Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions


Ratio Mathematica                                                Volume 37, 2019, pp. 61-68   

 

              
 

61 

 

 

 

On cyclic multigroup family 

 

 

Johnson Aderemi Awolola* 

 

  
 

Abstract  

In this paper, the concept of cyclic muligroup is studied from the 

preliminary knowledge of cyclic group which is a well-known 

concept in crisp environment. By using cyclic multigroups, we 

then delineate a cyclic multigroup family and investigate its 

structural properties. It is observed that the union of class of cyclic 

multigroups generated by 𝓐 is a cyclic multigroup. However, the 
union is an identity cyclic multigroup. In particular, we obtain a 

series of class of cyclic multigroups generated by 𝓐. 
 

Keywords: Multiset, Multigroup, Cyclic multigroup, Cyclic 

multigroup family. 

 

2010 AMS subject classification: 08𝐴72, 03𝐸72, 94𝐷05.† 
 

 

 

 

 

 

 

 

 
 

 

 
* Department of Mathematics/Statisics and Computer Science, College of Science (University 

of Agriculure, Makurdi, Nigeria), remsonjay@yahoo.com, awolola.johnson@uam.edu.ng. 
† Received on August 23rd, 2019. Accepted on December 25th, 2019. Published on December 

31th, 2019. doi: 10.23755/rm.v37i0.476. ISSN: 1592-7415. eISSN: 2282-8214.  

©Johnson Aderemi Awolola. This paper is published under the CC-BY licence agreement. 

mailto:remsonjay@yahoo.com


J. A. Awolola 

 

62 

 

1  Introduction 
 

In set theory, repetition of objects are not allowed in a collection. This 

perspective rendered set almost irrelevant because many real life problems 

admit repetition. To remedy the inadequacy in the idea of sets, the concept of 

multisets was introduced in [6] as a generalization of sets by relaxing the 

restriction of distinctness on the nature of the objects forming a set. Multiset is 

very promising in mathematics, computer science, website design, etc. See [4, 

5] for details. 

Generalization of algebraic structures is playing a prominent role in the 

sphere of mathematics. One of such generalization of algebraic structures is 

the notion of multigroups. Multigroups are actually a generalization of groups 

and have come into the centre of interest. In [1], the multigroup proposed is 

analogous to fuzzy group [2] in that the underlying structure is a multiset. 

Although multigroup concept was earlier used in [9, 12] as an extension of 

group theory, however the recent definition of multigroup in [1] is adopted in 

this paper because it shows a strong analogy in the behaviour of group and 

makes it possible to extend some of the major notions and results of groups to 

that of multigroups. Some of the related works can be found in [3], [7], [8], 

[10], [11] etc. 

 

The aim of this paper is to promote research and the development of 

multiset knowledge by studying cyclic multigroup family based on the 
sufficient condition for a multiset to be a cyclic multigroup. 

 

 

2  Preliminaries 

In this section, we give the preliminary definitions and results that will 

be required in this paper from [1, 8]. 
 

Definition 2.1 Let ℧ be a non-empty set. A multiset 𝐴 drawn from ℧ is 
characterized by a count function 𝐶𝐴 defined as 𝐶𝐴 : ℧ ⟶ 𝓓 , where 𝒟 
represents the set of non-negative integers. 

 

For each 𝑥 ∈ ℧, 𝐶𝐴(𝑥) is the characteristics value of 𝑥 in 𝐴 and 
indicates the number of occurrences of the element 𝑥 in 𝐴. An expedient 
notation of 𝐴 drawn from ℧ = {𝑥1,  𝑥2 , … ,  𝑥𝑛 } is [𝑥1,
𝑥2 , … ,  𝑥𝑛 ]𝐶𝐴(𝑥1), 𝐶𝐴(𝑥2) ,…, 𝐶𝐴(𝑥𝑛) such that 𝐶𝐴(𝑥𝑖 ) is the number of times 𝑥𝑖  

occurs in 𝐴, (𝑖 = 1, 2, … , 𝑛). 
 

The class of all multisets over ℧ is denoted by 𝑀𝑆(℧). 



