SQU Journal for Science, 2021, 26(1), 31-39  DOI:10.24200/squjs.vol26iss1pp31-39 

Sultan Qaboos University  

31 

 

Boolean Zero Square (BZS) Semigroups 

G.A. Pinto 

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

ABSTRACT: We introduce a new class of semigroups, that we call BZS - Boolean Zero Square-semigroups. A 

semigroup S with a zero element, 0, is said to be a BZS semigroup if, for every 𝑥 ∈ 𝑆, we have 𝑥 2 = 𝑥 or 𝑥 2 = 0. 
We obtain some properties that describe the behaviour of the Green’s equivalence relations ℛ, ℒ, ℋ and 𝒟. 
Necessary and sufficient conditions for a BZS semigroup to be a band and an inverse semigroup are obtained. A 

characterisation of a special type of BZS completely 0-simple semigroup is presented. 

 

Keywords: BZS semigroup; Green’s relations; Regular; Completely 0-simple; Inverse and band. 

 

 أشباه الزمر البوالنية ذات المربع الصفري 

 ج. أ. بينتو

, 0الذي يحتوي على صفر  Sاشباه الزمر البوالنيه ذات المربع الصفري". شبه الزمرة جديدا من اشباه الزمر و أطلقنا عليه اسم "لقد اوجدنا نوعا  :صلخمال

x  xيحقق  اما Sفي   xشبه زمرة البوالنيه ذو مربع صفري اذا كان كل يسمى 
2
xاو  =

2
. لقد اوجدنا بعض الصفات التي تصف تصرف عالقة جرين 0 =

شبه زمره باندية , شبه زمرة معاكسة. كما اوجدنا . كما اوجدنا  الشروط الالزمة و الشروط الكافية ليكون شبه الزمرة هذا 𝒟 و ℛ, ℒ, ℋالتكافؤية 
 اشباه الزمر البوالنيه ذات المربع الصفري وهو شبه الزمرة الصفري البسيط التام.التوصيف الكامل تنوع خاص من 

 

 البوالنيه ذو مربع صفري، عالقة جرين، منتظم، الصفري البسيط التام، معكوس و فرقة. شبه زمرة :مفتاحيةالكلمات ال

 

 

 

 

 

 

 

 

 

 

 

 

mailto:goncalo@squ.edu.om


G.A. PINTO 

 

32 

 

1. Introduction 

e shall use standard semigroup notation, that can be found, for example, in [1]. If S is a semigroup, 𝐸(𝑆) 
represents the set of idempotents of S, and 𝑉(𝑥) denotes the set of inverses of an element x in S. We recall that 

the natural order ≤𝑛  on the idempotents of a regular semigroup is defined by 
𝑒 ≤𝑛 𝑓 ⟺ 𝑒 = 𝑒𝑓 = 𝑓𝑒 

 

In [2], Farag and Tucci introduced the notion of a Boolean Zero Square (BZS) ring as an associative ring, not 

necessarily commutative and not necessarily with identity, such that every non-zero element of R is either idempotent 

or nilpotent of index 2, that is, 

(∀𝑥 ∈ 𝑅) 𝑥 2 = 𝑥 ∨ 𝑥 2 = 0 
 

The structure of BZS rings is investigated, in [2] and in [3]. 

It is possible to find in the literature several papers in Boolean Zero rings, and as a sample we refer to [4] in order to 

several constructions in this structure. 

Here we present a generalisation of these notions to semigroup theory, starting by introducing the following concept, 

that follows naturally from Ring theory. 

 

Definition. A semigroup 𝑆, with element zero 0, is said to be a BZS semigroup if, for every element 𝑥 in S, we have  
𝑥 2 = 𝑥 or 𝑥 2 = 0. 
 

In the ring case, there are more tools available due to the presence of two operations: addition and multiplication. This 

fact has, as a consequence, that the majority of the results obtained in [2] and [3] cannot be replicated to the semigroup 

case. 

 

One of the results obtained in [2] is that in a BZS ring, the set of nilpotent elements, is an ideal of the ring. This 

property does not hold in a general BZS semigroup, as it can easily be seen in Example 2 below. This happens because 

the result only mentions the multiplicative operation, although its proof uses the additive operation heavily. We obtain 

in Theorem 8 that in a BZS commutative semigroup the set of its nilpotent elements, is an ideal of the semigroup. 

 

In what follows, unless otherwise stated, S will always denote a BZS semigroup. We denote 

𝐸 = {𝑥 ∈ 𝑆|𝑥 2 = 𝑥}  and  𝑁 = {𝑥 ∈ 𝑆|𝑥 2 = 0} 
 

respectively, the set of idempotent elements and the set of nilpotent elements of S. 

 

Let us present some basic properties that hold in any such semigroup S. 

 

(∀𝑥 ∈ 𝑆) 𝑥 3 = 𝑥 2                      (1) 
 

For any 𝑥 ∈ 𝑆, we have two cases to consider: 
i) 𝑥 2 = 𝑥 ⟹ 𝑥 ⋅ 𝑥 2 = 𝑥 ⋅ 𝑥 ⟹ 𝑥 3 = 𝑥 2 
ii) 𝑥 2 = 0 ⟹ 𝑥 ⋅ 𝑥 2 = 𝑥 ⋅ 0 ⟹ 𝑥 3 = 0 ⟹ 𝑥 3 = 𝑥 2 

which proves the result. 

