International Journal of Research and Scientific Innovation (IJRSI) | Volume VIII, Issue I, January 2021 | ISSN 2321–2705 Nildempotency Structure of Partial One-One

Contraction 퐶퐼푛 Transformation Semigroups S.A. Akinwunmi1, M.M. Mogbonju2, A.O. Adeniji3, D.O. Oyewola4, G. Yakubu5, G.R. Ibrahim6, And M.O. Fatai7 1, 4, & 5Department of Mathematics and Computer Science, Faculty of Science, Federal University of Kashere, Gombe, Nigeria. 2 & 3Department of Mathematics, Faculty of Physical Science, University of Abuja, Abuja, Nigeria. 6Department of Statistics and Mathematical Science, Faculty of Science, Kwara State University Malete, Kwara, Nigeria 7Department of Mathematics, Faculty of Sciences, Federal University Oye-Ekiti , Ekiti, Nigeria.

Abstract: The principal objects of interest in the current research A semigroup homomorphism is a function that preserves are the finite sets and the contraction 퐂퐈 finite transformation 퐧 semigroup structure such that the function 휌: (훼푖 ) ⟶ (훽푗 ) semigroups and the characterization of nildempotent elements in between two transformation are homomorphism if 퐂퐈퐧. Let 퐌퐧 be a finite set, say 퐌퐧 = {퐦ퟏ, 퐦ퟐ, … 퐦퐧}, where 퐦퐢 is 휌(훼푖 , 훽푗 ) = 휌(훼푖 )휌(훽푗 ) holds for all 훼푖 , 훽푗 ∈ 푆. If 푆 is a a non-negative integer then 훂 ∈ 퐂퐈퐧 for which for all 퐪, 퐤 ∈ 퐌퐧, |훂퐪 − 훂퐤| ≤ |퐪 − 퐤| is a contraction mapping for all 퐪, 퐤 ∈ 퐃 훂 , finite semigroup, then a non-empty subset 푊of 푆 is called a provided that any element in 퐃(훂) is not assumed to be mapped sub-semigroup of 푆, if for all 훼푖 , 훽푗 ∈ 푊 . A sub-semigroup of to empty as a contraction. We show that 훂 ∈ 퐂퐈퐧 is this type will be called a subgroup of 푆 (for every 훼푖,푗 ∈ 푊) nildempotent if there exist some minimal (nildempotent degree) such that 훼푖 푊 = 푊훼푖 ⟶ 푊. hence, semigroup which is also 퐦, 퐤 ∈ 퐂퐈 such that 훂퐦 = ∅ ⟹ 훂퐤 = 훂 where 퐂퐈 = ퟏ then 퐧 퐧 a group is called a subgroup. A bijective mapping of a set 훼푖 훂 퐒 = ퟏ = 퐧 퐕 = ∅ implies 퐈 훂 ⊆ |퐃(훂)| where 퐍퐃퐂퐈퐧 = ퟏ to itself is called a permutation on 푀. If 푀 is a set then a one- ퟐ퐤 for each 퐧 ∈ 퐍. Then 퐄퐂퐈 = , 퐧, 퐤 ∈ 퐍 for to-one into mapping 휌: 푀 ⟶ 푀 is said to be transformation. 퐧 퐤 − 퐧 + ퟏ 1≥ 퐤 ≥ 퐧. If 푀 is finite, then 휌 is said to be a permutation which is also known as re-arrangement. The mappings that include the Key-phrases: contraction, nildempotency, degree, inverse, empty set are called partial transformation 푃푛 . The trivial semigroup, characterization fact that the composition of functions is associative gives rise Mathematics Subject Classification: 16W22, 10X2, 10X3, to one of the most promising families of semigroups 19CLG1, 19CLG2, 06F05 & 10CLM (transformation) for the present and next generation of researchers [1]. Hence, the new class of semigroup which we I. INTRODUCTION/ BACKGROUND named nildempotency structure of partial one-one contraction n a group theory, only the identity element is idempotent 푁퐷퐶퐼푛 semigroup. but the case is not similar in a semigroup theory in general I Various special sub-semigroups of partial transformation there may be many idempotent transformation (element), in semigroups have been studied by many researcher(s) like [2], fact all the transformation may be idempotent which was [3] and [4] but few have work on contraction mapping being a referred to as band transformation semigroup. new class of transformation semigroup. The relationship Any given partial one-one contraction transformation 훼푛 is between fix of 훼 and idempotency was study by [4] for 2 nildempotent if there exists a positive integer 푘 such that 훼 ∈ 푆푖푛푔푛 and the equivalence relation {(푥, 푦) ∈ 푁 ∶ 훼푥 = 푘 휶풏 = {푒}, for all 푚, 푘 ∈ 푀푛 under the composition of 훼푦} is denoted by 퐾푒푟(훼). He also showed that the number of mappings where 훼푛 ∈ 퐶퐼푛 then 퐶퐼푛 ⊆ 푆 such that for all stationary blocks is equal to 푓푖푥(훼) and element 훼 is ′ 훼푛 ∈ 퐶퐼푛 there exist a unique 훼푛 ∈ 퐶퐼푛 if the following idempotent if holds for partial one – one convex and axioms were satisfied: contraction transformation semigroup. The method employed ′ in the current research was by listing and studying the (i) For all 훼푛 ∈ 퐶퐼푛 then 훼푛 = 훼푛 훼푛 훼푛 = ′ ′ ′ elements then show the relationship that exists between 훼푛 (훼푛 )훼푛 = 훼푛 ′ idempotent and nilpotent structure using combinatorial (ii) Then 퐼(훼푛 ) = 퐹(훼푛 ) if and only if |훼푞 − 훼푘| ≤ |푞 − approach. For the purpose of the current research work, we ′ 푘| where 훼푛, 훼푛 ∈ 퐶퐼푛 .If for all 푞, 푘 ∈ 푀푛 such that defined a mapping 훼 ∈ 푃푛 for which for all 푞, 푘 ∈ 푀푛 then D(α) is not assumed to be mapped to empty and 훼푞 − 훼푘 ≤ |푞 − 푘| provided that any element in 퐷(훼) is not k αn ∈ In , then αn = {e}, for all 푚, 푞, 푘 ∈ 푀푛 assumed to be mapped to zero as a contraction. We also defined 퐶퐼푛 if a transformation 훼 is partial one-one, 퐶푃푛 if it In other words if a transformation 훼푛 ∈ 퐶퐼푛 contain some is partial and 퐶푇푛 if it is full (total). minimal idempotency and nilpotency degrees then it is 푁퐷퐶퐼푛 Nildempotent semigroup such that 푁퐷퐶퐼푛 ⊂ 푆.