On cyclic multigroup family 
 

63 

 

 

Definition 2.2 Let 𝐴, 𝐵 ∈ ℧. Then 𝐴 is a submultiset of 𝐵 written as 𝐴 ⊆ 𝐵 or 
𝐵 ⊇ 𝐴 if 𝐶𝐴(𝑥) ≤ 𝐶𝐵 (𝑥), ∀ 𝑥 ∈ ℧. Also, if  𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵, then 𝐴 is called 
a proper submultiset of 𝐵 and denoted as 𝐴 ⊂ 𝐵. 
 

Definition 2.3 Let 𝐴, 𝐵 ∈ 𝑀𝑆(℧). Then the union and intersection denoted by 
𝐴 ⋃ 𝐵 and 𝐴 ⋂ 𝐵 are respectively defined as follows:  

            𝐶𝐴 ⋃ 𝐵 (𝑥) = 𝐶𝐴(𝑥) ⋁ 𝐶𝐵 (𝑥) = 𝑚𝑎𝑥{𝐶𝐴(𝑥), 𝐶𝐵 (𝑥)} and             
 

      𝐶𝐴 ⋂ 𝐵 (𝑥) = 𝐶𝐴(𝑥) ⋀ 𝐶𝐵 (𝑥) = 𝑚𝑖𝑛{𝐶𝐴(𝑥), 𝐶𝐵 (𝑥)}, ∀ 𝑥 ∈ ℧. 
 

Definition 2.4  Let {𝐴𝑖 }𝑖∈Λ be an arbitrary family of multisets over ℧. Then for 
each 𝑖 ∈ Λ, ⋃𝑖∈Λ𝐴𝑖 = ⋁𝑖∈Λ𝐶𝐴𝑖 (𝑥) and ⋂𝑖∈Λ𝐴𝑖 = ⋀𝑖∈Λ𝐶𝐴𝑖 (𝑥). 

 

Definition 2.5 The direct product of multisets 𝐴 and 𝐵 is defined as  
𝐴 × 𝐵 =  {[𝑥, 𝑦]𝐶𝐴×𝐵 [(𝑥, 𝑦)]  ∣  𝐶𝐴×𝐵 [(𝑥, 𝑦)] = 𝐶𝐴(𝑥)𝐶𝐴(𝑦)}.  

 

Definition 2.6 Let ℧ be a non-empty set. The sets of the form  
𝐴𝑛 = {𝑥 ∈ ℧ ∣ 𝐶𝐴(𝑥) ≥ 𝑛,   𝑛 ∈ ℤ

+} are called the 𝑛 – level sets of 𝐴. 
 

Definition 2.7 Let ℧ and 𝜉 be two non-empty sets and 𝑓 ∶  ℧ ⟶ 𝜉 be a 
mapping. Then the image 𝑓(𝐴) of a multiset 𝐴 ∈ 𝑀𝑆(℧) is defined as     

𝐶𝑓(𝐴)(𝑦) = {
⋁ 𝐶𝐴(𝑥),              𝑓

−1(𝑦) ≠ ∅𝑓(𝑥)=𝑦

0,                                      𝑓 −1(𝑦) = ∅
 

 

Definition 2.8 Let 𝒳 be a group. By a multigroup over 𝒳 we mean a count 
function 𝐶𝐴 ∶  𝒳 ⟶ 𝒟 such that  
            𝐶𝐴(𝑥𝑦) ≥ 𝐶𝐴(𝑥) ⋀ 𝐶𝐴(𝑦), ∀ 𝑥, 𝑦 ∈ 𝒳 and 𝐶𝐴(𝑥

−1) ≥ 𝐶𝐴(𝑥), ∀ 𝑥 ∈ 𝒳. 
 

Moreover, an abelian multigroup over 𝒳 is defined as a multigroup 
satisfying the condition 𝐶𝐴(𝑥𝑦) ≥ 𝐶𝐴(𝑦𝑥), ∀ 𝑥, 𝑦 ∈ 𝒳. 

 

Let 𝑒 be the identity element of 𝒳. It can be easily verified that if 𝐴 is a 
multigroup over a group 𝒳, then 𝐶𝐴(𝑒) ≥ 𝐶𝐴(𝑥) and 𝐶𝐴(𝑥

−1) ≥ 𝐶𝐴(𝑥), ∀ 𝑥 ∈
𝒳. 

 

We denote the class of all multigroups over 𝒳 by 𝑀𝐺(𝒳). 
 

