SQU Journal for Science, 2019, 24(2), 139-146  DOI:10.24200/squjs.vol24iss2pp139-146 

Sultan Qaboos University  

139 

 

Eventually Pointed Principally Ordered 
Regular Semigroups 

G.A. Pinto 

Department of Mathematics, Sultan Qaboos University, P.O. Box 36, PC 123, Al- Khoud, Muscat 
Sultanate of Oman. Email: goncalo@squ.edu.om.  

ABSTRACT: An ordered regular semigroup, 𝑆, is said to be principally ordered if for every 𝑥 ∈ 𝑆 there exists 
𝑥 ∗ = max{𝑦 ∈ 𝑆|𝑥𝑦𝑥 ≤ 𝑥}. A principally ordered regular semigroup is pointed if for every element, 𝑥, we have 
𝑥 2 ≤ 𝑥. Here we investigate those principally ordered regular semigroups that are eventually pointed in the sense that 
for all 𝑥 ∈ 𝑆 there exists a positive integer, 𝑛, such that (𝑥 𝑛)2 ≤ 𝑥 𝑛 . Necessary and sufficient conditions for an 
eventually pointed principally ordered regular semigroup to be naturally ordered and to be completely simple are 

obtained. We describe the subalgebra of (𝑆,∗ ) generated by a pair of comparable idempotents 𝑒 and 𝑓such that 
𝑒 0 = 𝑓 0. 
 

Keywords: Regular semigroup; Strong Dubreil-Jacotin; Principally ordered; Naturally ordered; Pointed principally 

ordered; Green’s relations; Completely simple. 

لترتيب األساسي لشبه الزمر العاديةلالتوجه النهائي   

 بينتو  ج.أ

∗𝒙 لعنصر ا   𝑺 ϶ 𝒙، مرتبة أساسا إذا وجد لكل   𝑺يقال إن شبه الزمر العادية المرتبة، :صلخمال = {𝒚 ∈  𝑺 | 𝒙𝒚𝒙 ≤ 𝒙} .إلى مبدأ تتم اإلشارة 
𝒙𝟐.ايلي: م 𝒙 ر شبه الزمر العادية إذا كان لكل عنص ترتيب ≤ 𝒙  في النهاية بمعنى أنه لكل  هتوجنتحرى هنا عن شبه الزمر العادية المرتبة أساسا والتي

.   يكون ، بحيث𝒏عدد صحيح موجب، يوجد   𝑺 ϶ 𝒙عنصر  (𝒙𝒏)𝟐 ≤ 𝒙𝒏  إلى مبدأ يتم الحصول على الشروط الضرورية والكافية في نهاية األمر
 يدعىالتي تم إنشاؤها بواسطة زوج قابل للمقارنة  ∗,𝑺)  (الجبر الجزئي ل ببساطة كاملة. لقد وصفنا شبه الزمر العادية لتكون مرتبة طبيعيا ترتيب

𝒆𝟎يكون و سأيديمبوتنت = 𝒇𝒐..  
 

 جرين، بسيط متكامل. ساسا، مرتب طبيعيا، مرتب وموجه أساسا، عالقاتجاكوتن القوي، مرتب أ-دوبريلشبه الزمر العادية، : مفتاحيةالكلمات ال

1.  Introduction  

n ordered regular semigroup, 𝑆, is said to be principally ordered if for every 𝑥 ∈ 𝑆 there exists                               
𝑥 ∗ = max{𝑦 ∈ 𝑆|𝑥𝑦𝑥 ≤ 𝑥}. The basic properties of the unary operation 𝑥 → 𝑥 ∗ in such semigroups were 

established in [1] and [2] and are listed in [3, Theorem 13.26]. In particular, we recall for the reader’s convenience that, 

in such a semigroup, the following properties hold, and will be used throughout in what follows: 

 

(𝑃1) (∀𝑥 ∈ 𝑆) 𝑥 = 𝑥𝑥
∗𝑥 

(𝑃2) every 𝑥 ∈ 𝑆 has a biggest inverse, namely 𝑥
0 = 𝑥 ∗𝑥𝑥 ∗ 

(𝑃3) (∀𝑥 ∈ 𝑆) 𝑥
0 ≤ 𝑥 ∗ 

(𝑃4) (∀𝑥 ∈ 𝑆) 𝑥 ≤ 𝑥
∗∗ 

 

In [4] we introduced the class of pointed principally ordered regular semigroups. We say that a principally ordered 

regular semigroup, 𝑆, is pointed, if the classes modulo Green’s relations ℛ, ℒ, 𝒟 have biggest elements which are 
idempotent. In [4, Theorem 1] we proved, in particular, that a principally ordered regular semigroup S is pointed if and 

only if 𝑥 2 ≤ 𝑥, for every 𝑥 ∈ 𝑆. In this paper we investigate a generalization of pointed principally ordered regular 
semigroups. 