 

 

(∀𝑥 ∈ 𝑆) 𝑥 4 = 𝑥 2 and 𝑥 2 ∈ 𝐸(𝑆)                     (2) 
 

This follows immediately from (1). 
 

 

𝐸 ∩ 𝑁 = {0}                         (3) 
 

In fact, if 𝑥 ∈ 𝐸 ∩ 𝑁, we have 𝑥 = 𝑥 2 = 0, and the result follows. 
 

 

(∀𝑥, 𝑦 ∈ 𝑆) 𝑥𝑦 ∈ 𝐸\{0} ⟺ 𝑦𝑥 ∈ 𝐸\{0}                                   (4) 
 

Let us, assume that 𝑥𝑦 ∈ 𝐸\{0}, that is, (𝑥𝑦)2 = 𝑥𝑦, with 𝑥𝑦 ≠ 0. If 𝑦𝑥 = 0 then,  

  
𝑦𝑥 = 0 ⟹ 𝑥 ∙ 𝑦𝑥 ∙ 𝑦 = 𝑥 ∙ 0 ∙ 𝑦

⟹ (𝑥𝑦)2 = 0                  
  

 

 

W 



BOOLEAN ZERO SQUARE (BZS) SEMIGROUPS 

 

33 

 

which is a contradiction, and therefore we can conclude that 𝑦𝑥 ≠ 0. Also, if (𝑦𝑥)2 = 0 then, using (1), we have 
(𝑦𝑥)2 = 0 ⟹ 𝑥(𝑦𝑥)2𝑦 = 𝑥 ∙ 0 ∙ 𝑦

⟹ (𝑥𝑦)3 = 0                  

⟹
⟹

(𝑥𝑦)2 = 0            
𝑥𝑦 = 0                

     

  

which is also a contradiction. So, (𝑦𝑥)2 = 𝑦𝑥 ≠ 0, which means that, 𝑦𝑥 ∈ 𝐸\{0}. 
Similarly, we prove the reverse implication. 

 

 

(∀𝑥, 𝑦 ∈ 𝑆) 𝑥𝑦 ∈ 𝑁 ⟺ 𝑦𝑥 ∈ 𝑁                      (5) 
 

If 𝑥𝑦 ∈ 𝑁 then, by (3), 𝑥𝑦 ∉ 𝐸\{0} and therefore by (4), 𝑦𝑥 ∉ 𝐸\{0}, that is 𝑦𝑥 ∈ 𝑁. The converse 
implication follows similarly. 

 

Now, let us present some examples to illustrate this concept, which show that they can be found in a wide variety of very 

well-known classes of semigroups, such as bands, completely 0-simple semigroups and inverse semigroups. 

Example 1. Any band, B, with zero is clearly, a BZS semigroup, with 𝐸 = 𝐵 and 𝑁 = {0}. 
 

 

Example 2. In a context of ordered semigroup theory, Blyth and McFadden presented in [5] a semigroup which has 

proved to be very helpful in describing several classes of ordered semigroups. More details of the relevance and properties 

of this semigroup can also be found in [6]. It can be defined by 𝑁5 = {𝑢, 𝑒, 𝑓, 𝑎, 𝑏} with the following Cayley table: 
 

 u e f a b 

u u u f f b 

e e e a a b 

f u b f b b 

a e b a b b 

b b b b b b 

 

It follows directly from the table that 𝑁5 is a BZS semigroup, b is its zero element, 𝑁 = {𝑎, 𝑏} and 𝐸 = {𝑢, 𝑒, 𝑓, 𝑏}. This 
semigroup appears in a different context, as an example of a completely 0-simple semigroup that it is not orthodox. 

Routine calculations show that it is 0-simple 

 

𝑁5 = 𝑁5𝑢𝑁5 = 𝑁5𝑒𝑁5 = 𝑁5𝑓𝑁5 = 𝑁5𝑎𝑁5 
 

and, for example, 𝑓 is a primitive idempotent. Since 𝑒𝑓 ≠ 𝑓𝑒,  we can state that 𝑁5 is not an orthodox semigroup. 
 

 

Example 3. Consider the completely 0-simple semigroup 𝑆 = (𝐼 × 𝐺 × Λ) ∪ {0} with operation 
 

 (𝑖, 𝑎, 𝜆)(𝑗, 𝑏, 𝜇) = {
(𝑖, 𝑎𝑝𝜆𝑗 𝑏, 𝜇) if 𝑝𝜆𝑗 ≠ 0

0 if 𝑝𝜆𝑗 = 0
 

 

 (𝑖, 𝑎, 𝜆)0 = 0 = 0(𝑖, 𝑎, 𝜆) = 00 
 

where 𝐺 0 = 𝐺 ∪ {0} is a zero group, with 𝐺 = 〈𝑥〉 an order two cyclic group, 𝐼, Λ are non-empty index sets and 𝑃 = [𝑝𝜆𝑖 ] 
is a Λ × 𝐼 sandwich matrix with entries in 𝐺 0, and all the non-zero entries of 𝑃 are equal to x. Recall that every row and 
column of P has at least a non-zero entry. 

