Nildempotency Structure of Partial One-One Contraction 〖CI〗_n Transformation Semigroups

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

Nildempotency Structure of Partial One-One Contraction 〖CI〗_n Transformation Semigroups

S.A. Akinwunmi¹, M.M. Mogbonju², A.O. Adeniji³, D.O. Oyewola⁴, G. Yakubu⁵, G.R. Ibrahim⁶, and M.O. Fatai^{7
1, 4, & 5}Department of Mathematics and Computer Science, Faculty of Science, Federal University of Kashere,
Gombe, Nigeria.
^{2 & 3}Department 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 are the finite sets and the contraction 〖CI〗_n finite transformation semigroups and the characterization of nildempotent elements in 〖CI〗_n. Let M_n be a finite set, say M_n={m_1,m_2,…m_n}, where m_i is a non-negative integer then α∈〖CI〗_n for which for all q,k∈M_n, |αq-αk|≤|q-k| is a contraction mapping for all q,k∈D(α), provided that any element in D(α) is not assumed to be mapped to empty as a contraction. We show that α∈〖CI〗_n is nildempotent if there exist some minimal (nildempotent degree) m,k∈〖CI〗_n such that α^m=∅⟹α^k=α where |〖CI〗_n |=1 then α(S)=1=n(V)=∅ implies |I(α) |⊆|D(α)| where |〖NDCI〗_n |=1 for each n∈N. Then |〖ECI〗_n |=(█(2^k@(k-n)+1)) , n,k∈N for 1≥k≥n.

Key-phrases: contraction, nildempotency, degree, inverse, semigroup, characterization

Mathematics Subject Classification: 16W22, 10X2, 10X3, 19CLG1, 19CLG2, 06F05 & 10CLM

INTRODUCTION/ BACKGROUND

In a group theory, only the identity element is idempotent but the case is not similar in a semigroup theory in general there may be many idempotent transformation (element), in fact all the transformation may be idempotent which was referred to as band transformation semigroup.

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