 

 

A 

mailto:goncalo@squ.edu.om


 PINTO, G.A.  

140 

 

2. Eventually pointed principally ordered regular semigroups 

Definition Let S be a principally ordered regular semigroup. We say that S is eventually pointed if, for every 𝑥 ∈ 𝑆, 
there exists a positive integer, 𝑛, such that (𝑥 𝑛 )2 ≤ 𝑥 𝑛 , that is, 𝑥 2𝑛 ≤ 𝑥 𝑛. 
 

An immediate observation is that a pointed principally ordered regular semigroup is eventually pointed. Simply 

take 𝑛 = 1. 
Note that any finite ordered regular semigroup, 𝑆, is eventually pointed. In fact, there exists a positive integer 

power of each element 𝑆 that is an idempotent, and 𝑆 is eventually pointed. 
 

Under the notation of the previous definition, we have that 

 

𝑥 𝑛 ∙ 𝑥 𝑛 ∙ 𝑥 𝑛 = 𝑥 2𝑛 ∙ 𝑥 𝑛 ≤ 𝑥 𝑛 ∙ 𝑥 𝑛 = 𝑥 2𝑛 ≤ 𝑥 𝑛 
 

from which we may conclude the following property 

 

(𝛼) (∀𝑥 ∈ 𝑆)(∃𝑛 ∈ ℤ+) 𝑥 𝑛 ≤ (𝑥 𝑛)∗ 
 

Now, let us present examples that show that we are, in fact, in the presence of a new class of semigroups. 

 

Example 1. Consider Mat2𝐵, the ordered regular semigroup of 2 × 2 matrices with entries in a Boolean algebra 𝐵, and 
matrix multiplication, which can be seen in [1, Example 13.1]). For the basic operations in 𝐵 we use the notation 𝑎 + 𝑏 
(for 𝑎 ∨ 𝑏) and 𝑎𝑏 (for 𝑎 ∧ 𝑏). The subset Mat2𝐵(0,1) where all the entries of the matrices are either 0 or 1, 
respectively the bottom and top element of 𝐵, is a semigroup for matrix multiplication. Routine calculations allow us to 
say that Mat2𝐵(0,1) is a principally ordered regular semigroup with Hasse diagram: 
 

 

( denotes idempotents) 

where 

[
0 1
1 0

]
∗

= [
0 1
1 0

] , [
1 0
0 1

]
∗

= [
1 0
0 1

] , [
1 0
1 1

]
∗

= [
1 0
1 1

] 

 

and 

[
1 1
0 1

]
∗

= [
1 1
0 1

] , [
1 1
1 0

]
∗

= [
0 1
1 1

] , [
0 1
1 1

]
∗

= [
1 1
1 0

] 



EVENTUALLY POINTED PRINCIPALLY ORDERED REGULAR SEMIGROUPS 

 

141 

 

and, for every other element [
𝑎 𝑏
𝑐 𝑑

] ∈ Mat2𝐵(0,1), 

 

[
𝑎 𝑏
𝑐 𝑑

]
∗

= [
1 1
1 1

]. 

For biggest inverses 

[
0 0
0 0

]
0

= [
0 0
0 0

] 

and for any other element in [
𝑎 𝑏
𝑐 𝑑

] ∈ Mat2𝐵(0,1), 

[
𝑎 𝑏
𝑐 𝑑

]
0

= [
𝑎 𝑏
𝑐 𝑑

]
∗

. 

 

Note that the element [
1 1
1 0

] in Mat2𝐵(0,1) is such that 

 

[
1 1
1 0

]
2

= [
1 1
1 0

] [
1 1
1 0

] = [
1 1
1 1

] > [
1 1
1 0

], 

meaning that this principally ordered regular semigroup is not pointed. However, since it is a finite semigroup, it is 

eventually pointed. 

 

Example 2. Consider the set S = 0,a,1,b{ } with the following Cayley table and Hasse diagram: 
 

 

Routine calculations allow us to conclude that, with the multiplication and partial order just defined, S is an 

ordered regular semigroup. In fact, it is a principally ordered semigroup, with 

 

0∗ = 𝑏, 1∗ = 1, 𝑏∗ = 𝑏 and 𝑎∗ = 𝑎. 

Note that S is not pointed since 𝑎2 = 1 is not comparable with 𝑎. As in the previous example, we can conclude 
that, since 𝑆 is a finite semigroup, 𝑆 is an eventually pointed semigroup. 

