Journal of Computer Science Original Research Paper On the Construction and Properties of Lattice-Group Structure in Cartesian Product Spaces Davronbek Malikov and Susmit Bagchi Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju, Republic of Korea Article history Abstract: The lattice theory and group algebra have several applications in Received: 24-12-2019 computing sciences as well as physical sciences. The concept of lattice- Revised: 06-03-2020 group structure is an interesting hybrid algebraic structure having potential Accepted: 3-4-2020 applications. In this paper, the algebraic construction of lattice-group structure is formulated and associated algebraic properties are established. Corresponding Author: Susmit Bagchi The proposed construction considers Cartesian product spaces. The concept Department of Aerospace and of two-dimensional monoid is formulated in Cartesian product spaces of Software Engineering real numbers and a related lattice-group structure is established in the space (Informatics), Gyeongsang having reduced dimension. The different categories of functions are National University, Jinju, employed for dimension reduction while establishing the lattice-group Republic of Korea structure. The proposed lattice-monoid and lattice-group structures are Email: [email protected] 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. Keywords: Lattice, Group, Lattice-Group, Partial Order, Monoid, Invertibility Introduction theory are developed to compute slices for temporal logic formulas (Sen and Garg, 2003). These algorithms Group algebra, especially finite field is the are useful in detecting temporal logic formulas in a fundamental part of Advanced Encryption Standard distributed computation (Mauricio et al ., 1999). In other (AES). The construction of variants of the Diffie– words, partial-order relation and lattices help in obtaining Hellman key agreement protocol become easy by using clear, concise and efficient formulations of problems group theory, where the nonAbelian groups can be requiring the ability to take transitive closures, solve applied in public key cryptography (Vasco and circular constraints and perform aggregate operations. Steinwandt, 2015; Tzu-Chun, 2018). Furthermore, group Lattice theory can be used in the implementation of a theory has applications in physics and particularly in knowledge representation language. For example, the condensed matter physics (Mildred et al ., 2010). knowledge base system is realized by processing a Moreover, the properties of partial order relation as sequence of terminological axioms by using Birkhoff’s well as lattice theory are widely applied to various Representation theorem and finite distributive lattices domains of computer science and distributed systems (Seymour and Marc, 2007; Frank, 2000). including programming languages (Vijay, 2015). The Interestingly, lattice theory plays a role in other applications of the lattice theory in distributed branches of mathematics such as, probability theory computing are comprised of vector clocks design and and graph theory (George, 2009; Louis, 2016). The global predicate detection (Lamport, 1978). The applications of lattice theory with other results lead to a properties of lattice linear predicates enable efficient decomposition technique that expresses all the trees of detection of global predicates in distributed systems a graph in the form of set unions of Cartesian products (Chase and Garg, 1995). The lattice agreement in of the sets of subgraphs of the component graphs asynchronous message passing systems is useful due to (Wen-Hai et al ., 1990). The algebraic relation between its applications in atomic snapshot objects and fault- lattice theory and group is an interesting topic having tolerant replicated state machines (Attiya et al ., 1995; several application possibilities. Thus, the algebraic Xiong et al ., 2018). Computational aspects of lattice interrelationship between lattice and groups needs © 2020 Davronbek Malikov and Susmit Bagchi. This open access article is distributed under a Creative Commons Attribution (CC-BY) 3.0 license. Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421 attention. The construction of hybrid algebraic structure Distributive lattice representation (Seymour and involving lattice theory and group algebra would be an Marc, 2007; Birkhoff, 1967; Laszlo, 2015) interesting topic to investigate. The purpose of this paper is inline to such An algebraic lattice L ∨ ∧),,( is a set with two motivation of formulating algebraic hybrid structures. binary operations meet and join ∧ ∨),( respectively, This paper proposes the construction and algebraic such that both operations are commutative and analysis of lattice-group structure in two dimensional associative, where absorption law holds, as mentioned Cartesian product spaces of real numbers. It is below (where ∀x,y,z∈X): considered that the lattice-group structure is finite in nature. First we introduce the concept of lattice and (a) x∧=∧ y yxx, ∨=∨ y yx , lattice-monoid structures in 2D space. Next, we x∧( yz ∧ )( = xy ∧ ), ∧ z construct the group structure under the influence of (b) (1) ∨ ∨ = ∨ ∨ various types of function mappings. The function x( yz )( xy ), z mappings reduce the dimension while transforming the (c) xx=∧( xy ∨ ) =∨ x ( xy ∧ ). 