 

Consider 𝑇 = {(𝑖, 𝑥, 𝜆) ∈ 𝑆} ∪ {0} a subset of S, and let (𝑖, 𝑥, 𝜆), (𝑗, 𝑥, 𝜇) ∈ 𝑇. We have the following possibilities for the 
element 𝑝𝜆𝑗 : 

 

 If 𝑝𝜆𝑗 ≠ 0, then (𝑖, 𝑥, 𝜆)(𝑗, 𝑥, 𝜇) = (𝑖, 𝑥𝑝𝜆𝑗 𝑥, 𝜇) = (𝑖, 𝑥𝑥𝑥, 𝜇) = (𝑖, 𝑥, 𝜇) ∈ 𝑇 

 

 If 𝑝𝜆𝑗 = 0, then (𝑖, 𝑥, 𝜆)(𝑗, 𝑥, 𝜇) = 0 ∈ 𝑇 

 

 

 

 

 



G.A. PINTO 

 

34 

 

and we can say that 𝑇 is a semigroup, with the induced semigroup operation. Also, for any (𝑖, 𝑥, 𝜆) ∈ 𝑇, we have that 
 

 (𝑖, 𝑥, 𝜆)2 = {

(𝑖, 𝑥, 𝜆) if 𝑝𝜆𝑖 ≠ 0

0 if 𝑝𝜆𝑖 = 0
 

 

Therefore, 𝑇 is a BZS semigroup. 
 

 

Example 4. Consider the following set of 2 × 2 real matrices 
 

𝑆 = {𝐼, 𝐴, 𝐸11, 𝐸12, 𝐸21, 𝐸22, 𝑂} = {[
1 0
0 1

] , [
0 1
1 0

] , [
1 0
0 0

] , [
0 1
0 0

] , [
0 0
1 0

] , [
0 0
0 1

] , [
0 0
0 0

]} 

 

It is well known that S with the usual matrix multiplication is an inverse semigroup (see, for example [7, Section 7.6, 

Exercise 1]). S is not a BZS semigroup, since 𝐴2 = 𝐴𝐴 = 𝐼 ≠ 𝐴, 𝑂. 
 

But if we consider the subset 

𝑇 = {𝐼, 𝐸11, 𝐸12, 𝐸21, 𝐸22 , 𝑂}, 
 

it gives us the following Cayley table 

 

 𝐼 𝐸11 𝐸12 𝐸21 𝐸22 𝑂 

𝐼 𝐼 𝐸11 𝐸12 𝐸21 𝐸22 𝑂 

𝐸11 𝐸11 𝐸11 𝐸12 𝑂 𝑂 𝑂 

𝐸12 𝐸12 𝑂 𝑂 𝐸11 𝐸12 𝑂 

𝐸21 𝐸21 𝐸21 𝐸22 𝑂 𝑂 𝑂 

𝐸22 𝐸22 𝑂 𝑂 𝐸21 𝐸22 𝑂 

𝑂 𝑂 𝑂 𝑂 𝑂 𝑂 𝑂 

 

It follows immediately from the table that T is a subsemigroup of S which is a BZS inverse semigroup, with             

𝐸 = {𝐼, 𝐸11, 𝐸22, 𝑂} and 𝑁 = {𝐸12 , 𝐸21, 𝑂}. 
 

2. Green’s Relations 
 

Let us now obtain some basic properties on the Green’s relations ℛ, ℒ, ℋ, 𝒥 and 𝒟 on a BZS semigroup S. 
 

𝑅0 = 𝐿0 = 𝐻0 = 𝐷0 = 𝐽0 = {0}                    (6) 
 

For any 𝑥 ∈ 𝑅0, we have that 𝑥 = 𝑥 ∙ 1 ∈ 𝑥𝑆
1 = 0𝑆1 = {0}, which immediately implies that            

𝑅0 = {0}. The other equalities follow similarly. 
 

 

For 𝑥, 𝑦 ∈ 𝐸 or 𝑥, 𝑦 ∈ 𝑁                    (7) 
𝑥ℛ𝑦 ⟹ 𝑥𝑦ℛ𝑦𝑥  
 

In fact, if 𝑥, 𝑦 ∈ 𝐸, then, since ℛ is a left congruence [1, Proposition 2.1.2], 

  𝑥ℛ𝑦 ⟹ {
𝑥𝑥ℛ𝑥𝑦
𝑦𝑥ℛ𝑦𝑦

⟹ {
𝑥ℛ𝑥𝑦
𝑦𝑥ℛ𝑦

⟹ 𝑥𝑦ℛ𝑦𝑥 

and, if 𝑥, 𝑦 ∈ 𝑁 then 

  𝑥ℛ𝑦 ⟹ {
𝑥𝑥ℛ𝑥𝑦
𝑦𝑥ℛ𝑦𝑦

⟹ {
0ℛ𝑥𝑦
𝑦𝑥ℛ0

⟹(6) 𝑥𝑦 = 0 = 𝑦𝑥 ⟹ 𝑥𝑦ℛ𝑦𝑥 

For 𝑥, 𝑦 ∈ 𝐸 or 𝑥, 𝑦 ∈ 𝑁                    (8) 
𝑥ℒ𝑦 ⟹ 𝑥𝑦ℒ𝑦𝑥  

 

This follows similarly as in (7). 