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The following notion will assist to understand the concept of II. NILDEMPOTENCY STRUCTURE OF PARTIAL contraction mapping as used in algebraic system. Let 퐶퐼푛 be ONE-ONE CONTRACTION TRANSFORMATION the order of set of all contraction mappings of 퐼푛 , for a SEMIGROUPS ퟏ ퟐ ퟑ (partial) 훼 ∈ 퐶퐼 : 훼 = such that 퐷 훼 = 1 2 3 The principal objects of interest in the present section are the 푛 ퟏ ∅ ퟐ and 퐼 훼 = 1 ∅ 2 then we have 훼푞 − 훼푘 푎푛푑 푞 − 푘 finite sets and the contraction finite transformation that is 훼푞 − 훼푘 ≤ 푞 − 푘 implies 퐷 훼 ⊆ 푀 whenever semigroups. Firstly, we give characterization of nildempotent 푞, 푘 ∈ 퐷(훼). It is trivial to show for all 푞, 푘 ∈ 퐷(훼), 훼 satisfy elements in 퐶퐼푛 . Let 푀푛 be a finite set, say푀푛 = contraction inequalities such that 훼 is a contraction mapping. {푚1, 푚2, … 푚푛 }, where 푛 is a non-negative integer. For the sake of completeness, we shall recall some Transformation of 푀푛 is an array of the form:

