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Journal of Computer Science

Original Research Paper On the Construction and Properties of - Structure in 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 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 is formulated in Cartesian product spaces of Software Engineering real numbers and a related lattice-group structure is established in the (Informatics), Gyeongsang having reduced . 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 , Monoid, Invertibility

Introduction theory are developed to compute slices for temporal logic formulas (Sen and Garg, 2003). These algorithms Group algebra, especially finite 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 and lattices help in obtaining Hellman key agreement protocol become easy by using clear, concise and efficient formulations of problems , where the nonAbelian groups can be requiring the ability to take transitive closures, solve applied in public key (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 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 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 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 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 in which for every lattice- 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) 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 , Group and (Scott, 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 * 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 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 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 ,  .  xx,≤ xx , ≤≤ ... xx , ≤ xx ,  ⇒ ()()ab cd( rq) () nn−1 

 xx≤∧≤ xx,...... xx ≤∧≤ xx  2 ()()()ac bd rn−1 () qn  xx, , xx , ⊂ X Suppose, {( αβ) ( γµ )} . In this case we say xx≤ xx xx≤ xx that, if ( αβ,) ( γµ , ) and ( γµ,) ( αβ , ) then ≤ X 2 then is a relation in . x x x x ( α, β ) and ( γ, µ ) are comparable. If all elements Remark (ordered elements) of X 2 are comparable, then it said to The relation ≤ is a partial order relation on X n ⊂ ℜn be totally ordered or linearly ordered (Seymour and Marc, 2007). if it satisfies following standard properties for n = 2: Lowest Upper Bound and Greatest Lower Bound in ∀ ∈2 ∈≤ (a) ( xxaa,) X ,( xx aa , ) 2D xx xx ⊂≤ (b) If {( ab,) ,( cd , )} then Let in the algebraic structure in 2D be {( xa, xb), ( xc, 2 xx xx xx xx ⊂ X ( ab,,) ≤( cd) ⇔¬( cd ,,) ≤( ab )  xd), ( xe, xf), ( xg, xh) ( xl, xm), ( xn, xp)} and it maintains following condition:

and, (3) xx,,≤ xx ≤ xx , ≤ xx ,, ≤ xx ≤ xx , . ( ab) ( cd) ( ef) ( gh) ( lm) ( np ) xx, ,, xx ,, xx ⊂≤ (c) If {()()ab cd( ef )} then: In this case:

xx,≤ xx , ∧ xx , ≤ xx ,  ⇒ 2 ()()()ab cd cd( ef )  X (a) The supremum of is given by: xx,≤ xx ,  . ()ab() ef  sup{(x , x x ,(), x x ,(), x )} = x ,( x ), ba dc fe hg We write (X 2 ≤), to specify the poset in Cartesian 2 product space. (b) Accordingly, the infimum of X is given by: Suppose > is a relation on a set X 2 ⊂ ℜ2 satisfying x x x x x x = x x following two properties. inf{( g , h l ,(), m n ,(), p )} ,( fe ) .

∀xx ∈¬ X2 xx < xx  (a) ( ab,) ,,( ab) ( ab , )  Meet and Join in 2D (b) If xx, ,, xx ,, xx ⊂< then (4) 2 {()()ab cd( ef )} Let (X ≤), be a poset. If every pair of elements have xx,< xx , ∧ xx , < xx ,  ⇒ sup and inf, then this partial order is called lattice in ()()()ab cd cd( ef )  Cartesian product space and it is denoted by (X 2 L ≤),, . xx< xx  ()ab,() ef , . 2   The meet and join of (X L ≤),, is given respectively as,

2 ∀xx, , xx , ∈ L In this case, the relation < is called quasi-order on X . ( αβ) ( γµ ) :

404 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

xx∧ xx  = xxxx Proof ( αβ,) ( γµ ,)  inf{( αβγµ ,,,) ( )} The associativity and identity preservations in and, algebraic form are given below. Proof of associativity and identity:

xx,∨ xx ,  = sup xxxx ,,, ( αβ) ( γµ)  {( αβγµ) ( )} . xx xx xx  ()(),∗2 , ∗ 2 , ab cd( ef ) 

=xx, ∗ xx , ∗ xx , = xx ,. Any linearly or totally ordered set is a lattice since: ()()ab2 cd  2 ()() ef acebdf

According to the definition: infxx ,,, xx= xx , {( αβ) ( γµ)} ( αβ ) ∗ =∗ =∗∗ ( xxab,) 2( ee ,,) ( ee) 2 ( xx abab ,) ( xexe ,.) and, However, inverses do not exist as: xx xx= xx sup{( αβ ,,,) ( γµ)} ( γµ , ) ∀xx ∈ Xxx2−1 ≠ xx− 1 − 1 ()()ab, ,, ab( ab ,,) x x ≤ x x X 2 L ≤ whenever ( α , β ) ( γ , µ ) in ( ),, . thus Binary Operation in 2D xx∗ xx−1 ≠ ee 2 2 2 ()()()ab,2 ab , ,. Let {(x , xba ,(), xx dc )}⊂ X be such that, X ⊂ ℜ and, ∗ : X 2 → X 2 is an algebraic operation in Cartesian 2 2 Hence, M = ( X , * 2) is indeed a monoid in 2D. product space. If xx ∗ xx ⇒ x x ∈ X 2 [( ba ), 2 ,( dc )] [( α β ), ], x = x ∗ x ∈ X x = x ∗ x ∈ X Induced Group and where α ( a c ) and β ( b d ) then 2 2 Let M = ( X , * 2) be a monoid in Cartesian product ∗2 is a binary operation on X having closure property. space. Let a function f: X 2 →X be defined such that, 2 Associativity and Identity in 2D ∀(xa, xb) ∈ X , ∃xα ∈X where, f(( xa, xb)) = xα then 2 2 = ∗ Gf M f ),,( is called group, if only if, it satisfies Let {( xa, xb), ( xc, xd), ( xe, xf)} ⊂ X where X be an arbitrary point set in Cartesian product space. The standard group axioms under :* X 2 → X . associativity and identity on X 2 is defined as: Remark

xx,∗ xx , ∗ xx ,  We prove associativity of group operation and ()()ab2 cd 2 ( ef )  (a) (5) existence of identity as well as inverse below considering =xx ∗ xx ∗ xx 2 ()()ab,2 cd ,  2 () ef ,, M = ( X , * 2) and its transformation under f (.). 2 2 ∈ (b) ∀(x , x )∈ X ∃ ee ),(, ∈ X , such that, Let (xa, xb), ( xc, xd), ( xe, xf) M be such that: a b x x ),( ∗ ee ),( = ee ),( ∗ x x ),( = x x ),( . ba 2 2 ba ba ,∗ ,  ∗ , fxx(()ab) fxx(() cd)  fxx(( ef )) (a) 2 =fxx, ∗ fxx , ∗ fxx ,,  The M = ( X , * 2) is called a monoid in Cartesian ()()ab()() cd()() ef  2 space. Note that, M = ( X , * 2) is not yet fully equipped ∃∈ ∗ = ∗ (b) (ee,,,) Mfee(( )) fxx(( ab ,)) fxx(( ab ,)) fee(( ,,)) with any lattice structure. Moreover, it is important to ∀ ∈ ∃ note that, in our proposed construction, we consider that: (c) x xba ),( M , f ((x−a , x−b )) such that, ∗ = f (( , xx ba )) f ((x−a, x−b )) f ee )),(( . xx−1 ≠ xx−1 − 1 ()ab,( ab ,) . (6) Thus, the algebraic structure Gf = ( M,*, f) is a group :* X 2 → X Before proceeding further, we present a proof that M under the influence of f (.), where . If Hf ⊂ Gf 2 is a nonempty subset of a group G = ( M, *, f) then, the = ( X , * 2) is indeed a monoid. f structure Hf = ( E, * H, f) is a subgroup of Gf iff Hf = ( E, *H, Proposition 1 f) maintains all group axioms as mentioned earlier, where 2 2 2 E = Y ⊂ X , : 2 → and ∗=∗ . M = ( X , * 2) is a monoid in 2D. ( ) Yf Y H

