⊕−irreducible elements of almost distributive lattices
Monikarchana Yathaluru285archana@gmail.comDepartment of MathematicsMohan Babu UniversityTirupati, Andhra Pradesh, 517102, India0000-0003-2146-4800View full profile → , Siva Mala Munnangisivamala@vrsiddhartha.ac.inDepartment of Mathematics(Deemed to be University)Siddhartha Academy of Higher EducationVijayawada, Andhra Pradesh, 520007, India0000-0003-2254-4650View full profile → , Rafi Noorbhasharafimaths@gmail.comDepartment of Mathematics(Deemed to be University)Siddhartha Academy of Higher EducationVijayawada, Andhra Pradesh, 520007, India0000-0003-2027-0533View full profile → , *Vasu Babu RajulapatiCorresponding authorvasukgr@svecw.edu.inDepartment of MathematicsShri Vishnu Engineering College for Women (A)Bhimavaram, Andhra Pradesh, 534202, India0000-0002-0443-1276View full profile → , Sulochana Nagamsulochananagam@gmail.comDepartment of MathematicsGovernment College for Men (A)Kadapa, Andhra Pradesh, 516004, India0009-0009-4181-2017View full profile → , T. Satishtsatishmaths@gmail.comDepartment of Engineering Mathematics and HumanitiesSRKR Engineering College (A)Bhimavaram, Andhra Pradesh, 534204, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Jan 2025
- Published Online:
- 21 Mar 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDSGT-2025-06002
- Pages:
- 31–41
Abstract
This study analyzes ADL frameworks by examining ⊕-divisibility alongside the distinct roles of ⊕-prime and ⊕-irreducible elements. A novel congruence relation is introduced via multiplier filters, leading to necessary and sufficient conditions under which the resulting quotient ADL assumes the structure of a Boolean algebra. Furthermore, the study provides a characterization of ⊕−prime and ⊕−irreducible elements through their intrinsic connection to multiplier filters. These theoretical developments have significant interdisciplinary potential, particularly in engineering fields such as digital circuit design, fault-tolerant computing, data encryption, and decision logic systems. The formalism and properties of ⊕−irreducible elements offer robust mathematical tools for enhancing the design and analysis of complex systems where logical structure, redundancy minimization, and computational integrity are paramount.
Keywords
Subject Classifications
References
[1] U. M. Swamy and G. C. Rao, “Almost distributive lattices,” Journal of the Australian Mathematical Society (Series A), vol. 31, pp. 77–91 (1981).
[2] G. C. Rao and M. Sambasiva Rao, “Annulets in almost distributive lattices,” European Journal of Pure and Applied Mathematics, vol. 2, no. 1, pp. 58–72 (2009).
[3] N. Rafi and R. K. Bandaru, “µ−filters of almost distributive lattices,” Chamchuri Journal of Mathematics, vol. 10, pp. 53–65 (2018).
[4] R. Vasu Babu, N. Rafi, R. V. N. Srinivasa Rao, Y. Monikarchana, and S. M. Shaw, “D-divisibility of almost distributive lattices,” Asia Pacific Journal of Mathematics, vol. 11, pp. 24 (2024).
[5] N. Rafi, R. K. Bandaru, and G. C. Rao, “On divisibility of almost distributive lattices,” Bulletin of the International Mathematical Virtual Institute, vol. 5, pp. 171–180 (2015).
[6] A. P. Phaneendra Kumar, M. Sambasiva Rao, and K. Sobhan Babu, “Divisibility and filters in distributive lattices,” Palestine Journal of Mathematics, vol. 11, no. 2, pp. 143–151 (2022).
[7] G. C. Rao, “Almost distributive lattices,” Ph.D. dissertation, Dept. of Mathematics, Andhra University, Visakhapatnam, India (1980).
[8] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ., vol. 25. Providence, RI: American Mathematical Society (1967).
[9] G. Grätzer, General Lattice Theory. New York, NY: Academic Press (1978).
[10] G. C. Rao and S. Ravi Kumar, “Minimal prime ideals in an ADL,” International Journal of Contemporary Sciences, vol. 4, pp. 475–484 (2009).