BOOLEAN ZERO SQUARE (BZS) SEMIGROUPS 

 

35 

 

 

Note that properties (7) and (8) do not hold if one element is in E and the other is in N. To see this, consider 𝑁5 in Example 
2, where the ℛ classes of 𝑁5 are 𝑅𝑢 = {𝑢, 𝑓}, 𝑅𝑒 = {𝑒, 𝑎} and 𝑅𝑏 = {𝑏}, while its ℒ classes are 𝐿𝑢 = {𝑢, 𝑒}, 𝐿𝑓 = {𝑓, 𝑎} and 

𝐿𝑏 = {𝑏}. From property (7) we have that 𝑒 ∈ 𝐸, 𝑎 ∈ 𝑁, 𝑒ℛ𝑎 but 𝑒𝑎 = 𝑎 which is not ℛ related with 𝑎𝑒 = 𝑏. Similarly, for 
property (8). 
 

Theorem 1. Let S be a BZS semigroup. 

(1) If 𝑎 ∈ 𝐸, then 𝑅𝑎 ∩ 𝐸 is a subsemigroup of S, which is a right zero semigroup. In particular, if 𝑅𝑎 ⊆ 𝐸 then 𝑅𝑎 is a right 
zero semigroup. 
(2) If 𝑎 ∈ 𝑁, then 𝑅𝑎 ∪ {0} is a subsemigroup of S. 
 

Proof. (1): It is clear that 𝑅𝑎 ∩ 𝐸 is non-empty, since 𝑎 ∈ 𝑅𝑎 ∩ 𝐸. For any 𝑏, 𝑐 ∈ 𝑅𝑎 ∩ 𝐸, we have, 
 𝑏ℛ𝑎 and 𝑐ℛ𝑎 ⟹ 𝑏ℛ𝑐 ⟹ 𝑏 = 𝑏𝑏ℛ𝑏𝑐 ⟹ 𝑏𝑐 ∈ 𝑅𝑏 = 𝑅𝑎 
which means that, 𝑅𝑎 is a subsemigroup of S. Since 𝑏 and 𝑐 are idempotents, we have by [1, Proposition 2.3.3], that 𝑏𝑐 = 𝑐 
and therefore 𝑅𝑎 is a right zero semigroup. 
(2): It is clear that 𝑅0 = {0} is a subsemigroup of S. So, it is enough to consider 𝑎 ∈ 𝑁\{0}, that is, 𝑎 ≠ 0 and 𝑎

2 = 0. For 
𝑏, 𝑐 ∈ 𝑅𝑎, there exist 𝑥, 𝑦, 𝑧, 𝑤 ∈ 𝑆

1 such that  

 𝑎 = 𝑏𝑥, 𝑏 = 𝑎𝑦, 𝑐 = 𝑎𝑧 and 𝑎 = 𝑐𝑤 

If, on one hand 𝑏2 = 0 , then 

 𝑎 = 𝑏𝑥 ⟹ 𝑏𝑎 = 𝑏(𝑏𝑥) = 𝑏2𝑥 = 0 ∙ 𝑥 = 0 ⟹ 𝑏𝑎 = 0 

and  

 𝑏𝑐 = 𝑏(𝑎𝑧) = (𝑏𝑎)𝑧 = 0 ∙ 𝑧 = 0 

If, on the other hand 𝑏2 = 𝑏, then by [1, Proposition 2.3.3], 𝑏𝑐 = 𝑐 ∈ 𝑅𝑎. 
Therefore, 𝑅𝑎 ∪ {0} is a subsemigroup of S. 
Note that in general, an ℛ class, 𝑅𝑥, is not a subsemigroup of S. In fact, if we consider the semigroup 𝑁5 of Example 2, and 
its ℛ class, 𝑅𝑒 = {𝑒, 𝑎}, where 𝑒 ∈ 𝐸 and 𝑎 ∉ 𝐸, then we have that 𝑎𝑒 = 𝑏 ∉ 𝑅𝑒, which means that, 𝑅𝑒  is not a 
subsemigroup of 𝑁5. 
 

Theorem 2.  Let S be a BZS semigroup. 

(1) If 𝑎 ∈ 𝐸, then 𝐿𝑎 ∩ 𝐸 is a subsemigroup of S, which is a left zero semigroup. In particular, if 𝐿𝑎 ⊆ 𝐸 then, 𝐿𝑎  is a right 
zero semigroup. 
(2) If 𝑎 ∈ 𝑁 then, 𝐿𝑎 ∪ {0} is a subsemigroup of S. 
 

Proof. Similar to the proof of Theorem 1. 

 

Like in the note to Theorem 2, we can use Example 2 to illustrate that an ℒ class of a BZS semigroup is not, in general, a 
subsemigroup of S. 

 

Theorem 3. Let S be a BZS semigroup. 

(1) If 𝑎 ∈ 𝐸, then 𝐻𝑎  is a group with only one element. 
(2) If 𝑎 ∈ 𝑆\𝐸, then 𝐻𝑎 ⊆ 𝑆\𝐸, (𝐻𝑎 )

2 = {0} and 𝐻𝑎 ∪ {0} is a subsemigroup of S. 
(3) If a 𝒟 class of S contains an idempotent, all its ℋ classes are singleton. 
 