405 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

Surjective – Homeomorphic Function positive discrete variety in real ℜn, ( n = 1,2) and investigate whether the group structure can be formed Let H be a subgroup of G , such that H < G . The f f f f from the monoid in Cartesian product space successfully function f (.) is called as surjective and homeomorphic in the presence of dimension reducing functional map. (SH) if it satisfies following condition: Accordingly, the set of natural numbers are considered

as the domain and codomain of the mapping function, ∀yy,,, yy ∈⊂∃∈ Y2 X 2 , yY ( ab) ( cd ) α where the domain is a monoid. The resulting behaviour is presented in next theorem. such that, Theorem 1   2 fyy(()()ab,,∗2 yy cd) = fyy(() ab ,) ∗ fyy(() cd , ) ⊆ ⋅   If Gf = ( M N , , f) then Gf is not a multiplicative X 2 2 =[]yα ∗ y α = y2 α ∈ Y , group under surjection f: ( ⊂ N ) → (X⊂N), where, f(( xa, xb)) = xa + xb ∈N. where, yα = f(( ya, yb)) = f(( yc, yd)). Proof Partial Order Monoid in 2D 2 2 2 Let Gf = ( M ⊆ N ,⋅, f), where M = ( X ⊆ N ,⋅2) be a X 2 X 2 monoid in the Cartesian product space under multiplication, Let ( ,≤) be a poset and, M = ( ,*2) be a monoid. 2 2 2 2 2 ⋅2: X → X . Let f:( X ⊂N ) → (X⊂N) be a surjection The structure (X ∗2 ≤),, is called a partial-order- where f(( xa, xb)) = xa + xb ∈N. First, we prove that monoid (p-Monoid) if it satisfies following property, associativity law holds and exists in such settings. ∀xx,,,,, xx xx ⊂ X 2 , {()()ab cd( ef )} If {xα, xa, xb, xβ, xc, xd, xγ, xe, xf} ⊂N such that, xα = xx ≤ xx ⇒ f(( xa, xb)), xβ = f(( xc, xd)), xγ = f(( xe, xf)) then xα .(xβ . ()()ab, cd ,  (7) x ) = ( x . x ) ⋅ x . xx xx xx xx  γ α β γ (),∗2 , ≤() , ∗ 2 , ∧ ab() ef cd() ef  If f(.) is surjection such that, xα = xβ then 2 2 2 xx,∗ xx , ≤ xx , ∗ xx ,  x. x= xx . x. x ≥ 0 ()ef2() ab() ef 2 () cd  α γ γα , where α γ . In this case, the identity element is {1} ⊂N because, Lattice Monoid in 2D xα .1 = 1 . xα = xα. x Nx−1 N 2 However, ∀i ∈, i ∉ , and hence, Gf does not A p-Monoid (X ∗2 ≤),, is called a lattice-monoid (l- 2 Monoid) if and only if it satisfies following conditions: have inverse. Thus, G f = (M ⊆ N ⋅ f ),, is not a . It indicates that, the dimension 2 (a) ( X , ≤) is a lattice having meet and join, reducing additive surjection of discrete variety with ∀xx xx xx ⊂∇∈∧∨ X 2 (b) {(ab , ),( cd , ),( ef , )} , {,}, domain cannot transform a monoid in xx∗ xx ∇ xx = Cartesian product space into a multiplicative group. (ab , )2 [(, cd )(, ef )] (b.1) and, However, if the function is not strictly surjective in [(xx , )∗ ( xx , )][( ∇ xx , ) ∗ ( xx , )], ab2 cd ab 2 ef domain, then it can successfully transform a xx∇ xx ∗ xx = [(,cd )(, ef )]2 ( ab , ) multiplicative group as presented in following example. (b.2) [(xx , )∗ ( xx , )][( ∇ xx , ) ∗ ( xx , )]. cd2 ab ef 2 ab Example: Multiplicative Group under Exponential

2 Map The l-Monoid structure is represented by ( X , L, * 2, ≤). Let there be a real valued function Lattice-Group x+ x = a b fxx(( a, b )) e where ∗ ≡ ⋅ (multiplication). It is Let = ∗ be a group under the influence of f: Gf M f ),,( not considered that f(.) is surjective. 2 2 X →X. If ∀xα, xβ ∈ f( X ), xα ∧ xβ = xα and xα ∨ xβ = xβ then, LG f = ( L, Gf, f, *, ≤) is a lattice- group, where L is a (a) Associativity: xx++ xxxx+ xxxx ++ xx+ lattice on X. eab⋅( ee cd ⋅e f ) =( ee abcd ⋅) ⋅ e e f (b) The existence of identity element: Analytical Properties xx+0 0 xx + xx + eab⋅=⋅ eee ab = e ab In this section, a set of algebraic properties are (c) The existence of inverse: presented in order to gain insight to the lattice-group and xx+ −+()()xxab −+ xx ab xx+ 0 associated structures. First, we consider the strictly eea b ⋅ = e ⋅= ea b e

406 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

−1 In this case Gf = ( M,⋅,f) is a multiplicative group As Hf 0, xb > 0). 2 2 It indicates that H < G is a relation center (R-center) ∗: X → X . Let {(x , x x ,(), x x ,(), x )} ⊂ X and be f f ba dc fe and it can be shifted on real line based on the nature of ∃xα, xβ, xγ ∈X such that xα = f(( xa, xb)), xβ = f(( xc, xd)), group operations. x = f(( x , x )). However, according to associativity law, γ e f Examples xα ∗(xα ∗ xα = () xα ∗ xα )∗ xα as f(.) is a surjection We will present two examples of R-centers of a indicating xα = xβ = xγ. However, if ∗ = + then ∃0∈X such that, group based on the nature of group operations. x ∗ 0 = x = 0∗ x ∀x ∈ X ∃ −x ∈ X α α α . In this case, α (, α ) such Example 1 that, x ∗ −x = −x ∗ x = . Moreover, following the α ( α () α ) α 0 R-center of . 2 commutativity law: Let X ⊆ ℜ and, − − )}4,5(),4,5{( ⊂ R ⊂ X . As Hf < −1 2 G , so ∃ − − )}5,4(),5,4{( ⊂ R ⊂ X . According to ∀xx, ∈ Xx , ∗=∗ x x x . f αβ αβ βα surjectivity, f((5,4)) = f((4,5)) = 9 and − − = − − = − 2 2 f (( ))4,5 f (( ))5,4 9 . However, f((5,4)) + f((-5, Hence, Gf = ( X ⊆ ℜ , +, f) is an Abelian additive 2 -4)) = f((0,0)) = 0 and thus {9, 0, -9} = H , which is a R- group under surjection f: ℜ →ℜ in Cartesian product space. f 2 2 center of group Gf = ( M, +, f). Similarly, if ∗ = ⋅ , then Gf = ( X ⊆ ℜ ,⋅, f) is also an 2 Abelian multiplicative group under surjection f: ℜ →ℜ Example 2 in Cartesian product space, where identity element is 1. 2 2 Thus, the algebraic structure Gf = ( X ⊆ ℜ ,*, f) is an R-center of multiplicative group. −1 −1 2 Abelian group. We will prove the remaining part of the Let − −3,2(),3,2{( )} ⊂ R ⊂ ℜ . As Hf < Gf, theorem in two parts considering two different group −1 −1 −1 2 hence ∃ − −2,3(),2,3{( )}⊂ R ⊂ ℜ . operations as given below. Following the surjectivity, f((2,3)) = f((3,2)) = 6 ∈ℜ+ +≡∗ −1 −1 −1 −1 −1 + Case 1: 2 (Additive Group) and f ((− −3,2 )) = f ((− −2,3 )) = 6 ∈ℜ . So, it 2 2 −1 −1 Let M = ( X ⊆ ℜ , +2) be a monoid in Cartesian product is true that, f ))3,2(( ⋅ f ((− −3,2 )) = f ))1,1(( . Thus, −1 + space where x x ),( + x ,( x ) = (x + x , x + x ) . Let, Gf H = ⊂ ℜ ba 2 dc a bc d f {6,0,6 } is a R-center in positive real of ⊂ X 2 = ( M, +, f) be a group and Hf = ( E , +, f) be a subgroup Abelian group, Gf = ( M,⋅,f). Hf < Gf where f: E → (Y⊂X) is a surjection. If a relation However, if the function f(.) is surjective and 2 −1 R⊂ X , then ∀ xa ,( xb )∈ R ∃ xb ,(, xa )∈ R such that, homeomorphic (SH variety), then a cyclic subgroup can X 2 f ((x , x )) = f ((x , x )) = x + x . be constructed considering a group Gf = ( ,*, f) under ba ab a b the abstract algebraic operation having closure in point x x xx ⊂ R ∪ R−1 ∈ Let {( , ba ,(), ab )} ( ) be such that, xα Hf set X. The cyclic subgroup generation requires subset of where f(( xa, xb)) = f(( xb, xa)) = xα. integer set spanning negative as well as positive ranges

