Measures of central tendency in the disparity of elements in full transformation semi groups

Introduction

The concept of Measures of central tendency dates back to ancient civilizations where mathematician and philosophers like Aristotle, Blaise Pascal, Isaac Newton used various methods to calculate the middle of a set of numbers. Measures of Central Tendency are statistical measure that represents the average values of sets of elements in a semigroup. Recently, some authors like James and Peter (2011), Rotimi Kehinde et.al (2020), Francis Obinna (2022) have used different measures of central tendencies such as distance, work done and average work done to determine the values of semigroups. In this research, we used the distance, mean and mode to describe the movement of element in the domain and co-domain of each semigroup. Finite element formulation ovides necessary flexibility in taking care of different behavior of distinctly different subregions.

The Distance

The distance covered between the movements of elements in this semigroup is described as

The Mean This is given by the sum of the distance covered in the sets as it varies with the order of semi groups and is given as:

The Mode

This is determined by the frequency of distance covered by the movements of elements in the semi group at any degree.

Methodology 2.1 Measures of central tendency in full transformation semi group 𝑫 𝒔 𝑻 𝒏

Distance

The distance covered between the movements of elements in the semi group is described as

For all i,j ∈ 𝐷 𝑠 𝑇 𝑛 Below are transformations on𝐷 𝑠 𝑇 2 :

We will compute the distance between each transformation using the definition of distance function above,

Now, the total distance in 𝐷 𝑠 𝑇 2 is obtained by summing up the distance covered by each transformation in 𝐷 𝑠 𝑇

Mode

Considering the elements in D𝐷 𝑠 𝑂 3 it is observed that𝑀𝐷 𝑠 𝑂 3 = 4.

Measures of Central Tendency in Full Transformation

Mode

From the elements in D𝐷 𝑠 𝐶 4 , it is observed that𝑀𝐷 𝑠 𝐶 4 = 6.

Results

3.1 Proposition: Letα ⊆ D s T n , then, there is a measure of central tendency 𝐶 𝑛 in each of the subsemigroup, α ⊆ S which is a semigroup such that Proof: Consider two arbitrary terms an + am which are both in 𝐷𝐷 𝑠 𝑂 𝑛 , then,

Results in 𝑫 𝒔 𝑻 𝒏

since the operation on the two terms are present in the sequence of the full transformation semigroup then it is closed under the operation + and contains an identity element 0 such that an + 0 = an and 0 + an = an this means that 0 leaves every element in the sequence unchanged under the operation + which is a binary operator that combines two elements to produce another element in the semigroup. Proof: Since the mode is the value that appears most frequently in the transformation, by assumption, m n has the highest frequency among all transformations on M and since f(m n ) > f(m 𝑙 ) for all 𝑙 ≠ 𝑛, m n is the unique transformation with the highest frequency. Therefore, m n = 2n -2 is the mode of the frequency in M ⊆ 𝐷 𝑠 𝐶 𝑛 .

Conclusion

In this research, we have been able to obtain the measures of central tendency (Distance, Mean and Mode) of the set of elements in the disparity of full transformation semigroup, its subsemigroups and the structures of movements in each set of elements in the semigroups. These results describes the movement of elements in the semigroups from the domain and co-domain. we also derived the values for the n th term of the sequences obtained, these results have a wide range of applications in Statistics, Social science, Engineering, Medicine computational algebra, coding theory,automata theory,game theory, it will also be of importance in some branches of physics and theoretical computer science.

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