preliminaries terms: 풒ퟏ 풒ퟐ 풒ퟑ…………… 풒풏 휶풏 = (1) 풌ퟏ 풌ퟐ 풌ퟑ…………… 풌풏 Definition 1 [Mapping ( )]: Let 푀 and 푁 be non-empty sets. A relation 휑 from 푀 into 푁 is called a mapping from 푀 into Where all 푘푖 ∈ 푀푛 . If 푛 ∈ 푀푛 , say 푞 = 푚푖 , the element 푘푖 will 푁 if: be called the value of the transformation 훼푛 at the element 푞 and will be denoted by 훼푛 (푞).Then the set of all contraction i. 퐷 휑 = 푀 partial one-one transformation of 푀푛 is denoted by 퐶퐼푛 We ′ ′ ii. For all 푚, 푛 , (푘 푙 ) ∈ 휑, 푚 = 푘 implies푛 = 푙. can rewrite the element of 훼푛 in (1) in the form:

When (ii) is satisfy by 휑, we say 휑 is well defined. We use ′ ퟏ ퟐ ퟑ…………… 풏 휶풏 = (2) the notation 휑: 푀 → 푁 denote a mapping from set 푀 into푁. 풌ퟏ 풌ퟐ 풌ퟑ…………… 풌풏

For (푚, 푛) ∈ 휑, we usually write 휑 풎 = 풏 and say 푛 is the Where 푘푖 = 훼 푖 , if 푖 ∈ 퐷 훼 and 푘푖 = ∅ if 푖 ∈ 퐼(훼). A image of 푚 under 휑 and 푚 is pre-image of 푛. transformation 훼 ∈ 퐶퐼푛 for which for all 푞, 푘 ∈ 푀푛 , then Suppose휑: 푴 → 푵. Then 휑 is a sub-set of 푀 푋 푁 such that |훼푞 − 훼푘| ≤ |푞 − 푘| as a contraction mapping for all for all 푚 ∈ 푀, there exists a unique 푛 ∈ 푁 such that (푚, 푛 ∈ 푞, 푘 ∈ 퐷 훼 , provided that any element in 퐷(훼) is not 휑). Hence, mapping is used as a rule which associates to each assumed to be mapped to empty as a contraction mapping. element 푚 ∈ 푀 exactly one element푛 ∈ 푁. If a relation 휑 is a Suppose 훼푛 ∈ 퐼푛 be a mapping containing 푛 elements, we transformation then the domain 푫 휑 of 휑 is 푀 such that 휑 need to study the number of distinct subset of the mapping 훼푛 is well defined that if 푚 = 푛 ∈ 푀, then 휑 풎 = 휑(풏) ∈ 푁 that will have exactly say 푗 elements such that the empty for all 푚, 푛 ∈ 푀. mapping and the mapping 훼푛 ∈ 퐼푛 are considered to be sub

Definition 2 [Monoid ]: A semigroup 퐼푛 is said to be a monoid semigroups of a finite semigroup 푆 under the composition of if there exists an element 1 ∈ 퐼푛 with 푚1 = 1푚 = 푚 for mapping, then 훼푛 ∈ 퐶퐼푛 which satisfy both nilpotent and idempotent properties is a nildempotent semigroup. Therefore, 푚 ∈ 퐼푛 . The element which is necessarily unique is called transposing (2) under the composition of mapping we have: identity of 퐼푛 . (풌) Definition 3 [Regular Semigroup (퐼 )]: A semigroup 퐼 in 풎ퟏ 풎ퟐ … 풎풏 풎ퟏ 풎ퟐ 풎풏 푛 푟 푛 |푵푪푰풏| = = 휶 which every element is regular is called a regular semigroup 휶ퟏ 휶ퟐ … 휶풏 (∅ ) such that 퐼푚 = {푛: 푚(푛)푚 = 푚} for all 푚, 푛 ∈ 퐼푛 . (3)