Proposition 2.1 Let 𝐴 ∈ 𝑀𝑆(℧). Then 𝐴 ∈ 𝑀𝐺(𝒳) if and only if 𝐶𝐴(𝑥𝑦
−1) ≥

𝐶𝐴(𝑥) ⋀ 𝐶𝐴(𝑦), ∀ 𝑥, 𝑦 ∈ 𝒳. 
 



J. A. Awolola 

 

64 

 

Proposition 2.2 Let 𝐴 ∈ 𝑀𝐺(𝒳). Then 𝐴𝑛 , 𝑛 ∈ ℤ
+ are subgroups of 𝒳. 

 

Proposition 2.3 Let 𝒳, 𝒴 be groups and 𝑓 ∶  𝒳 ⟶ 𝒴 be a homomorphism. If 
𝐴 ∈ 𝑀𝐺(𝒳), then 𝑓(𝐴) ∈ 𝑀𝐺(𝒴). 
 

 

3  Cyclic Multigroup Family 

Definition 3.1 Let 𝒳 = 〈𝑎〉 be a cyclic group. If 𝒜 = {[𝑎𝑛]𝐶𝒜 (𝑎𝑛) ∣ 𝑛 ∈ ℤ} is 

a multigroup, then 𝒜 is called a cyclic multigroup generated by [𝑎]𝐶𝒜 (𝑎) and 

denoted by 〈[𝑎]𝐶𝒜 (𝑎)〉.  

 

Proposition 3.1 If 𝒜 is a cyclic multigroup and 𝑚 ∈ ℤ+, then 𝒜𝑚 =

{([𝑎𝑛]𝐶𝒜 (𝑎𝑛))
𝑚

∣ 𝑛 ∈ ℤ}  is also a cyclic multigroup.  

 

Proof. Let us show that 𝒜𝑚 satisfies the two conditions in Definition 2.8. We 
can consider only its count function because the 𝑚 − 𝑡ℎ power of  𝒜 effects 
just only the count function of 𝒜𝑚. 
 

Since 𝒜 is a multigroup and 𝐶𝒜 (𝑎) ∈ 𝒟, we have 

(𝐶𝒜 (𝑎
𝑛1 𝑎𝑛2 ))𝑚 ≥ (𝐶𝒜 (𝑎

𝑛1 ) ⋀ 𝐶𝒜 (𝑎
𝑛2 ))

𝑚
= (𝐶𝒜 (𝑎

𝑛1 ) )𝑚 ⋀ (𝐶𝒜 (𝑎
𝑛2 ) )𝑚 

and consequently, (𝐶𝒜 (𝑎
−𝑛))

𝑚
≥ (𝐶𝒜 (𝑎

𝑛))
𝑚

. 

 

 This completes the proof of the proposition.  

 

Example 3.1 Let 𝒳 = 〈𝑎〉 be a cyclic group of order 12 such that 𝐶𝒜 (𝑎
0) =

𝑡0, 𝐶𝒜 (𝑎
4) = 𝐶𝒜 (𝑎

8) = 𝑡1, 𝐶𝒜 (𝑎
2) = 𝐶𝒜 (𝑎

6) = 𝐶𝒜 (𝑎
10) = 𝑡2,  𝐶𝒜 (𝑥) = 𝑡3 

for other elements 𝑥 ∈ 𝒳, where 𝑡𝑖 ∈ 𝒟, 0 ≤ 𝑖 ≤ 3 with 𝑡1 > 𝑡1 > 𝑡2 > 𝑡3. It 
is clear that 𝒜 is a multigroup over 𝒳. Thus, 𝒜 = {[𝑎𝑛]𝐶𝒜 (𝑎𝑛) ∣ 𝑛 ∈ ℤ} is a 

cyclic multigroup generated by [𝑎]𝐶𝒜 (𝑎). 

 

Definition 3.2 Let 𝑒 be the identity element of the group 𝒳. We define the 

identity cyclic multigroup ℰ by ℰ = {[𝑒]𝐶𝒜 (𝑒) ∣ 𝐶𝒜 (𝑒) ≥ 𝐶𝒜 (𝑎
𝑛),   𝑛 ∈ ℤ}. 

 

Proposition 3.2 If 𝑚 ≤ 𝑛, then the multigroup 𝒜𝑛 is a submultigroup of 𝒜𝑚. 
 