407 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

2 symmetrically. However, the identity in 2D should also However, let ∃{a1, a2, a3…, an}⊂ X , be such that be mapped through the surjective and homeomorphic (y1, y2) = a1, ( y3, y4) = a2,…. ( yn-1, yn) = an, then: function. This property is presented in next theorem. faaa∗∗.... ∗= a faafa ∗∗ ∗∗ .... a . Theorem 3 ( 12223 2n ) ( 122) ( 32 2 n )

If f(.) is surjective and homeomorphic to the However, 2 abstract point set X, then Hf = ( E⊂ X ,*, f) is a cyclic ∗ ∗= ∗∗ ∗∗ group, where Hf

Let X be a point set and Gf = ( M,*, f) be a group under ∗∗.... ∗ =∈ faaa( 12223 2 an) y n β X abstract algebraic operation ∗: X 2 → X and H = f ⊂ X 2 ∃ ⊂ ∃ ⊂ (E ,*, f) be a subgroup Hf

1  and, fyy//,= fyy // , == ...  ∈ X . (10) ()()12()() 34   yβ  = ∗ fyy(()acbd,) fyy(() ab ,) fyy(() cd , ) (8) = ∗ = ∈ yα y α y2 α X . It indicates that,

It is clear that, according to the surjectivity one can 1  //,∗ // ,... ∗ // , =∈  . consider, fyy(()()()12234 yy 2 yyn− 1 n   X ynβ  ,= , = , fyy(()ab) fyy(() cd) fyy(( ef )) X 2 =y ∈ X . Let ∃(e,e)∈ , be such that f(( e,e)) = e∈X and, α ∗−1 = ynβ y n β e . Hence, Hf < Gf is a cyclic group if Hf = 2 i However, (E⊂ X ,*, f) and f(E\{( e,e)}) = {( ynβ) : n∈V⊂Z, i∈{1,- 1}}, where the identity is, (e,e)∈E. ,∗ , = ∗∗ , fyy(()cd2 ( yy ef)) fyyyy(( cedf )) The nature of function mapping affects the resulting

= ∗ group structure. For example, an arbitrary function map fyy()()cd, fyy()() cd , may not produce a valid group structure. This property is

presented in next theorem. It is important to note that, we in the algebraic structure under consideration. This have not used any specific definition of function and, we indicates, have used generalized classification of two function varieties while constructing the theorem. ,= ∗ = ∈ , fyy(( ce df )) yyyα α2 α X Theorem 4 = ∗ = ∗ where, yce y c y e and ydf y d y f . 2 2 If fe: X →X is an even function and fo: X →X is an Moreover, odd function then, algebraic structure G = ( M,+, f ) is an fe e additive group and Gfo = ( M,+, fo) is not a group in ∗ 2 2 fyy(()ac, bd) fyy ce , df X (( )) Cartesian product space, where M = ( ⊂ℜ ,+ 2). = ∗ = ∈ . y2α y 2 α y 4 α X Proof

2 2 Following the property of associativity, Let M = ( X ⊂ℜ ,+ 2) be a monoid and Gf = ( M,+, f) be an algebraic structure in Cartesian product space. fyy,∗ yy , ∗ yy , + ((()()ab2 cd) 2 ( ef )) Let ∃A⊂X such that A = {xαi = i ,2 in ∈ Z , ni ∈ Z} where = ∗ ∗∗ f : X 2 →A is an even function. Let ∃B⊂X be such that fyyyy()acbd,2 () yy ef , (9) e () 2 fo: X →B is an odd function where B = { xβi = 2 ni +1, = ∗ =∗=∈ + fyy()()ac, bd fyy() e , f y2α yy α 3 α X . () i∈Z , ni∈Z}.

408 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

2 2 2 2 Evidently, fe( X )∩ fo( X ) = φ, fe( X )∪ fo( X )⊆X. Following the property of associativity under Let the algebraic structures generated by fe(.) and fo(.) surjective-homeomorphic fe (.) , = + = + be given by, Gfe( M, , f e ) and Gfo( M, , f o ) ,⋅ , ⋅ , fe((()() yy ab2 yy cd) 2 ( yy ef )) 2,2n n∈ A , 2n+ n ∈ A . respectively. If i j then ( i j ) Let 3 (11) =, ⋅ ,2 =∈⊂ℜ . fyye()() acbd fyy e()() ef() n i X 2nk ∈ A and by the property of associativity:

2 2 As M=( X , ⋅ 2 ) is a monoid, hence, ∃(1,1) ∈ X in 2n+ 22 nn + = 22 nn + + 2. n i( jk) ( ij) k M= X 2 ⋅ = ∈ ( ,2 ). If Gf is a group then, fe ((1,1)) 1 X such that the following condition is maintained within Moreover, ∃f((0,0)) = 0 ∈A, such that 2ni +0 = 0 + the structure: 2ni, 2 ni∈A. nA∈ ∃− n ∈ A ∀∈yy, Xfyy2 , , ⋅= f 1,1 fyy , Given, 2i ,2( i ) , such that, 2ni + (-2ni) = ( ab) eabe(( )) (( )) eab(( ))

= ⋅ (-2ni) + 2 ni = 0 ∈A. fe()()1,1 fyy eab()() , = + Hence, Gfe( M, , f e ) is an additive and indicating the existence of identity in 2D. commutative group under the influence of fe (.) . However, this leads to a contradiction because n+ n + ∈ B 2 Again, let 2i 1,2 j 1 and hence fe ((1,1))≠ 1 . Thus, it is true that, ∀(,)ya y b ∈ X , ⋅ ≠ 21212n++ n += nn ++∉ 1. B fyyeab(( , )) f e ((1,1)) fyy eab (( , )). ( i) ( j) ( ij ) It indicates that fo(.) =, ⋅ , G G Hence, Gfe( M f e ) is not a group. is not closed under addition in fo . Hence, fo is not a On the other hand, if fo (.) is a surjective and group under the influence of fo(.). Interestingly, the combined varieties of function maps considering homeomorphic function then: surjective-homeomorphic as well as odd and even ⋅ = classifications influence the construction of group fyyo(( ab , )2 ( yy cd , )) fyy o (( acbd , )), structure from the monoid in Cartesian product space. We show that, none of the hybrid varieties of functions and furthermore: successfully constructs a multiplicative group from the = ⋅ monoid structure in Cartesian product space. fyyo(( acbd,)) fyy o(( ab ,)) fyy o(( cd , ))