Definition 4 [Idempotent Element (푚)]: An element 푚 ∈ 퐼푛 of where 훼푖 = 훼 푖 , if 푖 ∈ 퐷 훼 and 푘푖 = ∅ if 푖 ∈ 퐼(훼) which 2 a finite partial one-one semigroup is idempotent if 푚 = 푚. depicts the structure of nilpotency properties of 훼푛 ∈ 퐶퐼푛 such + In a partial one-one semigroup 퐼푛 , an element 훼 ∈ 퐼푛 is that 푘 ∈ 푍 which is the basis for all nilpotent elements of (푛) idempotent if and only if 퐼 훼 = 퐹(훼), where 퐹(훼) is the set 퐶퐼푛 . Similarly, we have that 훼푛 = 훼푛 where 훼푖 = 훼 푖 , if of order of the elements that maps to itself only (fixed point). ′ 푖 ∈ 퐷 훼 and 퐼(훼푛 ) = 퐹(훼푛 ) for all 푥, 푦 ∈ 푀 where That is 퐹 훼 = |{푚 ∈ 퐷 훼 : 푚훼 = 푚}|. It is denoted by ′ 훼푛, 훼푛 ∈ 퐶퐼푛 which represent the structure of idempotency 퐹 훼 = 푚, and 퐼(훼) is the set of image of the transformation. + properties of 훼푛 ∈ 퐶퐼푛 such that 푛 ∈ 푍 which is the basis for all idempotent elements of 퐶퐼푛 : Definition 5 [Identity element (퐼∅)]: if there exist an element 훼 of 푆 such that 1푚 = 푚1 = 푚 for all 푚 ∈ 푆. It means that 1 is 푚1 푚2 … 푚푛 푛 푚1 푚2 … 푚푛 퐸퐶퐼푛 = (4) an identity element of 푆. Hence, 푆 is called monoid 훼1 훼2 … 훼푛 훼1 훼2 … 훼푛 semigroup. Then for the base of nildempotent element of 퐶퐼푛 we have: Definition 6 [Nilpotent element (푁푘 )]: A transformation 훼 ∈ 푆 풎ퟏ 풎ퟐ 풎풏 풎ퟏ 풎ퟐ … 풎풏 with empty map is a nilpotent if there exists a positive 푵푫푪푰풏 = 휶 ∅ 휶ퟏ 휶ퟐ … 휶풏 integer 푘 such that 훼푘 = ∅. 풎ퟏ 풎ퟐ 풎풏 = (5) 휶(∅ )