Proof. (1): In fact, by [1, Corollary 2.2.6], 𝐻𝑎  is a subgroup of S. We need to prove that 𝐻𝑎  has a unique element. For 𝑎 = 0 
this is obvious, by (6). Let us now assume that 𝑎 ∈ 𝐸\{0}, and consider 𝑏 ∈ 𝐻𝑎 . We have that 
 

 𝑏ℋ𝑎 ⟹ {
𝑏ℛ𝑎
𝑏ℒ𝑎

⟹ {
𝑏𝑏ℛ𝑏𝑎
𝑏𝑎ℒ𝑎𝑎

⟹ {
𝑏2ℛ𝑏𝑎
𝑏𝑎ℒ𝑎

 

If 𝑏2 = 0 then 

 𝑏ℋ𝑎 ⟹ {
0ℛ𝑏𝑎
𝑏𝑎ℒ𝑎

⟹ {
𝑏𝑎 = 0
𝑎 ∈ 𝐿𝑏𝑎

⟹ 𝑎 ∈ 𝐿𝑏𝑎 = 𝐿0 = {0} ⟹ 𝑎 = 0 

which is a contradiction. Therefore, we can conclude that 𝑏2 = 𝑏, and 𝑏 is an idempotent. Using again [1, Corollary 2.2.6], 
we conclude that 𝑏 = 𝑎, and 𝐻𝑎  is a singleton subgroup of S. 
(2): Let 𝑎 ∈ 𝑆\𝐸, and consider 𝑦 ∈ 𝐻𝑎 , which cannot be equal to 0, by (6). Then, 

 

 𝑦ℋ𝑎 ⟹ {
𝑦ℛ𝑎
𝑦ℒ𝑎

⟹ {
𝑦𝑦ℛ𝑦𝑎
𝑦𝑎ℒ𝑎𝑎

⟹ {
𝑦2ℛ𝑦𝑎
𝑦𝑎ℒ0

⟹ 𝑦2 ∈ 𝑅0 = {0} ⟹ 𝑦 ∈ 𝑁 



G.A. PINTO 

 

36 

 

and therefore 𝐻𝑎 ⊆ 𝑆\𝐸. 
Again, with 𝑎 ∈ 𝑆\𝐸 we have that 
 

 𝑏ℋ𝑎 ⟹ {
𝑏ℛ𝑎
𝑏ℒ𝑎

⟹ {
𝑎𝑏ℛ𝑎𝑎
𝑏𝑎ℒ𝑎𝑎

⟹ {
𝑎𝑏ℛ0
𝑏𝑎ℒ0

⟹ {
𝑎𝑏 = 0
𝑏𝑎 = 0

 

Therefore, for 𝑏, 𝑐 ∈ 𝐻𝑎 , we can say that  
 

 𝑎𝑏 = 0, 𝑏𝑎 = 0, 𝑎𝑐 = 0 and 𝑐𝑎 = 0 
 

Since 𝑏ℋ𝑐, we have that 𝑏ℛ𝑐, and 
 

 𝑏ℛ𝑐ℛ𝑎 ⟹ 𝑐𝑏ℛ𝑐𝑐ℛ𝑐𝑎 = 0 ⟹ 𝑐𝑏ℛ0 ⟹ 𝑐𝑏 = 0 
 

from which, we conclude that (𝐻𝑎 )
2 = {0} and that 𝐻𝑎 ∪ {0} is a subsemigroup of S. 

(3): This follows by [1, Lemma 2.2.3] and (1). 

 

Note, that in Theorem 3(2) we verified that, for every 𝑎 ∈ 𝑁, the ℋ class 𝐻𝑎  is a subset of N. The same property does 
not hold for the ℛ and ℒ classes. In fact, the semigroup 𝑁5 of Example 2 is such that 𝑎 ∈ 𝑁, but 𝑅𝑎 = {𝑒, 𝑎} ⊄ 𝑁, as 
well as 𝐿𝑎 = {𝑓, 𝑎} ⊄ 𝑁. 
 

Theorem 4. Let S be a BZS semigroup. 

(1) If 𝑎 ∈ 𝑆\{0} and 𝐷𝑎 ⊆ 𝐸, then 𝐷𝑎  is a subsemigroup of S. 
(2) If 𝑎 ∈ 𝑆\𝐸, then 𝐷𝑎 ∪ {0} is a subsemigroup of S. 
 