=2 + + Theorem 5 2() 2ni 2 n i 1,

If fo(.) and fe(.) are surjective-homeomorphic, then where yac= y a ⋅ y c and ybd= y b ⋅ y d . algebraic structures Gfe = ( M,⋅,fe) and Gfo = ( M,⋅,fo) can not 2 Following the property of associativity under be groups under multiplication, where fe : X→ X is even surjective-homeomorphic function f (.) : function and f: X2 → X is odd function and X ⊂ ℜ . o o ,⋅ , ⋅ , Proof fo((()() yy ab2 yy cd) 2 ( yy ef )) : 2 → =, ⋅ , Let fe X X be even function such that fo()() yy ac bd f o()() yy e f (12) 2 = ∈ ⊂ℜ 3 2 ∀(ya,yb)∈X , fyyeab(( ,)) 2 nX i where ni∈Z. Let = + + +∈⊂ℜ 24631()nnni i i X . 2 2 fo: X →X be odd function and ∀(ya,yb)∈ X , = +∈ ⊂ℜ n∈ Z 2 2 fyyoab(( , )) 2 n i 1 X , where i . As M=( X , ⋅ 2 ) is a monoid, so ∃(1,1) ∈ X in

Let two algebraic structures be, Gfe = ( M,⋅,fe) and Gfo 2 2 M=( X , ⋅ 2 ) and, ∀(ya , y b ) ∈ X : = ( M,⋅,f ), where the monoid in Cartesian product space o M= X 2 ⊂ℜ⋅ 2 is given by, ( , 2 ) . ⋅ fo(( yy ab,)) f o (( 1,1 )) (13) Let in the algebraic structure {( ya, yb), ( yc, yd), ( ye, yf)} = ⋅ = 2 fo()()1,1 fyy oab()() , fyy oab()() , . ⊂ X and fe(.) be a SH variety such that 2 2 (,)(,)(xx⋅ xx =⋅ xxxx , ⋅ ) in M=( X ⊂ℜ⋅ , ). This ab2 cd acbd 2 Hence, f((1,1))= 1 ∈ X , which satisfies the required ⋅ = =∈2 o indicates, fyyeab(( , )2 ( yy cd , )) fyy eacbd (( , )) 4 nX i , G condition for odd function. Furthermore, if fo is a where y= y ⋅ y and ybd= y b ⋅ y d . ac a c group, then given:

409 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

fyyoab,= 21 n i + ∈ X ⊂ℜ , xx + xx ∨ xx + xx  (( )) ( ) ()()()ab,2 cd ,  ab , 2 ( ef , )   ∃/ / = +∈⊂ℜ−1 fyyoab,() 21 n i X , =+xxxx, +∨ xxxx + , +  ()() ()acbd  () aebf  (17)

=x + xx, + x  / /∈ 2 ()a eb f  where ( ya, y b ) X such that,

Thus, the (16) and (17) lead to: ⋅ / / fo(() yy ab,) f o(( yy ab , )) (14) / / xx xx xx  = ⋅ = ()(),+2 , ∨ , fyyoab, fyy oab() , f o () 1,1 . ab cd( ef )  ()() () () =+xxxx, +∨ xxxx + , +  . ()acbd  () aebf  However, as, fo (.) is odd function, hence / / ≠ + −1 x x ∈ fyyoab(( ,)) () 21. n i Again, in another case let us consider that ( a, b) 2 x ∈ℜ\ ℜ + x ∈ℜ + X such that a and b and {( xc, xd), ∀/ / ∈ 2 (x , x )} ⊂ (ℜ+)2. By following the lattice-monoid Thus,()yyab , ,( yy ab ,) X , e f (15) definition one can conclude that: ⋅/ / ≠ fyyoab()(), fyy oab()() , f o ()() 1,1 . xx,+ xx ,, ∨ xx  =++ xxxx , ()()ab2  cd( ef)  ( aebf ) = ⋅ Hence, Gfo( M, , f o ) is not a group. The algebraic structure can be further enriched by and, incorporating the partial ordering relation in the xx, + xx , ∨ xx , + xx ,  Cartesian product space. We introduce the lattice ()()()ab2 cd   ab 2 ( ef )  (18) structure into the partially ordered monoid to form a =x + xx, + x hybrid lattice-monoid structure in 2D. In the next ()a eb f theorem we illustrate that the 2-dimensional lattice- monoid is easy to formulate considering whole real line Hence,

ℜ generating combinatorial forms of ordered pairs in xx,+ xx , ∨ xx ,  Cartesian product space. ()()ab2  cd( ef ) 

=xx, + xx , ∨ xx , + xx ,  Theorem 6 ()()()ab2 cd   ab 2 () ef  2 ∗ ≡ + Let ( X ,* 2, ≤) be a partial order monoid. If , + + x ∈ℜ\ ℜ , x ∈ℜ {(xx , ),( xx , )}⊂ ( ℜ + ) 2 X2 L + ≤ if a b and cd ef . then, there exists lattice-monoid ( , ,2 , ) in x ∈ℜ + , x ∈ℜ\ ℜ + x x x +: X2 → X 2 Likewise, if a b and {( c, d), ( e, Cartesian product space, where 2 and + 2 2 x )} ⊂(ℜ ) then also the lattice-monoid structural +:X → X . f property is maintained. Hence, there exists lattice- 2 Proof monoid (X ,, L +2 ,) ≤ in Cartesian product space in all Let (X 2 ,∗ ,) ≤ be a partial order monoid in Cartesian cases. However, if the lattice-monoid is a multiplicative 2 variety and the partially ordered elements are not strictly ≤ product space, where is a partially ordered relation in in positive real , then the lattice-monoid structure 2D. First, we will consider the case where elements are fails to exist. The observation is presented in next lemma. + x x x x negative reals as, x ,{ xba } ⊂ ℜ \ ℜ and {( c, d), ( e, f)} ⊂ ℜ+ 2 Lemma 6.1 ( ) . If ∗ = + 2 then by following the definition of 2 lattice-monoid, one can derive as follows: In (X 2 ≤⋅ ),, partially ordered monoid, if xx xx xx⊂ X2 ∃ xx ∈ℜ 2 ℜ + 2 {(,abcdef ),(, ),(, )} ,(, ab ) \( ), then 2 xx xx xx⊂ XL ≤ 2 ()()ab, ,, cd ,, ef (,,), { ( )} (X ,,,) L ⋅2 ≤ does not exist in Cartesian product space.   ()()xx,+2 xx , ∨ xx , =() xx , + 2 xx ,, (16) ab cd() ef  ab() ef Proof xx+ xx =+ xxxx + ()ab,2 ()() ef , aebf , 2 Let (X ,,)⋅2 ≤ be a partially order monoid in 2D,

2 2 2 2 2 X ⊂ ℜ ⋅: X → X Furthermore, in (X L ≤),, the following equality where and 2 be such that xx⋅ xx =⋅ xxxx ⋅ ≤ holds: (,)(,)(ab2 cd acbd , ). Let be a partial order

410 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

+ relation in 2D. Let {,}xa x b ⊂ ℜ \ ℜ and, let whether the lattice-monoid can retain the distributive {(xx , ),( xx , )}⊂ ( ℜ + ) 2 . property and, if so then what the required condition is. cd ef This property is analyzed in next theorem. We show that, According to the definition of lattice-monoid, the every lattice-monoid is distributive in nature, if exists. following relation is established considering xx≤ xx (,cd ) (, ef ) : Theorem 7 Every lattice-monoid (X2 ,,,) L ∗ ≤ is distributive in xx,⋅ xx , ∨ xx ,  = xx , ()()ab2  cd( ef)  ( aebf ) (19) Cartesian product space, if exists. Proof Moreover, the following algebraic relation can be X X2 L ∗ ≤ established: Let be a point set and ( ,,2 ,) be a lattice- 2 2 monoid in Cartesian product space, where ∗2 : X → X xx,, ⋅ xx ∨ xx ,, ⋅ xx  = xx , 2 ≤ ()()()ab2 cd   ab 2 ( ef)  () acbd (20) and ∗: X → X are abstract algebraic operations and is a partial order relation in 2D. x=⋅ xxx =⋅ xxx =⋅ xx x= x ⋅ x xx xx xx where, ac a cae, a ebd , b d and bf b f . Let us consider the elements as, {(ab , ),( cd , ),( ef , )} 2 2 Hence, in Cartesian product space: ⊂ X . As (X ,, L ∗2 ,) ≤ is a lattice-monoid, hence 2 ∀(xxab , ),( xx cd , ) ∈ X , there exists in 2D. xx,⋅ xx , ∨ xx ,  ()()ab2  cd( ef )  However, as, ≤ is a relation in 2D, hence:

≠xx, ⋅ xx , ∨ xx , ⋅ xx ,.  ()()()ab2 cd   ab 2 () ef  ( xxab,) ≤( xx cd ,) ⇒≤∧≤( xx ac) ( xx bd ) .    2 + However, let (xa , x b ) ∈ X be such that xa ∈ℜ\ ℜ , Moreover, following the distributive law it can be x ∈ℜ + and {(xx , ),( xx , )}⊂ ( ℜ + ) 2 . According to the b cd ef derived as: definition of lattice-monoid as mentioned earlier: xx,∧ xx , ∨ xx ,  = xx , ()()ab cd( ef)  () ab (24) xx,⋅ xx , ∨ xx ,  = xx , ()()ab2  cd( ef)  ( aebf ) (21) Furthermore, the following is maintained in Again, it can be verified that, the lattice-monoid under partial order,

xx, ⋅ xx , ∨ xx , ⋅ xx ,  xx,, ∧ xx ∨ xx ,, ∧ xx  = xx , (25) ()()()ab2 cd   ab 2 ( ef )  ()()()ab cd   ab( ef)  () ab (22) =xx, ∨ xx ,  ()ac bd() ae bf  Hence, it is verified that:

However, [( xac , xbd ) ∨ (xae , xbf )] ≠ (xae , xbf ) since + xx,∧ xx , ∨ xx ,  = xx ,. x x x x ⊄≤ x ∈ ℜ , ()()ab cd( ef)  () ab {( ac , bd (), ae , bf )} . Furthermore, let a + x ∈ℜ\ ℜ be in X. According to the lattice–monoid b However, one can verify that definition, holds under lattice join operation as given below:

xx,⋅ xx , ∨ xx ,  = xx , and, ()()ab2  cd( ef)  ( aebf ) xx,∨ xx , ∧ xx ,  = xx ,, ()()ab cd( ef)  () cd xx, ⋅ xx , ∨ xx , ⋅ xx ,  (23) ()()()ab2 cd   ab 2 () ef  =xx, ∨ xx ,  and, ()ac bd() ae bf 

xx, ∨ xx , ∧ xx , ∨ xx ,  = xx ,. xx∨ xx ()()()ab cd   ab( ef)  () cd Similarly, in this case also, [(ac , bd ) ( ae , bf )] ≠ x x X 2 ⋅ ≤ (ae , bf ) . Hence, ( ,,)2 is not a lattice-monoid in Hence, every lattice-monoid (X2 ,, L ∗ ,) ≤ is Cartesian product space under multiplication operation 2 + 2 distributive in Cartesian product space. except the case, where (,)(x x ∈ ℜ ) and, a b There is an interplay between lattice-group, arbitrary xx xx ⊂ ℜ + 2 {(cd , ),( ef , )} ( ) . The interesting question is function mapping and distributive property. It is possible

411 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421 to construct a lattice-subgroup considering an arbitrary requirement of . Moreover, according function mapping having domain in 2D. We illustrate the to the distributive law: interaction between lattice-group and an arbitrary ∧ ∨ = ∧ ∨ ∧ function in the next theorem. yα( yy αα) ( yy αα) ( yy αα ). Theorem 8 ∈ Similarly for yβ, y γ f (.) the distributive law holds. G=( M ,, ∗ f ) ⊂2 → ⊂ In f if fYX:( ) ( YX ) is any = ∗ It indicates that LHf(, L H f ,,) f is a distributive arbitrary function, then LH=(, L H ,,) f ∗ is a f f lattice-subgroup where f(.) is surjective. distributive lattice-subgroup, where H< G . f f Case 2: f(.) is Injective

Proof 2 Let fYX:(⊂ ) → ( YX ⊂ ) be injective such that We will prove this theorem in three parts considering ⊂ 2 ≠ ≠ {,,}yyyα β γ fY () and, yα y β y γ where, {yα , y β , y γ } variations of mapping such as surjective, injective and is a subset of Gf. According to the associativity property bijective f(.). 2 of groups with respect to algebraic operation ∗: X → X : Let G=( M ,, ∗ f ) be a group under the influence of f = ∗ H< G f (.) . Let Hf ( E ,, f ) be a subgroup ( f f ) where, ∗ ∗ = ∗ ∗ yα( yy βγ )( yy αβ ). y γ E= Y2 ⊂ X 2 and fYX:(⊂ )2 → ( YX ⊂ ) be any arbitrary function. In the monoid in Cartesian product space, M= X2 ∗∃ eeX ∈ 2 Case 1: f(.) is Surjective ( ,2 ), ( , ) such that: = ∗ As, Gf ( M ,, f ) is a group, hence it is true that, fyy,= fee , ∗ fyy , ∀ ∈2 ⊂ ∃ ∈ (( 12)) (( )) (( 12 )) (,),(,yyab yy cd ),(, yy ef ) YY , X , yα, y β , y γ Y , where =, ∗ , = = = f()() yy1 2 fee()() fyy((ab , )) yfyyα , (( cd , )) yfyy β , (( ef , )) y γ and

{y , y , y } is a subset of G . α β γ f and, 2 However, if fYX:(⊂ ) → ( YX ⊂ ) is surjective then: ∀ ∈∃/ / ∈ fyy(()12,) Yfyy ,(( 12 , )) Y fyy,= fyy , = fyy ,. = y (()ab) (() cd) (( ef )) α such that, Thus, by following the group axioms it can be concluded that: ∗/ / = fyy(()12,) fyy(( 12 ,)) fee(() , )

∗ ∗ = ∗ ∗ yα( yy αα) ( yy αα) y α . indicating Gf is a group under f (.). / / Hence, if f(( y1 , y 2 )) is in Gf then, However, as M is a monoid so, ∃(,)ee ∈ Y2 , ∃∈ e X , fyy(( , ))≠ fyy ((/ , / )) maintaining injectivity of f (.). such that f(( ee , )) = e and: 12 12 Hence, if H=( EY ⊂ ,,) ∗ f where, the algebraic f operation * is closed in E and Hf maintains other group yα∗=∗ eey α = y α . axioms including f(( ee , ))∈ E , then H=( E ,, ∗ f ) is a f = ∗ H< G ⊂ ∀ ∈ subgroup of Gf ( M ,, f ) . Furthermore, according to Hence, f f if {e } H f and fyy((1 , 2 )) Y , ∧ ∨ = ∧ ∨ ∧ / / / / the distributive law, yα( yy βγ )( yy αβ )( yy αγ ) ∃fyy((1 , 2 )) ∈ Y such that, f(( y1 , y 2 )) ∗ f(( y1 , y 2 )) ≤ ≤ = because yα y β y γ . f(( e , e )) . Moreover, if f(( y1 , y 2 )) is in Hf then it / / = ∗ Thus, LHf(, L H f ,,) f is a distributive lattice- indicates that f(( y1 , y 2 )) is also in Hf. This is to note that, we are not enforcing any surjectivity subgroup where f (.) is injective. = under relational symmetry fyy((12 , )) fyy (( 21 , )) Case 3: f(.) is Bijective maintaining generality. = ∗ = ∗ Let fYX:(⊂ )2 → ( YX ⊂ ) be a bijective map and, Hence, Hf ( E ,, f ) is a subgroup of Gf ( M ,, f ) G=( M ,, ∗ f ) E⊂ Y under the influence of surjective map without the f be a group. If such that,