We can obtain this semigroup as a subsemigroup of Mat2𝐵(0,1), in Example 1. In fact, 
 

𝑇 = {[
0 0
0 0

] , [
0 1
1 0

] , [
1 0
0 1

] , [
1 1
1 1

]} 

 

has the same Cayley table and Hasse diagram, denoting elements of T respectively, by 0, a, 1, b. 

Alternatively, if we take a Boolean Algebra, 𝐵, and any element in it, 𝑥, different from 0 and 1, then 
 

𝑈 = {[
0 0
0 0

] , [
0 𝑥
𝑥 0

] , [
𝑥 0
0 𝑥

] , [
𝑥 𝑥
𝑥 𝑥

]} 

 

with matrix multiplication and the Cartesian order, we have the same Cayley table and Hasse diagram, and therefore 

the same ordered semigroup.  

 

Example 3. Consider the subset 

𝐺 = {[
0 1
1 0

] , [
1 0
0 1

]} 

 

of Mat2𝐵(0,1) described in Example 1. Obviously, it is a totally unordered group: 
 

 

 0 a 1 b 

0 0 0 0 0 

a 0 1 a b 

1 0 a 1 b 

b 0 b b b 



 PINTO, G.A.  

142 

 

We have that [
0 1
1 0

]
∗

= [
0 1
1 0

] and [
1 0
0 1

]
∗

= [
1 0
0 1

], meaning that 𝐺 is principally ordered. 

 

However, it is not pointed since [
0 1
1 0

]
2

= [
1 0
0 1

] is not comparable with [
0 1
1 0

]. By a similar argument to the one 

used in Example 1, we conclude that 𝐺 is eventually pointed. 
 

Example 4.  Consider the principally ordered band, constructed in [4, Example 1], 𝐿[2] = {(𝑥, 𝑦) ∈ 𝐿 × 𝐿|𝑦 ≤ 𝑥}, 
where 𝐿 is a lattice and the multiplication is defined by 
 

(𝑥, 𝑦)(𝑎, 𝑏) = (𝑥 ∨ 𝑎, 𝑦 ∧ 𝑏). 
 

The Cartesian product with the Cartesian order of 𝐿[2] and 𝑆 in Example 2 is an infinite principally ordered 
regular semigroup that is eventually pointed, but not pointed. 

Example 3 above provides an example of a principally ordered regular semigroup which is eventually pointed, 

that has a subgroup which is totally unordered and whose elements have finite order (one and two). This is always the 

case, as we can see in the next result, due to T. S. Blyth. 

 

Theorem 1. If 𝑆 is an eventually pointed principally ordered regular semigroup and if 𝐺 is any subgroup of S, then 𝐺 
is totally unordered and every 𝑥 ∈ 𝐺 is of finite order. 
Proof. For every 𝑥 ∈ 𝐺 there exists a positive integer, 𝑛, such that 𝑥 2𝑛 ≤ 𝑥 𝑛 . Let  𝐺 + = {𝑥 ∈ 𝐺|1𝐺 ≤ 𝑥} be the 
positive cone of 𝐺. If 𝑥 ∈ 𝐺 +, then we have that 1𝐺 ≤ 𝑥 ≤ 𝑥

2 ≤ ⋯ ≤ 𝑥 𝑛 ≤ ⋯ ≤ 𝑥 2𝑛 ≤ ⋯ whence 𝑥 𝑛 = 𝑥 2𝑛 and so 
𝑥 𝑛 is an idempotent in 𝐺. Thus 𝑥 𝑛 = 1𝐺 , consequently 𝑥 = 1𝐺  and 𝐺

+ = {1𝐺 }. The order in 𝐺 is then given by 
 

𝑔 ≤ ℎ ⇔ 1𝐺 ≤ ℎ𝑔
−1 ⇔ ℎ𝑔−1 ∈ 𝐺 + = {1𝐺 } ⇔ ℎ = 𝑔. 

 

Thus, 𝐺 is totally unordered. Then, for every 𝑥 ∈ 𝐺, it follows from 𝑥 2𝑛 ≤ 𝑥 𝑛, that  𝑥 2𝑛 = 𝑥 𝑛  whence 𝑥 𝑛 = 1𝐺  and so 
𝑥 is of finite order. 

 

Several of the results in [4], or similar ones, can be obtained in this new class of eventually pointed principally 

ordered regular semigroups. For this reason, it may be necessary to provide a new proof, or adjust the existent one for 

pointed principally ordered regular semigroups. 