Proof. Clearly  𝒜𝑛 and 𝒜𝑚 are multigroups by Definition 2.8. For every 𝑎 ∈
𝒟, 𝑎𝑚 ≤ 𝑎𝑛 implies 𝒜𝑚 ⊆ 𝒜𝑛 (since 𝐶𝒜𝑚 (𝑎) ≤ 𝐶𝒜𝑛 (𝑎)  ∀ 𝑎 ∈ 𝒳). 
 



On cyclic multigroup family 
 

65 

 

Proposition 3.3 If 𝒜𝑖 and 𝒜𝑗  are cyclic multigroups, and 𝑖 < 𝑗, then 
𝒜𝑖  ⋃ 𝒜𝑗   is also a cyclic multigroup for any 𝑖, 𝑗 ∈ ℤ+. 
 

Proof. It is sufficient to consider only count functions. Without loss of 

generality, let 𝑖 ≤ 𝑗. Since 𝒜𝑖 ⊆ 𝒜𝑗 , we have  
𝐶𝒜𝑖 ⋃ 𝒜𝑗 (𝑎

𝑛𝑎𝑚) = 𝐶𝒜𝑖 (𝑎
𝑛𝑎𝑚 ) ⋁ 𝐶𝒜𝑗 (𝑎

𝑛𝑎𝑚) = 𝐶𝒜𝑗 (𝑎
𝑛𝑎𝑚) 

                                                                                      ≥ 𝐶𝒜𝑗 (𝑎
𝑛) ⋀ 𝐶𝒜𝑗 (𝑎

𝑚) 
                                                                        = 𝐶𝒜𝑖 ⋃ 𝒜𝑗 (𝑎

𝑛) ⋀ 𝐶𝒜𝑖 ⋃ 𝒜𝑗 (𝑎
𝑚) 

and 

 

   𝐶𝒜𝑖 ⋃𝒜𝑗 (𝑎
−𝑛) = 𝐶𝒜𝑖 (𝑎

−𝑛) ⋁ 𝐶𝒜𝑗 (𝑎
−𝑛)  

                                  = 𝐶𝒜𝑖 (𝑎
𝑛) ⋁ 𝐶𝒜𝑗 (𝑎

𝑛) = 𝐶𝒜𝑖 ⋃ 𝒜𝑗 (𝑎
𝑛)        

 

Hence, 𝒜𝑖  ⋃ 𝒜𝑗 is a cyclic multigroup. 
 

Proposition 3.4  If 𝒜𝑖 and 𝒜𝑗  are cyclic multigroups, then 𝒜𝑖  ⋂ 𝒜𝑗   is also a 
cyclic multigroup. 

 

Proof. Similar to Proposition 3.3.  

 

Remark 3.1 Since a cyclic group is an abelian group, it is obvious by 

Definition 2.8 that the cyclic multigroups 𝒜𝑚, 𝒜𝑖  ⋃ 𝒜𝑗  and 𝒜𝑖  ⋂ 𝒜𝑗 are 
also abelian multigroups. 

 

Definition 3.3 Let 𝒜 be a cyclic multigroup, then the following class of cyclic 
multigroups {𝒜,  𝒜2,  𝒜3, … ,  𝒜𝑚, … , ℰ} is called the cyclic multigroup 
family generated by 𝒜 and denoted by 〈𝒜〉. 
 

Proposition 3.5 Let 〈𝒜〉 = {𝒜,  𝒜2,  𝒜3, … ,  𝒜𝑚, … , ℰ}. Then ⋃ 𝒜𝑛 = 𝒜∞𝑛=1  
and ⋂ 𝒜𝑛∞𝑛=1 = ℰ. 
 

Proof. The proof is immediate from Propositions 3.3 and 3.4.  

 

Proposition 3.6 Let 𝒜 be a cyclic multigroup. Then 𝒜 ⊆  𝒜2 ⊆   𝒜3 ⊆ ⋯ ⊆
 𝒜𝑛 ⊆ ⋯ ⊆  ℰ. 
 

Proof. It is known that 𝐶𝒜 (𝑎) ∈ 𝒟. Hence, 

𝐶𝒜 (𝑎) ≤ (𝐶𝒜2 (𝑎))
2

,  𝐶𝒜 (𝑎
2) ≤ (𝐶𝒜2 (𝑎

2))
2

, … ,  𝐶𝒜 (𝑎
𝑛) ≤ (𝐶𝒜2 (𝑎

𝑛))
2
.  

