Source of $\Gamma$-Semigroups' Primeness

Didem Yeşil; Rasie Mekera

doi:10.53570/jnt.1732995

Source of $\Gamma$-Semigroups' Primeness

Abstract

This study aims to define the source of a $\Gamma$-semigroup $S$'s primeness and to research some of its basic properties and the $P_{S_{\Gamma}}$ subset of S as $P_{S_{\Gamma}}=\{b\in S : S\Gamma S \Gamma b=(0)\}$, for the $\Gamma$-semigroup $S$ with zero. Afterward, it investigates the relationships between $\lvert P_{S_{\Gamma}}\rvert$-reduced, $\lvert P_{S_{\Gamma}}\rvert$-idempotent, $\lvert P_{S_{\Gamma}}\rvert$-strongly idempotent, and $\lvert P_{S_{\Gamma}}\rvert$-regular $\Gamma$-semigroup structures as follows: (\emph{i}) If $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-idempotent $\Gamma$-semigroup, then $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-regular $\Gamma$-semigroup, (\emph{ii}) If $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-idempotent $\Gamma$-semigroup, then $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-reduced $\Gamma$-semigroup, (\emph{iii}) If $S$ is an idempotent (regular, reduced) $\Gamma$-semigroup, then $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-idempotent (regular, reduced) $\Gamma$-semigroup, (\emph{iv}) If $S$ is a $\lvert P_{S_{\Gamma}}\rvert$- strongly idempotent (regular) $\Gamma$-semigroup, then $A$ is a $\lvert P_{A_{\Gamma}}\rvert$- strongly idempotent (regular) $\Gamma$-semigroup, and (\emph{v}) If $S$ is a commutative $\lvert P_{S_{\Gamma}}\rvert$-regular $\Gamma$-semigroup, then $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-reduced $\Gamma$-semigroup. Moreover, this study explores the connections between the source of $\Gamma$-primeness of a $\Gamma$-semigroup and the aforementioned $\Gamma$-semigroup structures and clarifies some theoretical parts of the study with several examples.

Keywords

References

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Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

Didem Yeşil ^*
0000-0003-0666-9410
Türkiye

Rasie Mekera
0000-0002-0092-2991
Türkiye

Early Pub Date

September 30, 2025

Publication Date

September 30, 2025

Submission Date

July 2, 2025

Acceptance Date

September 28, 2025

Published in Issue

Year 2025 Number: 52

DOI

https://doi.org/10.53570/jnt.1732995

IZ

https://izlik.org/JA95RU36FE

Cite

RIS / Bibtex

APA

Yeşil, D., & Mekera, R. (2025). Source of $\Gamma$-Semigroups’ Primeness. Journal of New Theory, 52, 27-37. https://doi.org/10.53570/jnt.1732995

AMA

1.Yeşil D, Mekera R. Source of $\Gamma$-Semigroups’ Primeness. JNT. 2025;(52):27-37. doi:10.53570/jnt.1732995

Chicago

Yeşil, Didem, and Rasie Mekera. 2025. “Source of $\Gamma$-Semigroups’ Primeness”. Journal of New Theory, nos. 52: 27-37. https://doi.org/10.53570/jnt.1732995.

EndNote

Yeşil D, Mekera R (September 1, 2025) Source of $\Gamma$-Semigroups’ Primeness. Journal of New Theory 52 27–37.

IEEE

[1]D. Yeşil and R. Mekera, “Source of $\Gamma$-Semigroups’ Primeness”, JNT, no. 52, pp. 27–37, Sept. 2025, doi: 10.53570/jnt.1732995.

ISNAD

Yeşil, Didem - Mekera, Rasie. “Source of $\Gamma$-Semigroups’ Primeness”. Journal of New Theory. 52 (September 1, 2025): 27-37. https://doi.org/10.53570/jnt.1732995.

JAMA

1.Yeşil D, Mekera R. Source of $\Gamma$-Semigroups’ Primeness. JNT. 2025;:27–37.

MLA

Yeşil, Didem, and Rasie Mekera. “Source of $\Gamma$-Semigroups’ Primeness”. Journal of New Theory, no. 52, Sept. 2025, pp. 27-37, doi:10.53570/jnt.1732995.

Vancouver

1.Didem Yeşil, Rasie Mekera. Source of $\Gamma$-Semigroups’ Primeness. JNT. 2025 Sep. 1;(52):27-3. doi:10.53570/jnt.1732995

Cited By

On Ordered-Primeness and Ordered-Semiprimeness in Ordered Semigroups

Journal of Advanced Research in Natural and Applied Sciences

https://doi.org/10.28979/jarnas.1919891