2D lattice into a lattice-group structure. The associated algebraic properties of the multidimensional lattices, A lattice is a partially ordered set in which for every lattice-monoids and lattice-groups are presented in the two elements a and b the least upper bound (called join, paper. The potential applications of the proposed denoted by, (a∨b)) and the greatest lower bound (called algebraic structure as well as analysis can be made in meet, denoted by, (a∧b)) exist. formulation of model of distributed computing. A lattice is distributive if it satisfies distributive law The rest of paper is organized as follows. Second given by, x∧( yz ∨ )( = xy ∧ )( ∨ xz ∧ ) . section presents preliminary concepts. Third section Interestingly, a lattice can be represented presents a set of definitions intended to the constructed graphically. The 2D representation of finite partially 2 structures. Fourth section presents a set of analytical ordered set X (with an implied upward orientation) properties of the algebraic structures and fifth section is the directed graph whose vertices are the elements presents a set of illustrative examples. The comparative X 2 evaluation with the other contemporary works in the of and also, there is a directed edge from (xa, xb) domain is presented in sixth section. Finally, seventh to (xc, xd) in the Cartesian product space such that, (xa, ≤ X 2 section concludes the paper. xb) (xc, xd) in . The 2D lattice representation is a type of Hasse Preliminaries diagram, which represents the elements of lattice in 2 X . We note that 2D lattice representation of poset In this section, we introduce basic definitions and (X 2 ≤), need not to be connected. properties related to lattice theory, posets and group algebra. Binary Operation, Group and Abelian Group (Scott, Binary Relation and Poset (Seymour and Marc, 1987; Herstein, 1975; Milne, 2013; Robert, 1969) 2007; Thomas, 2004; Dushnik and Miller, 1941; Bernd, 2016) If X ≠ φ and a binary operation * is defined as ∗: X 2 → X then the set is called closed under *. A group ⊂ ⊂ Let X be a point set and, A X and B X be such is given by G = X ∗),( such that, the following properties that, A∩B = φ. A binary relation R is an ordered pair hold: such that, R⊂A×B. For any set A, a subset of the An Cartesian product set is called an n-ary relation on (a) ∀xyzGx,, ∈ ,( ∗∗=∗∗ yz )( xy ) z A, where n∈Z+. Let ℜ be denoting set of real numbers. (b) ∀∈xGeGex, ∃∈ : ∗=∗= xe x (2) A partially ordered set or poset is a set P together ∀∈∃∈xGx−1 Gxx ∗ − 1 = x − 1 ∗= xe with a binary relation ≤ such that, the following (c) , : conditions are satisfied for all x,y,z∈P: A group G = ( X, *) is called Abelian if it satisfies the (a) x ≤ x (Reflexivity), commutative law given by, ∀xy, ∈ Gx , ∗=∗ y y x . (b) If x ≤ y and y ≤ x, then x = y (Anti symmetry), (c) If x ≤ y and y ≤ z, then x ≤ z (Transitivity) Functions and Invertibility (Seymour and Marc, 2007; William, 2013; Walter, 1976) An element x of a poset P is said to be a lower bound A function f: A→ B is one-to-one if af )( = af / )( for the subset S ⊂ P if x ≤ s for every s∈S. The element x / is a greatest lower bound of set S if x is a lower bound of implies a = a . It is said to be an onto function if each S and y ≤ x for any lower bound y of set S. element of B is in the image of some element of A 403 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421 meaning, f(A) = B. A function f : A → B is invertible Remark if the function both one-to-one and onto. Interestingly, there is a close relationship between partial orders and quasi orders. For example, if ≤ is a Definitions 2 partial order on X , then we can define: In this section, a set of definitions are constructed in xx xx xx xx relation to multidimensional lattices, lattice-monoids and ( ab,,) <( cd) ⇔( ab ,,) ≤( cd ) ∧ lattice-group structures. A few of the proposed ()()xxab, ≠ xx cd , . definitions are direct extension of one-dimensional construction into the multidimensional space. 2 Similarly, if < is quasi order on X , then we can Relation, Partial Ordering and Quasi Order in 2D xx≤ xx define ( ab,) ( cd , ) as: Let us consider, {( x , x ), ( x , x ), ( x , x ),… ( x , x ), a b c d e f r q ⊂ X 2 (xn-1, xn)} and if it is true that: xx xx xx xx ( ab,,) ≤( cd) ⇔( ab ,,) <( cd ) ∨ xx xx ()()ab, = cd , .