412 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

∀ ∈ ∃ ∈ ∗ x≥ x ≥ x fyy((1 , 2 )) E , f(( ee , )) E where, fyy((1 , 2 )) fee (( , )) α β γ due to monotonically decreasing function, = E ∗ ∧ ≠ ∨ ≠ f(( y1 , y 2 )) then, ( , ) is a monoid in 1D. Moreover, hence xα x β x α and xα x β x β . ∀ ∈ ∃/ / ∈ = ∗ ≤ fyy((1 , 2 )) E , fyy((1 , 2 )) E such that: Thus, it indicates that LGf(, L G f ,,,) f is not a lattice-group under the influence of f(.), where f(.) is ∗ / / monotonically decreasing. f(() yy12,) fyy(( 12 , )) The determination of homeomorphism in between =/ / ∗ = fyy()()12, fyy()() 12 , fee()() , two lattice-groups is important in order to determine similarities in structures. However, the lattice property and, H< G inherent to the lattice-groups and the multilevel f f mappings between spaces having different lead to difficulty in determining homeomorphism = ∗ where, Hf ( E ,, f ) considering * is closed in E. Again, as between lattice-groups. The next theorem illustrates ∀ ∈ ≠ . f (.) is bijective, hence, yα, y β E , yα y β Furthermore, that, the pair of mappings in two different dimensions by following the distributive lattice property if y≤ y ≤ y helps in determining the homeomorphism between two α β γ lattice-groups. ∧ ∨ = ∧ ∨ ∧ then, yα( yy βγ )( yy αβ )( yy αγ ) . Theorem 10 Hence, Hf < Gf and LH f is a distributive lattice- subgroup. Thus, = ∗ is a distributive = ∗ ≤ = ◊ ≤ LH f ,( HL f f ),, Let LGf1(, L 1 G f 11 ,,,) f and LGf2(, L 2 G f 22 , f ,,) : → lattice-subgroup independent of the nature of function be two lattice-groups. If g LGf1 LG f 2 is a mapping. This indicates that, formation of distributive 2 2 and hfX:()→ fX () , then the pair of mappings lattice-subgroup structure is possible irrespective of 1 2 <, > LG LG invertibility of the mapping from Cartesian product g h is a homeomorphism between f 1 and f 2 space to 1D. However, the characteristics of such iff h(.) is a bijection and group homeomorphism. mapping are important to maintain the resulting = ∗ Proof lattice-group structure, LGf(, L G f ,,) f . = ∗ ≤ The monotonicity of the function plays an important Let X be a point set and LGf1(, L 1 G f 11 ,,,) f and role in establishing the lattice-group structure as = ◊ ≤ LGf2(, L 2 G f 22 , f ,,) be two lattice-groups, where indicated in next theorem. 2 2 ∗: X → X and ◊: X → X are abstract algebraic Theorem 9 operations and ≤ is a partial order relation in Cartesian product space. If f(.) is monotonically decreasing function in group ∀ x x = ∗ = ∗ ≤ Let in the algebraic structure be, {(a , b ), Gf ( M ,, f ) , then LGf(, L G f ,,,) f is not a lattice- (x , x )}⊂ X 2 , ∃xxxx, , , ∈ X such that: group. c d α β η θ

Proof = = fxx1(( ab,)) xfxxα ,, 1 (( cd )) x β , (27) Let X be a point set and Gf = ( M,*, f) be a group, = = fxx2()()ab, xfxxη ,, 2 ()() cd x θ . where ∗: X 2 → X is an abstract algebraic operation and ≤ is a partial order relation in Cartesian product 2 As G and G are groups in 1D, hence if space. Let f: X →X be a monotonically decreasing f 1 f 2 2→ 2 function such that: hfX:()1 fX 2 () is a bijection and group 2 2 homeomorphism such that, 1 (Xf ) ∩ f 2 (X ) = φ then in 2 hx∗ x = hx ◊ hx ∀,,,,, ⊂ , this case, (αβ ) ()(). α β It indicates that: {()()xxab xx cd( xx ef )} X (26) ,≥ , ≥ , fxx()()ab fxx()() cd fxx()() ef ∗ hf( 1(() xxab,) f 1 (() xx cd , )) (28) = ◊ xx≤ xx ≤ xx hf1() xxab, hf 1 () xx cd , . where, (ab , ) ( cd , ) ( ef , ). ()() ()()

: → However, according to the lattice-group definition, Moreover, let g LGf1 LG f 2 be a bijection such that: ∀ ∈ 2 ∧ = x∨ x = x xxα, β fX ( ), xα x β x α and α β β , where = = =  = fxx((a , b )) x α and, fxx((c , d )) x β . In this case, as ( fgxx2)(( ab,)) xfgxxη ,( 2 )(( cd ,)) x θ ,

413 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

where, ° is the composition operation. iff hx(1∗ x 2 ) = x ∂ indicating x3◊ x 4 = x ∂ in the lattice- 2→ 2 ∃h−1 LG h x= x h x= x As hfX:()1 fX 2 () is a bijection hence, group structure f 2 , where (1 ) 3 and, (2 ) 4 . hxx−1◊ = hx − 1 ∗ hx − 1 such that, (ηθ ) () η (). θ This leads to the following conclusion:

Moreover, if (,)xxab≤ (, xx cd ) then f1(( xa , x b )) ∗ ≤ f(( x , x )) and, gxx(( , ))≤ gxx (( , )). Furthermore, hf( 1(() xxab,) f 1 (() xx cd , )) 1 c d ab cd (30) ≤  = ◊  (fgxx2 )((ab , )) ( fgxx 2 )(( cd , )) maintaining lattice- ()()fgxx2()ab,()() fgxx 2 () cd , . LG LG group properties of both f 1 and f 2 . Hence, according to the lattice-group definition, Moreover, as ≤ is a partial order relation maintaining 2 ∀∈ ∧= ∨= lattice-group properties of both LG f1 and LG f2, hence xxηθ, gXx ( ), ηθηηθθ x xx , x x because, both ∀ ∈ 2 LG LG x3, x 4 gX ( ), it satisfies lattice-groups properties, f 1 and f 2 are lattice-groups. x∧= x xx, ∨= x x . Thus, in this case is a Thus, if we denote as a pair of mappings 3 4 33 4 4 homeomorphism between two lattice-groups in Cartesian between the structures, LG and LG satisfying all the f 1 f 2 product space. above mentioned properties in the structures, then represents the homeomorphism between two Numerical Examples lattice-groups in Cartesian product space. In this section, a set of numerical examples are given Remark to illustrate the construction of lattices and their We show that, the homeomorphism orientations in spaces. Moreover, the formations of lattice-monoid and lattice-group structures are presented between two lattice-groups can also be formulated in a through numerical computational examples as well as more uniform expression without involving ∃h−1 associated graph representations following the proposed between the two spaces. However, we derive such algebraic structures. expression assuming a conditional homeomorphic + 2 mapping between two spaces. Oriented Lattice in (ℜ ) Let LG=(, L G ,,,) f ∗ ≤ and LG=(, L G , f ,,) ◊ ≤ f1 1 f 11 f2 2 f 22 Let ⊂ X 2 where 2 2 {(1,2),(2,3),(3,4),(3,6),(3,9),(5,12)} , be two lattice-groups and ∗: X → X and ◊: X → X are 2+ 2 X ⊂( ℜ ) . The structure of oriented lattice in positive abstract algebraic operations and ≤ is a partial order real Cartesian product space is given in Fig. 1. relation in Cartesian product space. Let xxxx, , , ∈ X 1 2 3 4 It is evident in Fig. 1 that, the lower bound (LB) = be such that: {(1,2),(2,3),(3,4)} and upper bound (UB) =

