On Maximal Ideals of Compact Connected Topological Semigroups

Abstract

Several results concerning ideals of a compact topological semigroup 𝑆 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.

1. Introduction

First, we list some standard definitions which can be found in [1–3].

Definition 1.1. A topological semigroup is a topological space 𝑆 together with a continuous function 𝑚∶𝑆×𝑆→𝑆 such that 𝑆 is Hausdorff and 𝑚 is associative.
A subsemigroup 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 element 𝑒 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 of 𝑆 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 subset 𝐴 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) ideal 𝑀 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.
A maximal (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. Let 𝐴 be a subset of a topological semigroup 𝑆, then 𝐽₀(𝐴) is defined as follows: 𝐽₀(𝐴)=∅if𝐴containsnoidealof𝑆,∪{𝐼∶𝐼isanidealof𝑆and𝐼⊂𝐴}. (1.1)

Theorem 1.4. Let 𝑆 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. Since 𝑆 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 𝑒_𝐾∈𝐾∩𝐸. Then, 𝐾⧵𝐻(𝑒_𝐾)≠∅. Otherwise, 𝐾=𝐻(𝑒_𝐾) 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, since 𝐾∩𝐸 is finite and 𝐾⧵𝐻(𝑒_𝐾)=∪{𝐻(𝑒)∶𝑒∈𝐾∩𝐸,𝑒≠𝑒_𝐾}, it follows that 𝐾⧵𝐻(𝑒_𝐾) and 𝐻(𝑒_𝐾) 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. Let 𝑆 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. Let 𝑆 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. If 𝑆 is a compact connected topological semigroup and 𝐽 a maximal ideal of 𝑆, then 𝐽 is dense in 𝑆.

Theorem 2.4. Suppose that 𝑆 is a compact topological semigroup and 𝑆²≠𝑆. (a)For each 𝑎∈𝑆⧵𝑆², 𝑆⧵{𝑎} is a maximal ideal of 𝑆. (b)If 𝑆 has more than one connected maximal ideal, then, 𝑆 is connected.

Proof. (a) Let 𝑎∈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 condition 𝑆 having more than one connected maximal ideal is a necessary condition for Theorem 2.4(b).

Example 2.5 2.1. Let 𝑆=[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 Theorem 2.4.3 of [3], and hence the proof is omitted.

Theorem 2.6. If 𝑆 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 Theorem 2.6 the component ideal of 𝐼.

Theorem 2.7. Let S be a compact connected topological semigroup and𝐶= ⋃{𝑀_𝐶∶𝑀_𝐶is 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 ideal 𝑀 of 𝑆, let 𝑀_𝐶 be its component ideal. Since 𝐾 is the kernel and 𝐾⊂𝑀_𝐶 for each 𝑀_𝐶, 𝐶=⋃{𝑀_𝐶∶𝑀_𝐶 is the ideal component of a maximal proper ideal 𝑀} is a connected ideal.
Suppose that there is a connected ideal 𝐼 such that 𝐶⊂𝐼⊊𝑆, then 𝐼 is contained in a maximal ideal 𝑀 of 𝑆. Since 𝐾⊂𝐼∩𝑀_𝐶, 𝐼∪𝑀_𝐶 is a connected ideal of 𝑆 and is contained in 𝑀, and hence 𝐼∪𝑀_𝐶⊂𝑀_𝐶⊂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 𝑀_𝐶 be the component ideal of 𝑀. Then, 𝑀_𝐶=𝐶.

Lemma 2.8. Let 𝑆 be a compact connected topological semigroup, 𝑀 a maximal ideal of 𝑆, and 𝑀_𝐶 the component ideal of 𝑀. If 𝑆²≠𝑆, then 𝑀_𝐶 is not closed in 𝑆.

Proof. If 𝑀_𝐶=𝑀, then the result follows from Theorem 2.4(b).
If 𝑀_𝐶⊊𝑀, then 𝑀=𝑀_𝐶∪𝐾_𝑀 where 𝐾_𝑀 is the union of all components of 𝑀 except 𝑀_𝐶. If 𝑀_𝐶 were closed in 𝑆, then 𝐾_𝑀 is open in 𝑆 because 𝐾_𝑀=𝑀∩(𝑆⧵𝑀_𝐶) and 𝑀 are both open. Therefore, for 𝑎∈𝑆⧵𝑀, 𝑆=𝑀=𝑀_𝐶∪(𝐾_𝑀∪{𝑎}), and hence 𝑆 is disconnected, which is a contradiction.

The next theorem provides a necessary and sufficient condition for a compact connected topological semigroup 𝑆 satisfying 𝑆²≠𝑆 by means of the component ideals of its maximal ideals.

Theorem 2.9. Let 𝑆 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⊂𝑀_𝐶 where 𝑀_𝐶 is a component ideal of 𝑀.

Proof. Suppose that 𝑆²≠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 that 𝑆/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⊂𝑀_C⊂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 ideal 𝑀_𝐶 of a maximal ideal 𝑀 can be 𝑀 itself.

Example 2.10 2.10.. Let 𝑆=[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 𝑀_𝐶=[0,1/2)=𝑀 and 𝑀^#_𝐶=[0,5/16)⊊𝑀^#.
The next theorem is Theorem 1.40 of [1], and hence the proof is omitted.

Theorem 2.11. Let 𝑆 be a compact connected topological semigroup. Then, 𝑆²=𝑆 if and only if each dense (left, right) ideal (containing 𝐾) is connected.
When 𝑆²=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 that 𝑆 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)If 𝑃≠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∩𝑃}𝐻(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) Since 𝑃 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 that 𝑆⧵𝑃⊂𝐽₀(𝑆⧵{𝑎}) for every 𝑎∈𝑃.

The following example shows that the condition 𝑆≠𝑃 is necessary for Theorem 2.12(b) and (c).

Example 2.13. Let 𝑆=[0,1] with the usual topology and the multiplication 𝑥𝑦=𝑦 for 𝑥,𝑦∈𝑆. Then, 𝑆=𝑃=𝐾.

Definition 2.14. A topological semigroup 𝑆 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. Let 𝑆 be a compact connected topological semigroup. Then, the following are equivalent. (a)There is an idempotent 𝑒 such that 𝑒∈𝑆⧵𝑀 for every maximal ideal 𝑀 of 𝑆. (b)The semigroup 𝑆 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}).
Let 𝑎∈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 Theorem 1.4.12 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 semigroup 𝑆 has the maximal ideal property is that 𝑆 has at least one idempotent 𝑒 with 𝑆=𝑆𝑒𝑆 and 𝑆 is not simple.