An Introduction to HP-Algebras: Axiomatic Foundations and Basic Theory

Dr Anjay Kumar Mishra; K.  Perarasan; M.  Vasuki; B. Mohamed  Harif; A. Dinesh  Kumar; Mbonigaba  Celestin

Authors

Dr Anjay Kumar Mishra Dean, Madhesh University, Birgunj, Nepal http://orcid.org/0000-0003-2803-4918
K. Perarasan Annai Vailankanni Arts and Science College (Affiliated to Bharathidasan University), Thanjavur, Tamil Nadu, India
M. Vasuki Srinivasan College of Arts and Science (Affiliated to Bharathidasan University), Perambalur, Tamil Nadu, India
B. Mohamed Harif Government Arts College (Affiliated to Bharathidasan University), Tiruchirappalli, Tamil Nadu, India
A. Dinesh Kumar Khadir Mohideen College (Affiliated to Bharathidasan University), Adirampattinam, Tamil Nadu, India
Mbonigaba Celestin Brainae Institute of Professional Studies, Brainae University, Delaware, United States of America

Keywords:

HP-Algebra, HP-Sub algebra, Algebraic Equivalence, Asymmetry Axiom.

Abstract

This research introduces Algebras, a novel algebraic framework defined by a set and a binary operation that adheres to the principles of Asymmetry, Transitive Difference, and Left Identity/Dominance. The study systematically develops Subalgebras as closed subsets, showing that they uphold all the axioms of Algebras and that their non-empty intersections also constitute subalgebras. This concept leads to the notion of a uniquely smallest generated subalgebra. Moreover, the study introduces Normal Sub algebras, a specific type of sub algebra that exhibits a "Left Absorption Property," meaning that for any element x from the main algebra and n from the normal sub algebra, indicating strong left-invariance. Additionally, Isomorphisms are defined as bijective mappings that preserve operations between Algebras. Theorems demonstrate that these isomorphisms maintain all fundamental axioms and the subalgebra structure. It is also established that the inverse of an Isomorphism is itself an Isomorphism, confirming algebraic equivalence. This foundational research aims to classify and understand the structural relationships within Algebras.

How to cite this article:

Perarasan, M. Vasuki, Mishra A k , B. Mohamed Harif, A. Dinesh Kumar, Celestin M, An Introduction to HP-Algebras: Axiomatic Foundations and Basic Theory, J Adv Res Appl Math Stat, Vol 11, Issue 1&2 2026: Pg. No. 14-25.

DOI : https://doi.org/10.24321/2455.7021.202606

Author Biography

Dr Anjay Kumar Mishra, Dean, Madhesh University, Birgunj, Nepal

Post Doctoral Research Scholar, Srinivas University, India and Associate Professor, Madan Bhandari Memorial Academy Nepal, Urlabari 3, Morang, Nepal.

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