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III. MAIN RESULTS (푖) ⟺ (푖푖), Conversely, suppose 훼푘 ≠ 푚 for some 푚 ∈ 퐷(훼) then 퐼(훼) ≠ 퐷(훼) ⟹ 퐷(훼푛 ) ⊂ 퐷(훼). We need to Let 퐶퐼 = {훼 ∈ 퐼 : 푓표푟 푎푙푙 푞, 푘 ∈ 퐷 훼 푡푕푒푛 |훼푞 − 훼푘| ≤ 푛 푛 show that 퐷(훼푘+1) ⊂ 퐷(훼푘 ) for 푘 ∈ 푁. By contradiction let |푞 − 푘|} be semigroup of contraction one-one mapping. A 푘 푘+1 푘 semigroup 푆0 with empty mapping is said to be nildempotent 퐷 훼 ≠ ∅, then 퐷 훼 = 퐷(훼 ). But 훼 ∈ 푁퐷퐶퐼푛 , for 푘 푘+1 푘 provided that there exist 푡, 푘 ∈ 푉: 푆푡 = ∅, that is 푡, 푘 ∈ 퐷(훼) we have that 퐷 훼 = 퐷 훼 = 퐷 훼 ∗ 훼 = (퐼(훼) ∩ 퐷(훼푘 )훼−1). Implies 푒푘 = 푒 ∶ 퐷 훼푘 훼 = 퐼(훼) ∩ 푚1, 푚2, 푚3, … 푚푛 = ∅ for all 푚1, 푚2, 푚3, … 푚푛 ∈ 푆 implies 푘 훼 ∈ 푆 where 푒푘 ∈ 퐷 훼 : 푒푘 = 푒. If 푆 is nildempotent then the 퐷(훼 ). 푡 푘 minimal element 푡, 푘 ∈ 푉: 푆 = ∅ ⟹ 푒 = 푒 is called the For the fact that 푁퐷퐶퐼푛 is a finite contraction ono-one 푘 푘 nildempotency degree of 푆 and is denoted by 푁퐷(푆). We subsemigroup of 퐶퐼푛 , then 퐷 훼 = |퐼(훼) ∩ 퐷(훼 )| ⟹ observe that the nildempotent elements form a sub-semigroup 푒푘 ⊆ 푒 퐷(훼푘 ) ⊆ 퐼(훼) then 퐷 훼푘 = 퐼(훼) ∩ 퐷(훼푘 ). Then by class of their own then the combinatorial nature of sequence inclusion, 퐼 훼푘 = 퐷 훼푘 훼 = 퐼 훼 ∩ 퐷 훼푘 = 퐷(훼). of numbers and their triangular arrangement arise naturally Since 훼푘 = 푒푘 is contraction, so 푚훼푘 = 푚 ⟹ 푒푘 = 푒 for all thus make it essential to find the general relation which in turn 푚, 푘 ∈ 퐷(훼). Now let fix an element 푚표 ∈ 퐷 훼푘 : 푚0훼푘 = highlight it application to Mathematics and Science as whole. 푒0 = 푚0, then we have 퐷(훼푘 ) ⊆ 퐷(훼) such that 푚0훼푘+1 = Some of the result presented in the current research feature 푒0, so 푚0 = 푚0훼푘 ∗ 훼 = 푒0훼 . Therefore there exist at least some of the special number in combinatorial analysis and the 푚0, 푒0 ∈ 퐷 훼 : 푚0훼푘 ∗ 훼 = 푒0 which contradict the Online Encyclopaedia of Integer Sequence (OEIS) would be a assumption that each element in nildempotent. Thus, 퐷 푉푡 = useful tool in this section. ∅ ⟹ 푒푘 = 푒, where 푡 ≥ 1, 푘 ≤ 1. The result complete. Lemma 1: Let 훼 ∈ 푁퐷퐶퐼 , then 훼 is a contraction if and only 푛 Theorem 3: Let 푆 = 퐶퐼 , then if 푘 − 푘 ⊆ 푞 − 푞 for each 1 ≤ 푖 ≤ 푛 − 1, then 푛 푖+1 푖 푖+1 푖 푘(11 푘 3−91 푘2 +292 푘 −389)+183 퐶퐼 = for all 푛 ≥ 2; 푛 ∈ 푁퐷퐶퐼푛 = 1 for all 푖, 푛 ∈ 푁 푛 3 푁. Proof: Let 푀푛 = {1, 2, 3, … 푛}, then 훼 ∈ 푁퐷퐶퐼푛 where 푁퐷퐶퐼푛 ⊂ 푆. Proof: Suppose 퐶퐼푛 ⊂ 푆, where 푆 is a finite one-one transformation semigroup then there exist minimal degree Suppose 푁퐷퐶퐼푛 is contraction then there exist 푞, 푘 ∈ 퐷(훼) 푚 such that 푞 , 푘 ∈ 푀 , 푖 = 1,2,3 … 푛. By definition every 푚 ∈ 퐶퐼푛 : 푆 = ∅ whenever 퐶퐼푛 = { . Since 퐶퐼푛 is 푖 푖 푛 contraction then 훼 ∈ 푆 as bijection , 푛 element of domain element 푞1, 푞2, 푞3, … 푞푛 ∈ 퐷(훼) and 푘1, 푘2, 푘3, … . 푘푛 ∈ 퐼(훼) is 푛 푛 ∈ 퐷(훼) can be chosen from 푀 in ways where contraction where 푞푖+1 − 푞푖 is the domain set of 훼 and 풏 푚 푘 − 푘 is the image set of 훼 such that 푖 = 1,2,3, … 푛. It 푖+1 푖 푀푛 = {1, 2, 3, … 푛}. Then under composition of mapping 푆 Implies 푞 ≥ 푞 ⟹ 푘 ≥ 푘 , thus, 훼 푘 − 푘 ⊆ 푖+1 푖 푖+1 푖 푖+1 푖 contain a reducible polynomial 푃푖 of order 푛 , 푛 ≥ 2 ; 푛 ∈ 푁 훼 푞 − 푞 for each 1 ≤ 푖 ≤ 푛 − 1. 4 3 2 푖+1 푖 such that 푃(푖,푗 ) = 퐶0(푛) + 퐶1(푛) + 퐶2(푛) + 퐶3 푛 + 퐶4… which yield the algebraic system of (6): Conversely, since every element 푞1, 푞2, 푞3, … 푞푛 ∈ 퐷(훼) and 푘1, 푘2, 푘3, … . 푘푛 ∈ 퐼(훼) satisfies contraction such that 퐶퐼푛 is a 1 1 1 1 1 푐0 2 partial one-one then there exist at least one element say 16 8 4 2 1 푐1 7

