if there exists a maximal left (right) ideal L* (R*) containing every proper left (right) ideal of S.
In [3], Paalmande Miranda presented several results showing how a compact connected topological semigroup S with the left or right maximal property is related to the condition S=aS, where a∈S. In the same spirit of these results and Theorem 2.11, the following theorem characterizes a compact connected topological semigroup satisfying the maximal property and the condition S=Sa∪aS∪SaS by means of its maximal ideals.Theorem 2.15.
LetS be a compact connected topological semigroup. Then, the following are equivalent.  (a)
There is an idempotente such that e∈S∖M for every maximal ideal M of S.(b)
The semigroupS has the maximal property and S=Sa∪aS∪SaS for some a∈S.Proof.
(a)⇒ (b) Since K⊂S∖{e} and I⊂J0(S∖{e}) for every proper ideal I of S, S has the maximal property with the maximal ideal J0(S∖{e}).
Leta∈S∖J0(S∖{e}). Then, J0(S∖{e}) is properly contained by the ideal Sa∪aS∪SaS∪{a}. Hence, Sa∪aS∪SaS∪{a}=S. Since S is connected and Sa, aS, SaS, and {a}  are closed, a∈Sa∪aS∪SaS, and hence, S=Sa∪aS∪SaS.
(b)⇒ (a) Suppose that S has the maximal property with the maximal ideal M* and S does not satisfy the condition in part (a). Then, E⊂M*, and hence it follows from Theorem 2.9 that S2⊂M*. On the other hand, S=Sa∪aS∪SaS⊂S2⊂M*, which contradicts M* being the maximal ideal of S.
The following corollary to Theorem1.4.12 of  [3] implies that the maximal ideal M in Theorem 2.9 is not unique.Corollary 2.16.
A necessary and sufficient condition that a compact connected topological semigroupS has the maximal ideal property is that S has at least one idempotent e with S=SeS and S is not simple.



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