Mathematics and Statistics Vol. 12(6), pp. 537 - 544
DOI: 10.13189/ms.2024.120604
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Left Noetherian Ternary Semigroup and Its Direct Product

Abin Sam Tharakan , G Sheeja ^*
Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur 603203, Tamil Nadu, India

ABSTRACT

A ternary semigroup is termed strongly left Noetherian if every left congruence on can be generated by a finite set. This work examines the fundamental characteristics of left Noetherian ternary semigroups, considering the relationships between the semigroups and their substructures about the left Noetherian condition. Additionally, the study investigates whether this property is retained in the direct product of such semigroups. This work offers a comprehensive characterization of strongly left Noetherian ternary semigroups. It is established that the homomorphic image of a Noetherian ternary semigroup retains the Noetherian property, and an alternative approach is introduced for demonstrating that a ternary semigroup is strongly left Noetherian. The relationship between ternary semigroups and their subsemigroups is explored, showing that a ternary semigroup is strongly left Noetherian if at least one of its subsemigroups is left Noetherian. A necessary characterization for inverse ternary semigroups is also presented. In addition, the strongly left Noetherian property is the direct product of ternary semigroups, identifying conditions that preserve this property. A necessary and sufficient condition is established for the direct product of an infinite ternary semigroup and a ternary monoid to be strongly left Noetherian, along with findings on the left Noetherian property in direct products of various ternary semigroups.

KEYWORDS
Strongly Left Noetherian Ternary Semigroup, Left Congruence, Finitely Produced, Direct Product

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Abin Sam Tharakan , G Sheeja , "Left Noetherian Ternary Semigroup and Its Direct Product," Mathematics and Statistics, Vol. 12, No. 6, pp. 537 - 544, 2024. DOI: 10.13189/ms.2024.120604.

(b). APA Format:
Abin Sam Tharakan , G Sheeja (2024). Left Noetherian Ternary Semigroup and Its Direct Product. Mathematics and Statistics, 12(6), 537 - 544. DOI: 10.13189/ms.2024.120604.