joint work with Hee Sik Kim, and Akbar Rezaei, (2025), submitted. 

Abstract. We introduce and study a novel partial order on the complex plane, called the tetra order, motivated by algebraic symmetries and cardinal directions such as identity, negation, duality, and transpose. We show that this order defines a partially ordered groupoid structure on C, satisfies compatibility with addition, and induces posets that decompose into ordinal sums of antichains. We illustrate structural connections between this ordering and finite BCK-algebras.

joint work with Akbar Rezaei, (2024), accepted. Proceedings of the 11th National Mathematics Conference of Payame Noor University.

Abstract. In this note, we consider a triple construction $(\ad;\star,\epsilon(0))$ on a $d$-algebra $(A;\ast,0)$  and investigate some of  their properties. Applying this construction to a $d$-transitive $d$-algebra, we show that $(\ad; <)$ is a poset, which induces a $BCK$-algebra.

joint work with Hee Sik Kim, (2023). Submitted. Preprint.

Abstract. Right feeble groups are defined as groupoids (X,*) such that (i) x, y ∈ X implies the existence of a, b ∈ X such that a*x = y and b*y = x. Furthermore, (ii) if x, y, z ∈ X then there is an element w ∈ X such that x*(y*z) = w*z.  These groupoids have a "remnant" group structure, which includes many other groupoids.  In this paper, we investigate some properties of these groupoids. Enough examples are supplied to support the argument that they form a suitable class for systematic investigation.

joint work with Xiao, J.; Neupane, A. and Sun, W. (2022). Proceedings of the 8th International Conference on Information Systems Security and Privacy - ICISSP, ISBN 978-989-758-553-1, pages 396-403. 

DOI: 10.5220/0010843300003120 

Abstract. Our current key exchange protocols are at risk of failing to keep private data secret due to advancements in technology. Therefore, there is a need to develop an efficient and secure key exchange protocol which can function in the new computing era to come. In this paper, we propose and develop a novel key exchange protocol based on logic algebra for the factorization problem. Both the security analysis and experimentation evaluation demonstrate promising results of our proposed approach. 

(2020) Scientiae Mathematicae Japonicae Online, e-2020-13 and (2021) SCMJ, Vol.84-3, pp.155-180 https://doi.org/10.32219/isms.84.3_155 

Abstract. Let (X, •) be a groupoid (binary algebra) and Bin(X), denote the collection of all groupoids defined on X. We introduce two methods of factorization for this binary system under the binary groupoid product “<>” in the semigroup (Bin(X) , <>). We conclude that a strong non-idempotent groupoid can be represented as a product of its similar- and signature- derived factors. Moreover, we show that a groupoid with the orientation property is a product of its orient- and skew- factors. These unique factorizations can be useful for various applications in other areas of study. Application to algebras such as B/BCH/BCI/BCK/BH/BI/d-algebra are widely given throughout this paper. 

Communications of the Korean Mathematical Society 26 (2011), No. 2, pp. 163-168.

Abstract. In this paper we introduce the notion of the center ZBin(X) in the semigroup Bin(X) of all binary systems on a set X, and show that if (X, •) ∈ ZBin(X), then x ̸= y implies {x, y} = {x • y, y • x}. Moreover, we show that a groupoid (X, •) ∈ ZBin(X) if and only if it is a locally-zero groupoid.