Proof. (1): For any 𝑎 ∈ 𝑆\{0}, we have by (6), that 𝐷𝑎 ≠ {0}. The fact that 𝐷𝑎 ⊆ 𝐸 therefore implies that             
𝐷𝑎 ⊆ 𝐸\{0}. 
Considering any 𝑏, 𝑐 ∈ 𝐷𝑎 , there exist 𝑑 ∈ 𝑆, such that 𝑏ℒ𝑑ℛ𝑐. By [1, Propositions 2.1.2 and 2.3.3], we have that 

 𝑑ℛ𝑐 ⟹ 𝑏𝑑ℛ𝑏𝑐 ⟹ 𝑏ℛ𝑏𝑐 ⟹ 𝑏𝑐 ∈ 𝑅𝑏 ⊆ 𝐷𝑎  

Thus, 𝐷𝑎  is a subsemigroup of S. 
(2) Consider any 𝑎 ∈ 𝑆\𝐸, that is, 𝑎 ≠ 0 and 𝑎2 = 0. For any 𝑏, 𝑐 ∈ 𝐷𝑎 , there exist 𝑑 ∈ 𝑆, such that 𝑏ℒ𝑑ℛ𝑐, which 
means, in particular, that 𝑏 = 𝑥𝑑 and 𝑐 = 𝑑𝑦 for some 𝑥, 𝑦 ∈ 𝑆1. 
Also, 𝑑ℛ𝑐 implies 𝑏 = 𝑥𝑑ℛ𝑥𝑐 and therefore 𝑥𝑐 ∈ 𝑅𝑏. 
Then, 

𝑏𝑐 = (𝑥𝑑)(𝑑𝑦) = 𝑥(𝑑𝑑)𝑦  

If 𝑑2 = 𝑑, then 𝑏𝑐 = 𝑥𝑑2𝑦 = 𝑥𝑑𝑦 = 𝑥𝑐 ∈ 𝑅𝑏 ⊆ 𝐷𝑎 .  
If 𝑑2 = 0, then 𝑏𝑐 = 0. 
Thus, 𝐷𝑎 ∪ {0} is a subsemigroup of S. 

3. Special classes of BZS semigroups 

We now devote our attention to obtaining necessary and sufficient conditions for a BZS semigroup S to be a band 

or an inverse semigroup. A characterisation of some BZS completely 0-simple is presented. Also, the commutativity 

property will be approached. 

 

Theorem 5. Let S be a BZS semigroup. The following statements are equivalent: 

(1) S is a band; 
(2) (∀𝑥 ∈ 𝑆) 𝑥 3 = 𝑥. 
 

Proof. (1) ⟹ (2): The definition of a band tells us that 𝑥 2 = 𝑥 for all 𝑥 ∈ 𝑆. Then, 
𝑥 3 = 𝑥 2 ∙ 𝑥 = 𝑥 ∙ 𝑥 = 𝑥  

and the result follows. 

(2) ⟹ (1): Take an element 𝑥 ∈ 𝑆. Since S is BZS, 𝑥 2 = 𝑥 or 𝑥 2 = 0. If 𝑥 2 = 𝑥, there is nothing to prove. If 𝑥 2 = 0, 
then 𝑥 = 𝑥 3 = 𝑥 2𝑥 = 0 ∙ 𝑥 = 0 which, immediately implies that 𝑥 2 = 𝑥, for every element of S, that is, S is a band. 
 

In the following Theorem and its proof, we use the identification provided from Rees Theorem [1, Theorem 3.2.3], for 

a completely 0-simple semigroup S. Such S is isomorphic to 

(𝐼 × 𝐺 × Λ) ∪ {0}, 



BOOLEAN ZERO SQUARE (BZS) SEMIGROUPS 

 

37 

 

where G is a group, 𝐼 and Λ are non-empty index sets, and 𝑃 = [𝑝𝜆𝑖 ] is a Λ × 𝐼 sandwich matrix with entries in the zero group 
𝐺 0 = 𝐺 ∪ {0}. Also, every row and column of P has at least a non-zero entry. The semigroup operation is defined by 

(𝑖, 𝑎, 𝜆)(𝑗, 𝑏, 𝜇) = {
(𝑖, 𝑎𝑝𝜆𝑗 𝑏, 𝜇) if 𝑝𝜆𝑗 ≠ 0

0 if 𝑝𝜆𝑗 = 0
 

 

Theorem 6. Let S be a BZS semigroup. The following statements are equivalent: 

(1) S is a completely 0-simple semigroup with no zero entries in the sandwich matrix; 
(2) S is a rectangular 0-band. 
 

Proof. (1) ⟹ (2): Let S be a completely 0-simple semigroup. Considering an arbitrary element 𝑥 in 𝐺, for any 𝜆 ∈ Λ and 
𝑖 ∈ 𝐼, we have that 𝑝𝜆𝑖 ≠ 0. Then, 

 

(𝑖, 𝑥, 𝜆)2 = (𝑖, 𝑥, 𝜆)(𝑖, 𝑥, 𝜆) = (𝑖, 𝑥𝑝𝜆𝑖 𝑥, 𝜆) ≠ 0, 

which therefore implies, since S is a BZS semigroup, that (𝑖, 𝑥, 𝜆)2 = (𝑖, 𝑥, 𝜆). Thus, 
 

(𝑖, 𝑥, 𝜆)(𝑖, 𝑥, 𝜆) = (𝑖, 𝑥, 𝜆) ⟺ (𝑖, 𝑥𝑝𝜆𝑖 𝑥, 𝜆) = (𝑖, 𝑥, 𝜆)

⟺ 𝑥𝑝𝜆𝑖 𝑥 = 𝑥                     

⟺ 𝑝𝜆𝑖 = 𝑥
−1                     

 

In particular, if we replace x by the identity element of the group 1𝐺 , we obtain 𝑝𝜆𝑖 = 1𝐺 , and therefore 𝑥
−1 = 1𝐺  which, is 

equivalent to 𝑥 = 1𝐺 . So, G is the trivial group. 
Then, 𝑆 is isomorphic to {(𝑖, 1𝐺 , 𝜆): 𝑖 ∈ 𝐼 and 𝜆 ∈ Λ} ∪ {0} , whose elements verify 

 

 (𝑖, 1𝐺 , 𝜆)(𝑗, 1𝐺 , 𝜇) = (𝑖, 1𝐺 , 𝜇)  and  (𝑖, 1𝐺 , 𝜆) ∙ 0 = 0 = 0 ∙ (𝑖, 1𝐺 , 𝜆) 
 

That is, S is a rectangular 0-band. 

