IIARD International Institute of Academic Research and Development

INTERNATIONAL JOURNAL OF COMPUTER SCIENCE AND MATHEMATICAL THEORY (IJCSMT )

E-ISSN 2545-5699
P-ISSN 2695-1924
VOL. 10 NO. 1 2024
DOI: https://doi.org/10.56201/ijcsmt.v10.no1.2024.pg37.47

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A Study on the Relationship Between Minimal Generating Sets and Independence in Semigroups

Michael Nsikan John, Sampson, Marshal Imeh, C.F. Igiri†, O. G. Udoaka, L.E. Effiong, Jackson Ante.

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Abstract

This paperreviews the work ofM.I. Sampson et. Al (2023) [7], and delves into the intricate relationship between minimal generating sets and independence in semigroups by examining the comparability of elements induced by orderings on the semigroup. It demonstrates that the existence of a minimal generating set implies independence, and conversely, independence implies the existence of a minimal generating set. Additionally, the paper presents two new algorithms: one for determining minimal generating sets for countable systems of semigroups and another for any given semigroup. These algorithms offer practical solutions for semigroup theorists and researchers. Through rigorous mathematical analysis and proof, this paper sheds light on the fundamental properties of semigroups and their generating sets.

keywords:

Semigroups, Independence, Minimal Generating Sets, Orderings, Algorithms

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References:

[1] Araújo, J. F., &Kochloukov, P. (2019). "Minimal generating sets of semigroups." Semigroup
Forum, 98(1), 13-27.

[2] Howie, J. M. (1995). Fundamentals of Semigroup Theory. Oxford University Press.

[3] Lallement, G. (1979). Semigroups and Combinatorial Applications. John Wiley & Sons.

[4] Petrich, M. (1984). Inverse Semigroups. John Wiley & Sons.

[5] Sampson, M. I., Zsolt, L., Achuobi, J. O., Igiri, C. F., & Effiong, L. E. (2023). On the
Relationship Between Minimal Generating Sets and Independence in Semigroups. Journal of
Algebra, 45(3), 321-335.

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