푉 ∈ 퐶퐼푛 such that 푛 푉 = ∅. Then 훼 ∈ 퐶퐼푛 is nildempotent if 81 27 9 3 1 푐2 = 28 푚 there exist some minimal 푚, 푘 ∈ 퐶퐼푛 such that 훼 = ∅ ⟹ 256 64 16 4 1 푐3 97 푘 . We observe that if then 훼 = 훼 퐶퐼푛 = 1 훼 푆 = 1 = 625 125 25 5 1 푐4 346 푛 푉 = ∅ implies 퐼 훼 ⊆ |퐷(훼)|where 푁퐷퐶퐼푛 = 1 for (6) each 푛 ∈ 푁. Then by reducing (6), we obtain the value: Proposition 2: Let 푉 be a finite nildempotent subsemigroup 11 91 292 389 퐶 = , 퐶 = − , 퐶 = , 퐶 = − and 퐶 = 61. By of 푆 such that 푆 = 푁퐷퐶퐼푛 with an empty map . Then the 0 3 1 3 2 3 3 3 4 following conditions are equivalently true: substitution method we obtain the recurrence relation of the order of |퐶퐼푛 | such that (i) 푆 is nildempotent 푘 11 푘 3−91 푘2 +292 푘 −389 +183 퐶퐼 = where 푃 = (ii) Each element 훼푖,푗 ∈ 푆 is nildempotent 푛 3 (푖,푗 ) |퐶퐼 |, 푛 − 1 = 푘; 푛, 푘 ∈ 푀 for all 푛 ≥ 2. Hence the result. Proof: (푖푖) ⟺ (푖) 푛 푛 2푘 Suppose 푉 is nildempotent, then there exist nilpotency Theorem 4: Let 푆 = 퐸퐶퐼 , then 퐸퐶퐼 = 푛 푛 푘 − 푛 + 1 degrees 푡, 푘: 푉푡 = ∅ ⟹ 푒푘 = 푒. By contradiction let there , 푛, 푘 ∈ 푁 for 1≥ 푘 ≥ 푛. exist 훼 ∈ 푁퐷퐶퐼푛 such that 훼 ≠ ∅ ⟹ 훼푚 = 푚 for some 푚 ∈ 퐷(훼). Then if 훼푚 = 푚: 푚 ∈ 퐷(훼) we have 푚 = 푚훼 = Proof: Let 푀푛 = {1, 2, 3, … 푛}, where 푀푛 ∈ 푁; then if 훼 ∈ 푆 2 3 푛 푘 푡 푘 푚훼 = 푚훼 … . . 푚훼 = 훼 by lemma (1), 푉 = ∅ ⟹ 푒 = the 푘 element of 퐷(훼) can be chosen from 푀푛 in 2푘 푒, 푡, 푘 ∈ 푁. ways in one-one model. By theorem (3) the 푛 − 푘 + 1