(2) ⟹ (1): If S is a rectangular 0-band, then 
(∀𝑎 ∈ 𝑆)(∀𝑏 ∈ 𝑆\{0}) 𝑎2 = 𝑎 and 𝑎𝑏𝑎 = 𝑎 . 

Then, for any 𝑎, 𝑏 ∈ 𝑆 and 𝑏 ∈ 𝑆\{0}, we have that, 𝑎 = 𝑎𝑏𝑎 ∈ 𝑆𝑏𝑆 which implies that 𝑆 ⊆ 𝑆𝑏𝑆. 
Since, the reverse inclusion is always true, we can conclude that S is a 0-simple semigroup. 

 

Also, if in 𝑆\{0}, 𝑎 ≤𝑛 𝑏, then 𝑎𝑏 = 𝑏𝑎 = 𝑎. We have that 
 

𝑎𝑏 = 𝑏𝑎 ⟹ {
𝑎𝑏𝑎 = 𝑏𝑎𝑎
𝑏𝑎𝑏 = 𝑏𝑏𝑎

⟹ {
𝑎 = 𝑏𝑎
𝑏 = 𝑏𝑎

⟹ 𝑎 = 𝑏 

 

which, means that all non-zero idempotents are primitive, and therefore S is completely 0-simple. 

Also, if 𝑎, 𝑏 ∈ 𝑆\{0}, then if 𝑎𝑏 = 0 then 𝑎𝑏𝑎 = 0 ≠ 𝑎, which is a contradiction. So, all the entries of the sandwich matrix are 
not zero. 

 

It follows from the previous Theorem and its proof that for a BZS semigroup to be completely 0-simple where the sandwich 

matrix has no zero entries, it is necessary to have a singular group in the middle component of the Rees representation. 

In fact, we can say that a BZS semigroup is completely 0-simple where the sandwich matrix has no zero entries if, and only if, 

it is a completely simple semigroup with a zero adjoined. 

 

Theorem 7. Let S be a BZS semigroup. S is an inverse semigroup if, and only if, the following conditions hold: 

(1) S is regular; 
(2) (∀𝑥 ∈ 𝑆) 𝑥 ′𝑥 2𝑥′ = 𝑥 2, for any inverse 𝑥′ of 𝑥. 
 

Proof. Let S be an inverse semigroup. Any element x in S has a unique inverse denoted by 𝑥 −1. By [1, Theorem 5.1.1], an 
inverse semigroup is a regular one, where the idempotents commute. So, by (2), 𝑥 2 is an idempotent that, therefore, 
commutes with 𝑥𝑥 −1 and with 𝑥 −1 𝑥. Thus, 

 

𝑥 4 = 𝑥 2 ⟹ 𝑥 −1𝑥 4 = 𝑥 −1𝑥 2 ⟹ 𝑥 −1𝑥 ∙ 𝑥 2 ∙ 𝑥 = 𝑥 −1𝑥 2 ⟹ 𝑥 2 ∙ 𝑥 −1𝑥 ∙ 𝑥 = 𝑥 −1𝑥 2 
 

⟹ 𝑥(𝑥𝑥 −1𝑥)𝑥 = 𝑥 −1𝑥 2 ⟹ 𝑥 3 = 𝑥 −1𝑥 2 ⟹ 𝑥 3𝑥 −1 = 𝑥 −1𝑥 2𝑥 −1 
 

                       ⟹ 𝑥 2 ∙ 𝑥𝑥 −1 = 𝑥 −1𝑥 2𝑥 −1 ⟹ 𝑥𝑥 −1 ∙ 𝑥 2 = 𝑥 −1𝑥 2𝑥 −1 ⟹ 𝑥 2 = 𝑥 −1𝑥 2𝑥 −1 
  



G.A. PINTO 

 

38 

 

Conversely let us, assume that (1) and (2) hold. Let 𝑒  be an idempotent of S and  𝑒′ any inverse of 𝑒. By (2), we have 
that 𝑒 ′𝑒 2𝑒′ = 𝑒 2, that is, 𝑒 ′ = 𝑒. Thus, we can conclude that each idempotent in S has a unique inverse. Now, 
considering an element 𝑥 in 𝑆 and 𝑥 ′, 𝑥′′ inverses of 𝑥, we have that 𝑥𝑥 ′ and 𝑥𝑥 ′′ are idempotents and inverses of each 
other, as well as  𝑥 ′𝑥 and 𝑥 ′′𝑥. Thus, 𝑥𝑥 ′ = 𝑥𝑥 ′′ and 𝑥 ′𝑥 = 𝑥 ′′𝑥 and we can deduce that 
 𝑥 ′ = 𝑥 ′(𝑥𝑥 ′) = 𝑥 ′(𝑥𝑥 ′′) = (𝑥′𝑥)𝑥 ′′ = (𝑥′′𝑥)𝑥 ′′ = 𝑥′′ 

The result follows, since by [1, Theorem 5.1.1], a regular semigroup where each element has a unique inverse is an 

inverse semigroup. 

