Abstract
The lattice theory and group algebra have several applications in computing sciences as well as physical sciences. The concept of lattice-group structure is an interesting hybrid algebraic structure having potential applications. In this paper, the algebraic construction of lattice-group structure is formulated and associated algebraic properties are established. The proposed construction considers Cartesian product spaces. The concept of two-dimensional monoid is formulated in Cartesian product spaces of real numbers and a related lattice-group structure is established in the space having reduced dimension. The different categories of functions are employed for dimension reduction while establishing the lattice-group structure. The proposed lattice-monoid and lattice-group structures are finite in nature. The algebraic properties of lattice-group as well as associated structures are formulated. A set of numerical examples are presented in the paper to illustrate structural properties. Finally, the comparative analysis of the proposed structure with other contemporary work is included in the paper.
Highlights
Group algebra, especially finite field is the fundamental part of Advanced Encryption Standard (AES)
The construction of variants of the Diffie– Hellman key agreement protocol become easy by using group theory, where the nonAbelian groups can be applied in public key cryptography (Vasco and Steinwandt, 2015; Tzu-Chun, 2018)
The distributive lattice-subgroup can exist under specific condition and, in such case the mapping function can be arbitrary in nature
Summary
Especially finite field is the fundamental part of Advanced Encryption Standard (AES). The combined varieties of function maps considering surjective-homeomorphic as well as odd and even classifications influence the construction of group structure from the monoid in Cartesian product space. X θ such that: Let X be a point set and Gf = (M,*,f) be a group, where ∗: X 2 → X is an abstract algebraic operation and ≤ is a partial order relation in Cartesian product space. Let {(−9, −5),( − 4, − 3),( − 2,4),(0,0)} be the basis set in Cartesian product space (a subset of R2) having partial ordering relation, where the monoid operation is * = +. It satisfies the following latticemonoid property as:. The set given by, {(2,3), (4,5), (6,12), (9,12)}⊂N2⊂R2 does not satisfy the definition of partially ordered monoid
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