 

Theorem 2. Let 𝑆 be an eventually pointed principally ordered regular semigroup. If 𝑒 is a maximal idempotent of 𝑆, 
then 𝑒 is a maximal element of 𝑆. 
Proof. Let 𝑒 be a maximal idempotent of 𝑆 and consider any element 𝑥 in 𝑆 such that 𝑒 ≤ 𝑥. Since, 𝑆 is eventually 
pointed, there exists a positive integer, 𝑛, such that 𝑥 2𝑛 ≤ 𝑥 𝑛. Then, 𝑒 = 𝑒 𝑛 = 𝑒 … 𝑒 ≤ 𝑥 … 𝑥 = 𝑥 𝑛, and using (𝛼), 
𝑒 = 𝑒𝑒 ≤ 𝑥 𝑛𝑥 𝑛 ≤ 𝑥 𝑛(𝑥 𝑛)∗ ∈ 𝐸(𝑆), whence 𝑒 = 𝑥 𝑛(𝑥 𝑛 )∗. Now,  
 

𝑥 𝑛 = 𝑥 𝑛(𝑥 𝑛)∗𝑥 𝑛 = 𝑒𝑥 𝑛 ≤ 𝑥𝑥 𝑛 = 𝑥 𝑛+1 ≤ ⋯ ≤ 𝑥 2𝑛 ≤ 𝑥 𝑛, 
 

from which we conclude that 𝑥 𝑛 ∈ 𝐸(𝑆) and consequently 𝑒 = 𝑥 𝑛. It follows that  𝑒𝑥 = 𝑥 𝑛𝑥 = 𝑥 𝑛+1 = 𝑥 𝑛 = 𝑒, and 
similarly 𝑥𝑒 = 𝑒. Thus,  𝑥𝑒𝑥 = 𝑒𝑥 = 𝑒 ≤ 𝑥 gives 𝑒 ≤ 𝑥 ∗, whence 𝑒 = 𝑒𝑒 ≤ 𝑥𝑥 ∗ , which implies 𝑒 = 𝑥𝑥 ∗ and 
therefore 𝑥 = 𝑥𝑥 ∗𝑥 = 𝑒𝑥 = 𝑒 , meaning that 𝑒 is a maximal element of 𝑆. 
 

Theorem 3. An eventually pointed principally ordered regular semigroup, 𝑆, has at most one maximal idempotent. 
Proof. Let 𝑒, 𝑓 be maximal idempotents of 𝑆. Consider the element 𝑒𝑓 in 𝑆. Since 𝑆 is eventually pointed there exists a 
positive integer, 𝑛, such that (𝑒𝑓)2𝑛 ≤ (𝑒𝑓)𝑛 . From 
 

(𝑒𝑓)𝑛 𝑒(𝑒𝑓)𝑛 = (𝑒𝑓)𝑛 𝑒𝑒𝑓(𝑒𝑓)𝑛−1 = (𝑒𝑓)𝑛𝑒𝑓(𝑒𝑓)𝑛−1 = (𝑒𝑓)2𝑛 ≤ (𝑒𝑓)𝑛 

we obtain 𝑒 ≤ ((𝑒𝑓)𝑛 )∗ and similarly 𝑓 ≤ ((𝑒𝑓)𝑛)∗. Using Theorem 2, we have that both 𝑒 and 𝑓 are maximal 
elements of S and therefore 𝑒 = ((𝑒𝑓)𝑛)∗ = 𝑓. 
 

We recall that the natural order, ≤𝑛 , on the idempotents of a regular semigroup is defined by 
 

𝑒 ≤𝑛 𝑓 ⇔ 𝑒 = 𝑒𝑓 = 𝑓𝑒 
and that an ordered regular semigroup (𝑇, ≤) is said to be naturally ordered if the order extends the natural order, in 
the sense that if 𝑒 ≤𝑛 𝑓 then 𝑒 ≤ 𝑓. 
 



EVENTUALLY POINTED PRINCIPALLY ORDERED REGULAR SEMIGROUPS 

 

143 

 

Theorem 4 Let 𝑆 be an eventually pointed principally ordered regular semigroup. The following statements are 
equivalent: 

(1) 𝑆 is naturally ordered 
(2) 𝑆 has a biggest idempotent (in fact, a maximal element), 𝜉, and 𝜉 = 𝑒 ∗ for every idempotent 𝑒 of 𝑆. 

 

Proof. (1) ⇒ (2): By [1, Theorem 13.29(3)] we have that, for every idempotent e of S, (𝑒𝑒 ∗)∗ is a maximal idempotent. 
Thus, by Theorem 3, we can conclude that S has a biggest idempotent, 𝜉, which by Theorem 2 is a maximal element of 

𝑆. From (1) and [1, Theorem 13.28] we have that 𝑆 is a strong Dubreil-Jacotin semigroup and using                      