 



J. A. Awolola 

 

66 

 

By Definition 2.2, we have 𝒜 ⊆ 𝒜2 . By generalizing it for any 𝑖, 𝑗 ∈

ℤ+ with 𝑖 ≤ 𝑗, we then obtain (𝐶𝒜𝑖 (𝑎))
𝑖

≤ (𝐶𝒜𝑗 (𝑎))
𝑗

, (𝐶𝒜𝑖 (𝑎
2))

𝑖

≤

(𝐶𝒜𝑗 (𝑎
2))

𝑗

, … ,  (𝐶𝒜𝑖 (𝑎
𝑛))

𝑖

≤ (𝐶𝒜𝑗 (𝑎
𝑛 ))

𝑗

. So 𝒜𝑖 ⊆ 𝒜𝑗  for any 𝑖, 𝑗 ∈ ℤ+ 

with 𝑖 ≤ 𝑗, which means that 𝒜 ⊆  𝒜2 ⊆   𝒜3 ⊆ ⋯ ⊆  𝒜𝑛 ⊆ ⋯ . 
 

Finally, we have ℰ = ⋂ 𝒜𝑛∞𝑛=1 , which is immediate from Proposition 
3.5 since 

  

Lim
𝑛⟶∞

𝐶𝒜 (𝑎
𝑛) = {

𝑡0, 𝑖𝑓  𝑎 = 𝑒,
0, 𝑖𝑓  𝑎 ≠ 𝑒.

 

    

 

This completes the proof for the required relations.  

 

Corollary 3.1 Let 〈𝒜〉 = {𝒜,  𝒜2,  𝒜3, … ,  𝒜𝑚, … , ℰ}. Then  
𝒜 <  𝒜2 <  𝒜3 < ⋯ <  𝒜𝑚 < ⋯ <  ℰ.   

 

Proof. The proof is similar to Proposition 3.6.  

 

Proposition 3.7 Let 𝜑 be a group homomorphism of a cyclic multigroup 𝒜. 
Then the image of 𝒜 under 𝜑 is a cyclic multigroup.  
 

Proof. It is well known that in the theory of classical cyclic groups, the image 

of any cyclic group is a cyclic group and the homomorphic image of a 

multigroup is a multigroup (from Proposition 2.3). From these two results and 

Definition 2.8, it is clearly seen that the image of 𝒜 under 𝜑 is a cyclic 
multigroup. 

 

Proposition 3.8 Let 𝒳𝑛 be the 𝑛 − level set of the cyclic group 𝒳. If 𝑖, 𝑗 ∈ ℤ
+ 

such that 𝑖 < 𝑗, then 𝒜𝑛
𝑖  is a subgroup of  𝒜𝑛

𝑗
. 

 

Proof. It is obvious that sets 𝒳𝑛 and 𝒳𝑛
𝑚 are cyclic subgroups of 𝒳𝑛  in crisp 

sense. Since 𝑖 < 𝑗, then 𝒜𝑛
𝑗

(𝑎) ≥ 𝒜𝑛
𝑖 (𝑎) ≥ 𝑛, ∀ 𝑎 ∈ 𝒳𝑛

𝑗
. Thus, 𝒳𝑛

𝑖 ⊆ 𝒳𝑛
𝑗
. 

Therefore, 𝒳𝑛
𝑖  is a subgroup of 𝒳𝑛

𝑗
. 

 

Remark 3.2 From Propositions 3.6 and 3.8, we have that a normal series of 𝒳 
is a finite sequence 𝒳𝑛

𝑚, 𝒳𝑛
𝑚−1, … , 𝒳𝑛 of normal level subgroups of 𝒳 such 

that 𝒳𝑛
𝑚 > 𝒳𝑛

𝑚−1 > ⋯ >  𝒳𝑛 since 𝒳 is a cyclic group. 
 



On cyclic multigroup family 
 

67 

 

Proposition 3.9 Let {𝒜𝑚,  𝒜𝑚−1, … , 𝒜} be a finite cyclic multigroup family. 
Then 𝒜𝑚 × 𝒜𝑚−1 × … × 𝒜 = 𝒜𝑚. 
 

Proof. It is easily verified using the definition of product of multigroups and 

Proposition 3.6.  

 

 

4  Conclusion 
The paper introduced the concept of cyclic multigroup family and 

investigated its related structure properties. For future studies, one can extend 

this idea to other non-classical algebraic structures such as soft group, rough 

group, neutrosophic group and smooth group.  

 

 

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