 

Theorem 8. Let S be a BZS commutative semigroup. Then, 

(1) E is a subsemigroup of S; 
(2) N is an ideal of S; 
(3) If S is inverse then 𝑥 3 = 𝑥, for every 𝑥 ∈ 𝑆. 
 

 

Proof. Consider any elements 𝑥, 𝑦 ∈ 𝑆. 
(1):  If, on one hand, both belong to E, we have 𝑥 2 = 𝑥 and 𝑦2 = 𝑦, and therefore 

(𝑥𝑦)2 = (𝑥𝑦)(𝑥𝑦) = 𝑥(𝑦𝑥)𝑦 = 𝑥(𝑥𝑦)𝑦 = (𝑥𝑥)(𝑦𝑦) = 𝑥 2𝑦2 = 𝑥𝑦 

which means that 𝑥𝑦 ∈ 𝐸 , and therefore E is a subsemigroup of S. 
(2):  If, on the other hand, for example 𝑥 ∈ 𝑁, we have that 

 (𝑥𝑦)2 = (𝑥𝑦)(𝑥𝑦) = 𝑥(𝑦𝑥)𝑦 = 𝑥(𝑥𝑦)𝑦 = (𝑥𝑥)(𝑦𝑦) = 𝑥 2 ∙ 𝑦2 = 0 ∙ 𝑦2 = 0 

Thus, 𝑥𝑦 ∈ 𝑁 and we can conclude that N is an ideal of S. 
 

(3): If S is an inverse commutative semigroup, any 𝑥 ∈ 𝑆 has a unique inverse, 𝑥 −1, and we have by Theorem 7 (2), that 

𝑥 = 𝑥𝑥 −1𝑥𝑥 −1𝑥 = 𝑥(𝑥 −1𝑥𝑥𝑥 −1) = 𝑥 ∙ 𝑥 2 = 𝑥 3 
 

We have seen previously that  𝑥 3 = 𝑥 for all 𝑥 ∈ 𝑆 holds in any BZS semigroup that it is also a band, or a commutative 
inverse semigroup. It also holds for a BZS completely 0-simple semigroup, where the sandwich matrix has no zero 

entries. However, this property does not hold for all the BZS semigroups. To see this, let us consider 

 

𝑇 = {𝐼, 𝐸11, 𝐸12, 𝐸21, 𝐸22, 𝑂} 
 

of Example 4, which is an inverse BZS semigroup. Note that we have 

 

𝐼3 = 𝐼,   𝐸11
3 = 𝐸11,  𝐸22

3 = 𝐸22,  𝑂
3 = 𝑂, 

𝐸12
3 = 𝑂 ≠ 𝐸12,  𝐸21

3 = 𝑂 ≠ 𝐸21 
 

from which, we can deduce that the mentioned property does not hold in all the BZS semigroups. 

4. Conclusion 

In this paper, we introduce a new class of ordered semigroups: BZS - Boolean Zero Square semigroups. Several 

basic properties on Green’s relations are obtained. Necessary and sufficient conditions for a BZS semigroup to be a band 

and to be an inverse semigroup are obtained. A characterisation of a special type of BZS completely 0-simple semigroup 

is presented. 

Conflict of interest 

 The author declares no conflict of interest. 

Acknowledgment 

I thank Sultan Qabbos University for providing facilities and to the referees for their comments that improved the 

qulity of this paper. 

 

 

 



BOOLEAN ZERO SQUARE (BZS) SEMIGROUPS 

 

39 

 

References 

1. John, M. Howie, Fundamentals of Semigroup Theory, L.M.S. Monographs, 12, Oxford University Press, Oxford, 
1995. 

2. Farag, M. and Tucci, R.  BZS Rings, Palestine Journal of Mathematics 2019, 8(2), 8-14.doi:10.7151/dmgaa.1243. 
3. Farag, M.  BZS Rings II, 2020 (preprint). 
4. Bhavanari, S., Lungisile, G. and Dasari, N. Ideals and direct product of zero square rings, East Asian Mathematics 

Journal 24, 2008, 4, 377-387. 

5. Blyth, T.S. and  McFadden, R.  Naturally ordered regular semigroups with a greatest idempotent, Proceedings of 
the Royal Society of Edinburgh, 91A, 1981, 107-122. doi:10.1017/S0308210500012671. 

6. Blyth, T.S. Lattices and Ordered Algebraic Structures, (Springer 2005). doi:10.1007/b139095. 
7. Clifford, A.H. and Preston, G.B. The Algebraic Theory of Semigroups, volume II, A.M.S. Mathematical Surveys, 

1967. doi: 10.1090/surv/007.2. 

   
 

Received   25 June 2020   

Accepted   18 January 2021