[1, Theorem 13.20], 𝑒 = 𝑒(𝜉: 𝑒)𝑒 for every idempotent 𝑒 of  𝑆. Since, 𝑒𝜉 ≤ 𝜉,  it then follows that 
 𝜉 ≤ 𝜉: 𝑒 ≤ 𝑒 ∗, whence 𝜉 = 𝑒 ∗, using the fact that 𝜉 is a maximal element of S. 
(2) ⇒ (1): Let 𝑒 and 𝑓 be idempotents of 𝑆 such that 𝑒 ≤𝑛 𝑓, that is 𝑒 = 𝑒𝑓 = 𝑓𝑒. Then, using (2), we have that 
𝑒 ∗ = 𝜉 = 𝑓 ∗ , and therefore 
𝑒 = 𝑓𝑒𝑓 ≤ 𝑓𝑒 ∗𝑓 = 𝑓𝜉𝑓 = 𝑓𝑓 ∗𝑓 = 𝑓,  
from which we conclude that S is naturally ordered. 

 

Note that [4, Theorem 5] states that a pointed principally ordered regular semigroup, S, is naturally ordered if and 

only if it has a biggest element, 𝜉, and 𝜉 = 𝑥 ∗ for every 𝑥 in 𝑆. 
 

Corollary If 𝑆 is an eventually pointed principally ordered and naturally ordered regular semigroup, then Green’s 
relations 𝒟 and 𝒥 coincide. 
Proof. Let 𝑥 be an arbitrary element of 𝑆. Since 𝑆 is eventually pointed, there exists a positive integer, 𝑛, such that 
𝑥 2𝑛 ≤ 𝑥 𝑛. By [1, Theorem 13.27] we have that the unary operation 𝑥 → 𝑥 ∗ is antitone, whence (𝑥 𝑛)∗ ≤ (𝑥 2𝑛)∗. From 
Theorem 4, 𝑆 has a biggest idempotent, 𝜉, which is, in fact, a maximal element of 𝑆, and therefore 𝜉𝜉𝜉 = 𝜉 implies 
𝜉 ≤ 𝜉∗ whence 𝜉 = 𝜉∗. 

Now, using (𝛼), 𝑥 2𝑛 = 𝑥 𝑛𝑥 𝑛 ≤ 𝑥 𝑛 (𝑥 𝑛)∗ ≤ 𝜉, whence 𝜉 = 𝜉∗ ≤ (𝑥 2𝑛)∗, and consequently 𝜉 = (𝑥 2𝑛)∗, which 
implies 𝑥 𝑛 ≤ (𝑥 𝑛)∗ ≤ (𝑥 2𝑛)∗ = 𝜉, from which we obtain 𝜉 = 𝜉∗ ≤ (𝑥 𝑛)∗ and therefore 𝜉 = (𝑥 𝑛 )∗. Thus, 

 𝑥 2𝑛 = 𝑥 2𝑛(𝑥 2𝑛)∗𝑥 2𝑛 = 𝑥 𝑛𝑥 𝑛(𝑥 𝑛)∗𝑥 𝑛 𝑥 𝑛 = 𝑥 𝑛 𝑥 𝑛𝑥 𝑛 = 𝑥 3𝑛; that is, 𝑥 2𝑛 is an idempotent of 𝑆. Therefore 𝑆 is a 
group bound and [5, Theorem 1.2.20] allows us to conclude that Green’s relations 𝒟 and 𝒥 coincide. 

 

In the presence of an identity element, 1, an interesting situation can be described. 

 

Theorem 5. Let S be an eventually pointed principally ordered regular semigroup with an identity element, 1, and 
consider 𝑇 = {𝑥 ∈ 𝑆|1 ≤ 𝑥}. For every 𝑥 ∈ 𝑇, there exists a positive integer, 𝑛, such that 𝑥 𝑛 = (𝑥 𝑛)∗ = (𝑥 𝑛 )∗∗ is an 
idempotent of 𝑆. 
Proof. Since 𝑆 is eventually pointed, for every 𝑥 ∈ 𝑇 there exists a positive integer, 𝑛, such that 𝑥 2𝑛 ≤ 𝑥 𝑛 . We have 
that 1 ≤ 𝑥 and 1 = 1 … 1 ≤ 𝑥 … 𝑥 = 𝑥 𝑛  whence, by (𝑃4), 1 ≤ (𝑥

𝑛)∗∗. Then, (𝑥 𝑛)∗ ∙ 1 ∙ (𝑥 𝑛 )∗ ≤ (𝑥 𝑛)∗; that is, 
((𝑥 𝑛)∗)2 ≤ (𝑥 𝑛)∗ and therefore, using (𝛼), 
 

