SOME ALGEBRAIC PROPERTIES OF ORDER-PRESERVING FULL CONTRACTION TRANSFORMATION SEMIGROUP

Let Xn be a finite set, CTn the semigroup of full contraction and OCTn order-preserving full contraction transformation semigroup of a finite set. In this paper the Local and global U-depth as well as the status of OCTn were investigated where U is the generating set. The local and global depth were found from the known generating Set of OCTn and also, the status of the semigroup OCTn was obtained from the product of global depth and the order of generating of OCTn. For α ∈ OCTn, local depth of is equal to its defect, global depth and status of OCTn are n− 1 and 2(n− 1)2 respectively. We also look at the structure of Greens relations of order-preserving full contraction transformation semigroup.


INTRODUCTION
The algebraic theory of semigroup has been well studied during the second half of the twentieth century.Many ideas in study of semigroup were directly motivated by their analogues in the ring theory.However, certain important and interesting concepts such as the concepts of Green's relation, idempotent rank, nilpotent rank , product of idempotent and so on developed independent , just as the set of all permutation of a finite set proves to be an important source of examples in group theory, the set of all mappings of a finite set in to itself provide us with a corresponding objects in the semigroup theory.The importance of mappings in semigroup theory can be judged by the fact that every semigroup is isomorphic to a semigroup of mappings in this sense it turns out that the understanding of semigroup of mappings in semigroup theories is of paramount significance.Transformation semigroups are one of the most fundamental mathematical objects.They occur in theoretical computer science, where properties of language depend on algebraic properties of various transformation semigroups related to them.
Algebraic and Combinatorial properties of transformation semigroup have been studied over a long period and interesting results have emerged for example Ganyushkin and Mazorchuk (2009); Umar (1996).One of the reasons is due to the fact that every finite inverse semigroup can be embedded in a symmetric inverse semigroup.On studying a new class of semigroup it is mostly of interest to know the type of semigroup in question for example , whether the semigroup is regular, inverse (see Clifford and Preston, 1967).Another interest is in the characterisation of the five classical Green's relations, mostly when the semigroup is regular.For nonregular semigroup, generalisation of Green's relations like starred analogues are investigated ( Maclister 1976).Regular semigroups were introduced by Green (1951) and was the paper in which popular Green's relations were introduced.It was his study of regular semigroups which led Green to define relations.Recently, Garba et.al.(2017)characterised the Green's relations and starred Green's relations in CT n , CI n and also starred Green's relation in OCT n .

BASIC DEFINITIONS
and OCT n = CT n ∩ O n be the subsemigroups of T n \ S n consisting of all order-preserving maps, all contraction maps and all order -preserving maps contraction maps respectively, (S n issymmetricgroup) Definition 2.1: A non-empty subset U of semigroup S is said to be a generating set for S, written as U = S, if every elements of S is expressible as a product of elements in U. if S is finite, local U-depth of s ∈ S is the smallest integer k such that s = u 1 u 2 u 3 . . .u k for some u 1 u 2 . . .u k ∈ U and the global U-depth of S is the smallest integer in such that U (m) = S where Definition 2.2:.On the set X n , the equivalence

MAIN RESULTS
According to Garba et.al. (2017) the generating set of semigroup (OCT n ) is expressed as and from definition of generating set, any arbitrary α ∈ OCT n is expressed as product of β and Results on Depth and Status of Order-preserving Full Contraction Transformation Semigroup Theorem 4.1: Let S = OCT n , for each α ∈ OCT n , the local U-depth of α is equal to de f (α) .
 where A i < A j < . . .< A s are non singleton blocks in α and a i , a j , . . ., a s are the order of non single blocks respectively, that is Now, if α ∈ OCT n , there must exist a block A i for which |A i | ≥ 2. Three cases were considered to prove this theorem.
Case 2: If a 1 = 2, in this case α can be expressed as a product of β and γ as follow and if at least one of s + 1, s + 2 is not inA s Case 3: If a 1 ≥ 3, in this case α can be expressed as product of β and γ i.e.

Remark
We observe that as the value of a 1 increases, the more the factor γ appears.Theorem 4.3: Proof: Let U be the generating set, |U| be the order of U and ∆(U) be the global U-depth of OCT n .Then, |U| is equal to 2(n − 1) by corollary 2.1 and from theorem 3.2.above ∆(U) equal to n − 1.Since the status of semigroup is the least value of the product of order of generating set and global U-depth, that is Results on Green's Relations of order-preserving full contraction transformation semi-

CONCLUSION
In conclusion, we have shown that in the semigroup OCT n , the minimum length factorisation of each α ∈ OCT n is equal to the defect of α and the Status of semigroup OCT n were obtained using the global U − depth to be Stat(OCT n ) ≤ 2(n − 1) 2 with relevant examples.Also, the structure of Green's relations on semigroup OCT n were characterized.

Definition 2 . 3 :
The status of semigroup S is defined as the least value of the product of the order of generating set |U| and the global U-depth ∆ (U).That is,Stat (S) = min{|U|∆ (U) : U = S} Definition 2.4:The defect of α is define as non image point of α i.e n − |Im(α)|.

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d.)This follows from(Garba et.al.2017, )  in the characterisation of starred Green's relations of order-preserving partial one-one contraction transformation semigroup (OCI n ).