Exploring prime $ F- $filters of almost distributive lattices

Ali Yahya Hummdi; N. Rafi; M. Balaiah; Y. Monikarchana; Ali Yahya Hummdi; N. Rafi; M. Balaiah; Y. Monikarchana

doi:10.3934/math.20251210

AIMS Mathematics

2025, Volume 10, Issue 11: 27519-27534. doi: 10.3934/math.20251210

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Exploring prime $ F- $filters of almost distributive lattices

1.
Department of Mathematics, College of science, King Khalid University, Abha 61471, Saudi Arabia
2.
Department of Mathematics, Siddhartha Academy of Higher Education Deemed to be University, Vijayawada, Andhra Pradesh, 520007, India
3.
Department of Mathematics, Bapatla Engineering College, Bapatla, Andhra Pradesh, 522102, India
4.
Department of Mathematics, Mohan Babu University, A.Rangampet, Tirupati, Andhra Pradesh, 517102, India

Received: 14 July 2025 Revised: 23 October 2025 Accepted: 11 November 2025 Published: 26 November 2025
MSC : 06D15, 06D99

The concept of $ {F}- $filters is introduced in an almost distributive lattice(ADL) and their properties are studied. A set of equivalent conditions is established for every proper $ {F}- $filter of an ADL to become a prime $ {F}- $filter. For any $ {F}- $filter $ {U} $ of an ADL, $ \mathcal{O}^{{F}}({U}) $ is defined, and it is proved that $ \mathcal{O}^{{F}}({U}) $ is an $ {F}- $filter if $ {U} $ is prime. It is also derived that each minimal prime $ {F}- $filter belonging to $ \mathcal{O}^{{F}}({U}) $ is contained in $ {U} $, and $ \mathcal{O}^{{F}}({U}) $ is the intersection of all the minimal prime $ {F}- $filters contained in $ {U}. $ The concept of $ {F}- $normal ADL is defined and characterized in terms of the prime $ {F}- $filters and minimal prime $ {F}- $filters, as well as relative annihilators with respect to $ {F} $.
- almost distributive lattice,
- $ {F}- $filter,
- $ {F}- $normal,
- totally ordered set,
- comaximal
Citation: Ali Yahya Hummdi, N. Rafi, M. Balaiah, Y. Monikarchana. Exploring prime $ F- $filters of almost distributive lattices[J]. AIMS Mathematics, 2025, 10(11): 27519-27534. doi: 10.3934/math.20251210

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Abstract

The concept of $ {F}- $filters is introduced in an almost distributive lattice(ADL) and their properties are studied. A set of equivalent conditions is established for every proper $ {F}- $filter of an ADL to become a prime $ {F}- $filter. For any $ {F}- $filter $ {U} $ of an ADL, $ \mathcal{O}^{{F}}({U}) $ is defined, and it is proved that $ \mathcal{O}^{{F}}({U}) $ is an $ {F}- $filter if $ {U} $ is prime. It is also derived that each minimal prime $ {F}- $filter belonging to $ \mathcal{O}^{{F}}({U}) $ is contained in $ {U} $, and $ \mathcal{O}^{{F}}({U}) $ is the intersection of all the minimal prime $ {F}- $filters contained in $ {U}. $ The concept of $ {F}- $normal ADL is defined and characterized in terms of the prime $ {F}- $filters and minimal prime $ {F}- $filters, as well as relative annihilators with respect to $ {F} $.

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