(𝑥 𝑛)∗ = 1 ∙ (𝑥 𝑛)∗ ≤ (𝑥 𝑛)(𝑥 𝑛)∗ ≤ (𝑥 𝑛)∗(𝑥 𝑛)∗ = ((𝑥 𝑛)∗)2 ≤ (𝑥 𝑛)∗ 
 

which gives (𝑥 𝑛)∗ = (𝑥 𝑛 )(𝑥 𝑛)∗ , an idempotent of 𝑆. Similarly (𝑥 𝑛 )∗ = (𝑥 𝑛)∗(𝑥 𝑛) , from which we obtain 
 

𝑥 𝑛 = 𝑥 𝑛(𝑥 𝑛)∗(𝑥 𝑛)∗∗(𝑥 𝑛)∗𝑥 𝑛 = (𝑥 𝑛)∗(𝑥 𝑛)∗∗(𝑥 𝑛)∗ = (𝑥 𝑛)∗ 
 

allowing us to conclude that (𝑥 𝑛)∗ = (𝑥 𝑛)∗∗ and, therefore, 𝑥 𝑛 = (𝑥 𝑛)∗ = (𝑥 𝑛 )∗∗. 
 

In the special case where S is a pointed principally ordered regular semigroup, with an identity, 1, we have that 
the positive integer in the previous theorem is 𝑛 = 1. This means that, for these semigroups, we have 𝑥 = 𝑥 ∗ = 𝑥 ∗∗ for 
every element 𝑥 in 𝑆 such that 1 ≤ 𝑥. Thus, we obtain the following Corollary of Theorem 5. 

 

Corollary Let 𝑆 be a pointed principally ordered regular semigroup with an identity element, 1, and consider 𝑇 =
{𝑥 ∈ 𝑆|1 ≤ 𝑥}. For every 𝑥 ∈ 𝑇, 𝑥 = 𝑥 ∗ = 𝑥 ∗∗ is an idempotent of 𝑆. 
Proof. It is an immediate consequence of Theorem 5. 

From these results, we therefore obtain as a consequence [4, Theorem 7]: 

 

Theorem 6 Let 𝑆 be a pointed principally ordered regular semigroup with an identity element, 1. The set                
𝑆 ∗ = {𝑥 ∈ 𝑆|1 ≤ 𝑥} is a join semilattice in which 𝑥 ∨ 𝑦 = 𝑥𝑦. 
Proof. We have, for every 𝑥 ∈ 𝑆, that 𝑥 ∙ 1 ∙ 𝑥 = 𝑥 2 ≤ 𝑥 , which implies that 1 ≤ 𝑥 ∗ and, therefore,                          
𝑆 ∗ ⊆ {𝑥 ∈ 𝑆|1 ≤ 𝑥}. Conversely, for any 𝑥 ∈ {𝑥 ∈ 𝑆|1 ≤ 𝑥}, we obtain, by the Corollary to Theorem 5, that 𝑥 = 𝑥 ∗ , 
which means that {𝑥 ∈ 𝑆|1 ≤ 𝑥} ⊆ 𝑆 ∗. Thus, 𝑆 ∗ = {𝑥 ∈ 𝑆|1 ≤ 𝑥}. 



 PINTO, G.A.  

144 

 

The fact that 𝑆 ∗ = {𝑥 ∈ 𝑆|1 ≤ 𝑥} is a join semilattice in which 𝑥 ∨ 𝑦 = 𝑥𝑦 follows, using the same argument as in       
[4, Theorem 7]. 

 

Example 5. In Mat2𝐵(0,1) of Example 1, [
1 0
0 1

] is the identity element and we have 

 

(Mat2𝐵(0,1))
∗

= {[
1 0
0 1

] , [
1 0
1 1

] , [
1 1
0 1

] , [
1 1
1 1

] , [
0 1
1 0

] , [
1 1
1 0

] , [
0 1
1 1

]} 

while 

{[
𝑎 𝑏
𝑐 𝑑

] ∈ Mat2𝐵(0,1) |[
1 0
0 1

] ≤ [
𝑎 𝑏
𝑐 𝑑

]} = {[
1 0
0 1

] , [
1 0
1 1

] , [
1 1
0 1

] , [
1 1
1 1

]} 

This does not contradict Theorem 6, since Mat2𝐵(0,1) is not pointed. 
 

In [4, Theorem 8] it is proven that any two-comparable 𝒟-related idempotents in a pointed principally ordered 
regular semigroup are mutually inverse. This heavily depends on the fact that in such a semigroup 𝑥𝒟𝑦 if and only if 
𝑥 0 = 𝑦0. This is not necessarily true in eventually pointed principally ordered regular semigroups. To see this, let us 
present an example. 

 

Example 6. In Mat2𝐵(0,1) of Example 1, we have that [
1 0
0 1

] and [
0 1
1 0

] are 𝒟 related, but 

 

[
1 0
0 1

] = [
1 0
0 1

]
0

≠ [
0 1
1 0

]
0

= [
0 1
1 0

]. 