2 {(3,6),(3,9),(5,12)}. Moreover, in the lattice structure, ∀xx,,, xx ⊂ X , 2 2 {( ab) ( cd )} inf( X ) = (3,4) and sup( X ) = (3,6). = = fxx1()()ab, xfxx 11 ,,()() cd x 2 , (29) It can be observed from the oriented lattice structure + 2 = = that, it is monotonically increasing in (ℜ ) space. fxx2()()ab, xfxx 32 ,,()() cd x 4 . Oriented Non-Lattice in (ℜ+)2 If hfX:()2→ fX () 2 is a bijection and group 1 2 In this section, we present an example of non- 2 ∩ 2 = homeomorphism such that, 1 (Xf ) f 2 (X ) φ then lattice in positive real Cartesian product space. Let hx∗ x = hx ◊ hx G G 2 (12 ) ()() 1 2 because f 1 and f 2 are groups {(2,4),(4,3),(6,4),(4,4) ⊂} X be the set of points in : → + 2 in 1D. Furthermore, let g LGf1 LG f 2 be a bijection (ℜ ) . The oriented but non-lattice structure is such that: presented in Fig. 2. It is clear from Fig. 2 that, it is following the definition of the partial order relation ≤ =  = ( fgxx2)(( ab,)) xfgxx 32 ,( )(( cd ,)) x 4 . because, every pair of lattice elements satisfies following required condition: 2 If ∃x∂ ∈ f2 ( X ) such that: xx ≤ xx ⇒≤∧≤ xx xx ( ab,(,)) cd ( ac) ( bd )  . (31) ◊  = ( fgxx2)(( ab,)) ( fgxx 2 )(( cd , )) x ∂ However, in this case, it is not a lattice in 2D, since then: the first pair of elements does not satisfy relation ≤ as:

∗ = ◊  (4,3)≤ (2,4) ⇒¬ (4 ≤∧≤ 2) (3 4). hxx( 122) ( fgxx)(( ab,)) ( fgxx 2 )(( cd , )) [ ] [ ]

414 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

A lattice in Cartesian product space (positive reals) 12 'c:\x.dat'

10

8

Pointson y-axis 6

4

2 1 1.5 2 2.5 3 3.5 4 4.5 5 Points on x-axis

Fig. 1: An oriented lattice representation in real Cartesian product space (strictly positive)

A non-lattice in Cartesian product space (positive reals) 4 'c:\x.dat'

3.8

3.6

Pointson y-axis 3.4

3.2

3 2 2.5 3 3.5 4 4.5 5 5.5 6 Points on x-axis

Fig. 2: Example of non-lattice representation in real Cartesian product space (strictly positive)

ℜ2 partial ordering relation, where the monoid operation Oriented Partially Ordered Monoid in is * = +. Evidently, it satisfies the following lattice- In this section, we present an example of partially monoid property as: ordered monoid in real Cartesian product space. −−≤− − ⇒−−+− −− −− − [( 9, 5) ( 4, 3)] [ ( 9, 5)2 ( 2,4) Let {( 9, 5),( 4, 3),( 2,4),(0,0)} be the set (32) 2 ≤− − + − in Cartesian product space (a subset of ℜ ) having ( 4, 3)2 ( 2,4)] .

415 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

A partially ordered additive monoid in 2D (reals) 4 'c:\x.dat'

3

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-1 Points ony-axis

-2

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-5 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 Points on x-axis

Fig. 3: A partially ordered monoid in Cartesian product space (additive)

A partially ordered multiplicative monoid in 2D (reals) 12 'c:\x.dat'

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Fig. 4: Partially ordered monoid representation in Cartesian product space (multiplicative)

The representation of resulting partially ordered Not a Partially Ordered Monoid in ℜ2 monoid is given in Fig. 3. Again, if we select the basis set as {(-9, -5), (-4, -3), The representation of not a partially ordered monoid in 2D is given in Fig. 5, where {(2,3), (4,5), (6,12), (-2, 4), (6,8), (8,12), (1,1)} and we change the algebraic 2 2 operation as ∗ ≡ ⋅ (multiplicative), then the resulting 9,12)} ⊂N ⊂ℜ and ∗ ≡ ⋅ (multiplicative). We have partially ordered monoid is presented in Fig. 4. considered ≤ as the poset relation.

416 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

Not a partially ordered monoid in Cartesian product space 12 'c:\x.dat'

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9

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Fig. 5: Not a partially ordered monoid in real Cartesian product space

A lattice-monoid (LM) in strictly positive 2D space 7 'c:\x.dat'

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4 Pointson y-axis

3

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Fig. 6: A lattice-monoid representation in real Cartesian product space (strictly positive)

Following the definition of partially ordered monoid, in Cartesian product space, where ∗ ≡ ⋅ (multiplicative). (2,3),(4,5)∈ N 2 ⊂ ℜ 2 and, (-1,-2)∈ X2 ⊂ ℜ 2 . The following Evidently, the following lattice property is satisfied by it: computation illustrates the reasoning: (1,1)⋅ [(2,3) ∧ (2,7)] =⋅ [(1,1) (2,3)] ∧ [(2,3) ≤ (4,5)] ≡ ¬[((2,3) ⋅2 − − ))2,1( ≤ ((4,5) ⋅2 − −2,1( ))]. 2 2 ⋅ ⋅ = Hence, one can conclude that, [(1,1)2 (2,7)],(1,1) 2 (2,3) (2,3), and, (33) [(2,3) ≤ (4,5)] ≠ ¬ − −6),2([ ≤ − ,4( − )]10 . Thus, the set given (1,1)⋅2 [(2,3) ∨ (2,7)] = [(1,1) ⋅ 2 (2,3)] 2 2 ∨⋅ ⋅= by, {(2,3), (4,5), (6,12), (9,12)} ⊂N ⊂ℜ does not satisfy [(1,1)2 (2,7)],(1,1) 2 (2,7) (2,7). the definition of partially ordered monoid. Figure 6 presents the lattice-monoid in real Cartesian A Lattice-Monoid in 2D product space under multiplication. In this section, we present an example of lattice-monoid It can be observed in Fig. 6 that, the lattice-monoid structure in 2D. Let {(1,1),(2,3),(2,7),(5,7)} be the basis set structure in 2-dimensional positive real surface is

417 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421 oriented in nature. Moreover, the oriented structure is not Lattice-Group in Cartesian Product Space a monotonic increase or decrease as a chain on the positive real surface. In this section, we present an example illustrating lattice-group in Cartesian product space. We start with Not a Lattice-Monoid in 2D a simple of elements to illustrate the basic structure of lattice-group. We consider that In this section, we present a counter example LG=(, L G ,,,) f + ≤ is a lattice- group, where L is a illustrating a non-lattice-monoid in 2D space. f f 2 = + Let {(1,1),(1,2),(2,4),(3,9),(4,5)}⊂ X be in Cartesian lattice and Gf ( M ,, f ) is an additive group in X. product space. Figure 7 represents the example of non- Moreover, we consider surjective function defined as, lattice-monoid structure. The reason is that, in 2D, f(( ab , ))= a + b . {(3,9),(4,5)}⊄≤ .

Not a lattice-monoid (LM) in positive 2D space (reals) 9 'c:\x.dat'

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Fig. 7: A non-lattice-monoid representation in real Cartesian product space (strictly positive)

A lattice-group in Cartesian product space (reals) 2 'c:\x.dat'

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-0.5

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Fig. 8: An additive lattice-group representation in Cartesian product space