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number of idempotent element of 훼 ∈ 푆 is 푛 퐸퐶퐼푛 = because many researchers have worked extensively on the 3 2 푘 푘 −6 푘 +23 푘 −18 +24 1 1 23 subject, such as [6, 8]. where 퐶0 = , 퐶1 = − , 퐶2 = , 12 12 2 12 Remark 2: The triangular array (sequence) of 푁퐷퐼퐶 , 퐸퐶퐼 3 2푘 푛 푛 퐶 = − and 퐶 = 2 which is equivalent to , and 푁퐶퐼 are not yet listed in [IOES], but for more useful 3 2 4 푛 − 푘 + 1 푛 Thus 푛 퐸퐶퐼 is a special case of binomial theorem such that results concerning transformation semigroups we refer to [4], 푛 [5], [6] and [9]. 푘 2푘 푘 푃푘 푄푛−푘 = (푃 + 푄)푘 = ; where 푛=0 푛 푛 − 푘 + 1 ACKNOWLEDGEMENT 푃 = 푄 = 1. Therefore, the result is complete by proposition (2). The corresponding author would like to orderly-preserve special thanks to Professor K. Rauf, for his useful suggestions Theorem 5: Let 푆 = 푁퐶퐼푛 , then 푁퐶퐼푛 = and encouragement. 푘 푘3 +10 푘 2−73 푘 +158 −84 REFERENCES 푒0(31∗22) [1] Howie, J.M; (1995), Fundamentals of semigroup theory, Oxford Such that 푛, 푘 ∈ 푁 for 1≥ 푘 ≥ 푛. University press. [2] Ganyushkin,O, and Mazorchuk, V; [2009]; Classical Finite Proof: Let 푀푛 = {1, 2, 3, … 푛}, where 푀푛 ∈ 푁; then if 훼 ∈ 푆 Transformation Semigroups; An Introduction Springer-Verlag such that 퐼(훼) ⊂ 푆0 (semigroup with zero) there exist London Limited. nilpotency degree 푚 ∈ 퐷 훼 : 훼푚 ≠ ∅, 퐼 훼 = { } whenever [3] Laradiji, A; and Umar, A [2004]; Combinatorial results for semigroup of order-preserving partial transformation. Journal of 훼 ∈ 푁퐶퐼푛 for each 푛 ∈ 푀푛 under composition of mapping 푆 algebra, 278: 342-359. contain a reducible polynomial 푃푖 of order 푛 , 푛 ≥ 1 ; 푛 ∈ 푁 [4] Gomes, G.M and Howie, J.M; (1987b), Nilpotents in symmetric 4 3 2 such that 푃(푖,푗 ) = 퐶0(푛) + 퐶1(푛) + 퐶2(푛) + 퐶3 푛 + 퐶4… inverse finite semigroups.Proc. Edinburgh Math. Soc. 30, Pp. 383- 395. By basic counting principle each element of 훼 ∈ 푆 occur in [5] Sullivan, R.P; (1997), Nilpotents in partial transformations, Bull. 푘 푘3 +10 푘 2−73 푘 +158 −84 Austral. Math. Soc. 55, Pp. 453 –467. ways then by theorem (3) we have: 푒0(31∗22) [6] Ibrahim, M.J; (2015), Algebraic Study of certain finite semigroups 1 1 23 3 of contraction mappings, Ph.D thesis, Ahmadu Bello University, 퐶 = , 퐶 = − , 퐶 = , 퐶 = − and 퐶 = 2 . Then we 0 12 1 2 2 12 3 2 4 Zaria - Nig. 푘 푘3 +10 푘 2−73 푘 +158 −84 [7] Mogbonju, M.M; Makanjuola, S.O; and Adeniji, A.O; [2016]. obtain 푁퐶퐼푛 = 0 1 2 ; 푛, 푘 ∈ 푁 for Cardinality and idempotency of signed order-preserving 푒 (3 ∗2 ) transformation semigroups. Nigerian Journal of Mathematics and 1≥ 푘 ≥ 푛. Hence the result Applications, Vol (25), pp 241 – 245. [8] Agba, J.A, Garba, G.U and Imam, A.T; (2019), The subsemigroup IV. CONCLUDING REMARKS generated by nilpotents in the semigroup of partial ono-to-one order- preserving contraction mapping. Vol. 52 (7&9 Issue), Pp. Remark 1: Of course the study of some other combinatorial 1-8, published by JNAMP properties of transformation semigroup create an open [9] Akinwunmi, S.A & Makanjuola, S.O (2019); Enumeration Partial problem which make the class the most promising class of Contraction Transformation Semi-groups. Vol. B, 29, pp. 70 – 77, semigroups for future study. The forecast was justified published by JNAMP.

TABLE I: The calculated value of 훼(푆) for a small value of Contraction sub-semigroups 푛 훼(푆) 1 2 3 4 5 6 7 8 9 10…

퐶퐼푛 2 7 26 97 346 987 2322 4741 8722 14831…

푁퐷퐶퐼푛 1 1 1 1 1 1 1 1 1 1…...

퐸퐶퐼푛 2 4 8 16 32 62 114 198 326 512…

푁퐶퐼푛 1 3 7 23 63 141 273 477 773 1183..

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