Although the previous description of Green’s 𝒟 relation is not true in our situation, we can prove a generalization of  
[4, Theorem 8] to the wider class of ordered regular semigroups with biggest inverses. 

 

Theorem 7. If 𝑆 is an ordered regular semigroup with biggest inverses such that 𝑒 and 𝑓 are comparable idempotents, 
for which 𝑒 0 = 𝑓 0, then 𝑒 and 𝑓 are mutually inverse. 
 

Proof. Let us assume without loss of generality that the idempotents 𝑒 and 𝑓 are such that 𝑒 ≤ 𝑓 and 𝑒 0 = 𝑓 0. Then, 
 

𝑒 = 𝑒𝑒𝑒 ≤ 𝑒𝑓𝑒 ≤ 𝑒𝑓 0𝑒 = 𝑒𝑒 0𝑒 = 𝑒 ⇒ 𝑒 = 𝑒𝑓𝑒 
and 

 

𝑓𝑒𝑓 ≤ 𝑓𝑓𝑓 = 𝑓 = 𝑓𝑓 0𝑓 = 𝑓𝑒 0𝑓 = 𝑓𝑒 0𝑒𝑒 0𝑓 = 𝑓𝑓 0𝑒𝑒𝑒𝑓 0𝑓 ≤ 𝑓𝑓 0𝑓𝑒𝑓𝑓 0𝑓 = 𝑓𝑒𝑓 ⇒ 𝑓 = 𝑓𝑒𝑓 
 

which means that 𝑒 and 𝑓 are mutually inverse. 
 

In order to characterise, when an eventually pointed principally ordered regular semigroup is completely simple, 

we consider 𝐶 = {𝑥 ∈ 𝑆|𝑥 0 = 𝑥 ∗} the set of compact elements of 𝑆. 
 

Theorem 8. Let 𝑆 be an eventually pointed principally ordered regular semigroup. The following statements are 
equivalent: 

(1) 𝑆 is completely simple; 
(2) 𝑆 is naturally ordered and 𝐸(𝑆) ⊆ 𝐶; 
(3) 𝑆 has a biggest idempotent (in fact, a maximal element), 𝜉, and 𝜉 = 𝑒 0 for every idempotent 𝑒 of S. 

 

Proof. (1) ⇒ (2): If 𝑆 is completely simple then, since ≤𝑛  reduces to equality, 𝑆 is trivially naturally ordered. We 
have, by Theorem 4, that 𝑆 has a biggest idempotent, such that for every idempotent 𝑒 of 𝑆, 𝜉 = 𝑒 ∗. Now,  
 

𝑒 0 = 𝑒 ∗𝑒𝑒 ∗ = 𝜉𝑒𝜉 
and 

𝑒 0𝑒 0 = 𝜉𝑒𝜉𝜉𝑒𝜉 = 𝜉𝑒𝜉𝑒𝜉 = 𝜉𝑒𝑒 ∗𝑒𝜉 = 𝜉𝑒𝜉 = 𝑒 0 
 

gives that 𝑒 ∗ = 𝜉 and 𝑒 0 = 𝜉𝑒𝜉 are idempotents such that 
 

𝑒 0𝑒 ∗ = 𝜉𝑒𝜉𝜉 = 𝜉𝑒𝜉 = 𝑒 0 = 𝜉𝑒𝜉 = 𝜉𝜉𝑒𝜉 = 𝑒 ∗𝑒 0 

That is, 𝑒 0 ≤𝑛 𝑒
∗ and, since ≤𝑛 reduces to equality, 𝑒

0 = 𝑒 ∗, for every idempotent 𝑒 of 𝑆. Therefore, we can conclude 
that 𝐸(𝑆) ⊆ 𝐶. 



EVENTUALLY POINTED PRINCIPALLY ORDERED REGULAR SEMIGROUPS 

 

145 

 

(2) ⇒ (1): Suppose now that 𝑆 is naturally ordered and 𝐸(𝑆) ⊆ 𝐶. Assume that 𝑒 and 𝑓 are idempotents of 𝑆, such 
that 𝑒 ≤𝑛 𝑓 . Thus, in particular, 𝑒 ≤ 𝑓 and also 𝑒 = 𝑒𝑓 = 𝑓𝑒. 
By Theorem 4, 𝑆 has a biggest idempotent (in fact, a maximal element), 𝜉, such that 𝑒 ∗ = 𝜉 = 𝑓 ∗. Since 𝐸(𝑆) ⊆ 𝐶 we 
have that 𝑒 0 = 𝑓 0, whence, by Theorem 7, 𝑒 and 𝑓 are mutually inverse and, using the fact that 𝑒 ≤𝑛 𝑓,     
 

𝑓 = 𝑓𝑒𝑓 = 𝑒𝑓 = 𝑒. 
Hence, ≤𝑛  reduces to equality and 𝑆 is completely simple. 