418 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

Let {(−− 1, 2),(0,0),(1,2)} ⊂ℜ 2 be in 2D reals, where groups considering lattice structures. For example, the ∗ ≡ + and, f((−−=− 1, 2)) 3, f ((0,0)) = 0, f ((1,2)) = 3 and, is a lattice based structure, which is according to the lattice-group definition, finite and symmetric (Schattschneider, 1978). The −∧3 0 =− 3, −∨ 3 0 = 0. finiteness and symmetry of wallpaper group are maintained based on two dimensional repetitive Figure 8 illustrates the example of additive lattice- patterns preserving symmetries. Moreover, the group structure. Furthermore, in a more generalized wallpaper group is Abelian and cyclic in nature form, if X ⊆ Z then there exists at least one surjective f: 2 (Schattschneider, 1978). On the other hand, the X → X such that, LG f = ( Z, Gf, f, +, ≤) is a lattice- group, 2 proposed lattice-group constructs an Abelian structure where Z is a lattice-chain () and Gf = ( X ,+, f) is under any arbitrary function (i.e., surjective, injective an additive group in X. and bijective). The proposed lattice-group can be constructed as a if and only if the function Comparative Evaluation is surjective and homeomorphic. The In this section, we have presented the comparative (An) is a group variety based on even permutations and analysis of properties of lattice-group structure with the preserves finite and symmetry properties (Donold, other contemporary groups in the domain. The 1968). An alternating group can be Abelian if and only comparative analysis is based on a set of parameters such if the group order is n≤3. The Z3 type of alternating as, finiteness, compactness, commutativity, symmetry, group is cyclic in nature (Donold, 1968). However, the cyclicness, convexity, direction and orientation. First, we proposed lattice-group is a cyclic group under the present a set of properties of various group structures, influence of surjective functions. The simple which is summarized in Table 1. is a non-Abelian group structure and it is finite as well The lateral completion of an arbitrary lattice group as symmetric (Milne, 2013). The embedding of incorporates finite and Abelian as well as symmetry topological lattice-ordered group are locally compact, properties employing function (Bernau, 1975). where the group is Abelian (Robert, 1969). In this Moreover, it can be cyclic depending on the function group, the cyclicness and symmetry depend of function where it considers orientation and convexity. The mapping between spaces. If the function is surjective is finite and Abelian where it then there exists a relation-center in the lattice-group. is symmetric and it can be convex and directed in However, if the function is odd then the lattice-group does not exist under addition in Cartesian product nature (Shirshova, 2015). The ( Dn) considers structure, which is finite in space. Furthermore, if the odd function and even nature. The Abelian and cyclic properties of dihedral function are surjective as well as homeomorphic under group is dependent on group order. A dihedral group is multiplication then the lattice-group cannot be Abelian as well as cyclic if the group order is in {1,2} constructed. Interestingly, if the monotonically (Bilal et al ., 2013). Interestingly, if the group order decreasing function mapping between spaces is applied, then the lattice-group cannot exist in the Cartesian n≥3, then ( Dn) becomes a non-Abelian acyclic group. However, the Abelian and cyclic property of proposed product space. The proposed lattice-group under the lattice-group in this paper independent of group order. influence of function proposed in this paper is finite. In the lattice-group, the group order is determined by Finally, the lattice-groups exhibit the group the type of function mapping between spaces having homeomorphism under the influence of a pair of different dimensions. There are different varieties of mappings between spaces having varying dimensions.

Table 1: Comparison of various properties of different group structures (n is group order) Finiteness/ Group types compactness Commutativity Symmetry Cyclicness Convexity Direction/orientation The lateral completion of Finite Abelian Depends on Depends on Convexity Oriented an arbitrary lattice group function function Partially ordered group Finite Abelian Yes No Convexity Directed Dihedral group Finite Abelian (if n = 1,2) Yes Yes (if n = 1,2) Convexity Oriented in the three dimensional case Wallpaper group Finite Abelian Yes Yes Convexity Directed Alternating group Finite Abelian (if n ≤ 3) Yes Yes (if only Convexity Directed/oriented Z3 type) Simple lie group Finite Non-Abelian Yes No Convexity Oriented Embeddings of topological Locally Abelian Depends on Depends on Convexity Oriented lattice-ordered group compact function function Littice-group Finite Abelian Depends on Depends on Convexity Oriented function function

419 Davronbek Malikov and Susmit Bagchi / Journal of Computer Science 2020, 16 (4): 402.421 DOI: 10.3844/jcssp.2020.402 .421

Conclusion Ethics In this paper, we have constructed the algebraic Authors declare that there are no ethical issues in the structures hybridizing lattice, monoid and group paper. employing various mappings under certain conditions. First, we prepare the monoid structure in Cartesian References product space by using an arbitrary algebraic operation in space having closure property. In order to generalize Attiya, H., M. Herlihy and O. Rachman, 1995. Atomic the concept, the inverses of points in Cartesian product snapshots using lattice agreement. Distributed space are not considered to be same as the ordered pairs Comput., 8: 121-132. Bilal, N.A., A.A. Othman and S.A. Mousa, 2013. of respective individual inverses in the space. We m+1 illustrate that, the monoid in Cartesian product space can Dihedral groups of order 2 . Int. J. Applied Math., retain the poset relation successfully generating the 26: 1-7. partially ordered monoid. Before proceeding to the Birkhoff, G., 1967. Lattice theory. American lattice-group, the generalized 2D lattice structure is mathematical society. prepared and hybridized with the monoid to establish the Bernd, S., 2016. Ordered Sets: An Introduction with lattice-monoid in Cartesian product space. This paper Connections from to . illustrates that there exists lattice-group structure under 2rd Edn., Birkhauser, Basel, the influence of surjective functions. Moreover, the ISBN-10: 3319297880, pp: 420. extension of surjective function into surjective- Bernau, S.J., 1975. The lateral completion of an homeomorphic function generates cyclic group. arbitrary lattice group. J. Australian Math. Furthermore, the distributive lattice-subgroup can exist Society, 19: 263-289. under specific condition and, in such case the mapping Chase, C.M. and V.K. Garg, 1995. Efficient detection of function can be arbitrary in nature. Interestingly, the restricted classes of predicates. Proceedings of the monotonically decreasing function does not support International Workshop on Distributed Algorithms, existence of a lattice-group. If the group is Abelian and (WDA’ 95), France, Springer, pp: 303-317. is prepared employing surjective function, then a relation DOI: 10.1007/BFb0022155 center exists in the lattice-group. The relation center is Dushnik, B. and E.W. Miller, 1941. Partially ordered formulated by applying constructs of relational algebra sets. Am. J. Math., 63: 600-610. including inverse relations. Interestingly, the relation Donold, S.P., 1968. Permutation Groups. 1st Edn., centers exist in both additive and multiplicative lattice- Benjamin Company, New York, groups. The construction and analysis of group ISBN-13: 9780486485928. homeomorphisms have resulted in the formulation of a Frank, J.O., 2000. An application of lattice theory to function pair to establish the homeomorphism between knowledge representation. Theoretical Comput. Sci., two lattice-groups. The presented numerical examples 249: 163-196. DOI: 10.1016/S0304-3975(00)00058-X illustrate geometric representations of various structures. George, G., 2009. Lattice Theory: First Concepts and A set of counter examples are also presented in Distributive Lattices. 1st Edn., Dover Publications, numerical forms. The comparative analysis explains a set USA, ISBN-10: 048647173X, pp: 240. of distinctive characteristics of the proposed lattice- Herstein, I.N., 1975. Topics in Algebra. 1st Edn., John group in the family of other group varieties. Wiley Sons, USA, ISBN-10: 0471010901, pp: 388. Lamport, L., 1978. Time, clocks and the ordering of Acknowledgement events in a distributed system. Commun. ACM, 21: 558-565. Authors would like to thank Editors and anonymous Louis, N., 2016. An Introduction to Lattice based reviewers for valuable comments. probability theories. J. Math. Psychol., 74: 66-81. DOI: 10.1016/j.jmp.2016.04.013 Funding Information Laszlo, F., 2015. Abelian Groups. 1st Edn., Springer, Cham, ISBN-10: 3319194224, pp: 747. This work is partly funded by Gyeongsang National Milne, J.S., 2013. Lie Algebras. Algebraic Groups and University, Jinju, ROK. Lie Groups Version 2.00. Vasco, M.I.G. and R. Steinwandt, 2015. Group Theoretic Author’s Contributions Cryptography. 1st Edn., Chapman Hall/CRC, Boca First author (D. Malikov) proposed the concept, Raton, ISBN-10: 1584888377, pp: 244. proposed theorems, proofs and drafted the paper. Second Mildred, S.D., D. Gene and J. Ado, 2010. Group theory and corresponding author (S. Bagchi) refined the Applications of Physics of Condensed Matter. concept, refined theorems and, corrected the draft. Springer, Berlin, ISBN-10: 3642069452, pp: 582.

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