(2) ⟹ (3): From Theorem 4, 𝑆 has a biggest idempotent (in fact, a maximal element), 𝜉, and 𝜉 = 𝑒 ∗ for every 
idempotent 𝑒 of 𝑆. Since 𝐸(𝑆) ⊆ 𝐶, we obtain that 𝜉 = 𝑒 ∗ = 𝑒 0. 
(3) ⟹ (2): By (3), 𝜉 is a biggest idempotent (in fact a maximal element) of 𝑆, such that 𝜉 = 𝑒 0 for every idempotent 
𝑒 of 𝑆. By (𝑃3), we have that 𝜉 = 𝑒

0 ≤ 𝑒 ∗. Since 𝜉 is a maximal element of 𝑆, we obtain 𝜉 = 𝑒 ∗ = 𝑒 0 and, by 
Theorem 4, 𝑆 is naturally ordered and 𝐸(𝑆) ⊆ 𝐶. 

Using Theorem 7, we can prove as in [4, Theorem 9] the following result in an eventually pointed principally 

ordered regular semigroup, under the weaker conditions of 𝑒 and 𝑓 comparable idempotents such that 𝑒 0 = 𝑓 0. 
 

Theorem 9. Let 𝑆 be an eventually pointed principally ordered regular semigroup. If 𝑒 and 𝑓 are idempotents of 𝑆 
such that 𝑒 ≤ 𝑓 and 𝑒 0 = 𝑓 0, then the subalgebra 𝑇 of (𝑆,∗ ) generated by {𝑒, 𝑓} is a band having at most 10 elements. 
In the case where 𝑇 has precisely 10 elements it is represented by the Hasse diagram. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

in which elements joined by lines of positive gradient are ℛ-related, those joined by lines of negative gradient are      
ℒ-related, and the vertical line also indicates ≤𝑛. 
 

Example 7. Consider in Example 1 

 

𝑒 = [
0 0
0 1

] and 𝑓 = [
0 1
0 1

] 

 

that verify 𝑒 ≤ 𝑓 and 𝑒 0 = [
1 1
1 1

] = 𝑓 0. This gives a simplified version of the structure presented in Theorem 9, the 

four element set, with the induced order and matrix multiplication, 

 

𝑇 = {[
0 0
0 1

] , [
0 1
0 1

] , [
0 0
1 1

] , [
1 1
1 1

]}. 

 

To obtain, precisely, the diagram in Theorem 9, let us recollect that in [4, Example 4] we obtained this diagram, in a 

pointed principally ordered regular semigroup, with a semigroup denoted by 𝐸(𝐵2 ). If we define, in the Cartesian 
ordered set, 𝑇 × 𝐸(𝐵2 ), the Cartesian multiplication, we easily obtain an eventually pointed principally ordered regular 
semigroup with a copy of the diagram in Theorem 9. We just need to consider, for example, the Cartesian product, 

Cartesian order of {𝑒} by the band obtained in [4, Example 4]. 
 

 

e 

e*= f * 

ef fe 

f 

ff * 

ee* e*e 

f *f 

e º =f  
º 



 PINTO, G.A.  

146 

 

Conclusion  

In this paper we introduced a new class of ordered semigroups: Eventually Pointed Principally Ordered Regular 

Semigroups, as a generalisation of pointed principally ordered regular semigroups. Examples showing that the classes 

are distinct were presented, and it was possible to prove several of the known results, or similar versions of them, in 

this class of semigroups, with new and adjusted justifications. 

Conflict of interest  

The author declares no conflict of interest.  

Acknowledgement 

I thank Sultan Qaboos University for providing facilities.  

References 

1. Blyth, T.S. and Pinto, G.A. Principally ordered regular semigroups, Glasgow Mathematics Journal. 32 (1990) 
349-364. doi:10.1017/S0017089500009435. 

2. Blyth, T.S. and Pinto, G.A. Idempotents in principally ordered regular semigroups, Communications in Algebra 
19 (1991) 1549-1563. doi:10.1080/00927879108824220. 

3. Blyth, T.S. Lattices and Ordered Algebraic Structures, (Springer 2005). doi:10.1007/b139095. 
4. Blyth, T.S. and Pinto, G.A. Pointed principally ordered regular semigroups, Discussiones Mathematicae 36 

(2016) 101-111. doi:10.7151/dmgaa.1243. 

5. Higgins, P.M.  Techniques of Semigroup Theory (Oxford Science Publications, 1992) doi:10.1007/BF02573500.  

 
 

 

Received   29 May 2019          

Accepted   2 September 2019