Ratio Mathematica Volume 47, 2023 Elongation of Sets in Soft Lattice Topological Spaces G. Hari Siva Annam* T. Abinaya† Abstract The aim of this paper, we investigate some Lattice sets such as soft lat- tice exterior, soft lattice interior, soft lattice boundary and soft lattice border sets in soft lattice topological spaces which are defined over a soft lattice L with a fixed set of parameter A and it is also a general- ization of soft topological spaces. Further, we develop and continue the initial views of some soft lattice sets, which are deep-seated for further research on soft lattice topology and will consolidate the ori- gin of the theory of soft topological spaces. 2020 AMS subject classifications: 54A05, 54A10. 1 *Assistant Professor, PG and Research Department of Mathematics, Kamaraj College, Thoothukudi-628003, Tamil Nadu, India. hsannam84@gmail.com. Affiliated to Manonmaniam Sundaranar University, Tirunelveli-627012, Tamil Nadu, India. †Research Scholar [21212102092011], PG and Research Department of Mathematics, Kamaraj College, Thoothukudi, Manonmaniam Sundaranar University, Tirunelveli-627012, Tamil Nadu, India. rvtpraba77@gmail.com. 1Received on August 12, 2022. Accepted on January 2, 2023. Published on January 10, 2023. doi: 10.23755/rm.v41i0.838. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 206 G. Hari Siva Annam and T. Abinaya 1 Introduction The concept of soft theory was first originated by Molodstov in 1999, which is deal with unpredictable problems meanwhile modeling results in engineering cases such as medical sciences, economics, etc., In 2003, Maji. et. al.[8] stud- ied and discussed the fundamental ideas of soft theory. Following stage of soft set linked with netrosophic sets are introduced by Parimala Mani et. al.[9] in 2018 and Also, we introduced the new notion of neutrosophic complex αΨ-connectedness in neutrosophic complex topological spaces and investigate some of its properties in 2022[5]. In 2019[13], several new generalizations of nano open sets be intro- duced and investigated by Nethaji, Ochanan. The study of soft topological spaces (on short S.T.S) is instated by Shabir and Naz[14] in 2011. They discussed S.T on the collection θ on soft set (on short S.S) over U. Accordingly, they discussed fundamental notions of S.T.S such as soft open (on short S.O), soft closed (on short S.C), S closure, S neighborhood of a point, S Ti spaces, for (i =1, 2, 3, 4), S regular spaces, S normal spaces, and their specific features are also established. Therefore, in 2011[1], Naim Cagman, Serkan Karatas, and Serdar Enginoglu investigated a topology with S.S called S.T and its corresponding features. Then they present the foundation of the theory S.T.S. The S.T.S may be the initial stage for the concepts of the soft mathematical opinion of structures which are the foundation of S.S. theoretic operation. From the concept of S.S, the idea of soft lattices (on short S.L) has arisen. In 2010[7], F. Li studied and defined this conviction of S.L and primary operations of results on S.L. Additional, an application of S.S to lattices has executed by E. Kuppusamy in 2011. A different approach towards S.L can be seen in E. Kup- pusamy apart from what F. Li has done. Further, the operation and the properties of S.L were studied by V. D. Jobish. et. al.[4] in 2013. Many theorems related to various types of unions, intersections, and complements including De Margon’s Laws are obtained. In 2020[12], M. Parimala et. al explained the nIαg closed sets in nano ideal toplogical spaces with various prevailing closed sets. Currently, topology depends toughly on the thoughts of the soft theory. Recently, S.L.T.S was first investigated by Sandhya. et. al.[11] in 2021 that are discussed throughout an S.L ’L’ with a fixed set of parameters ’A’ and it is also a general- ization of S.T.S. They detailed discussed the concept of Soft L - open (on short S.L - O), soft L - closed (on short S.L - C), S.L - closure, S.L – interior point, and S.L - neighborhood. In this paper, we continue investigating a soft L – in- terior (on short S.L - I), soft L – exterior (on short S.L - E), soft L - boundary (on short S.L - B), and soft L - border (on short S.L - Bor) which are basics for stimulating research on S.T.S and will build up the fountain of the theory of S.T.S. 207 Elongation of Sets in Soft Lattice Topological Spaces 2 Preliminaries Definition 2.1 (5,7). Let’s take U be a whole set and A be a set parameters. A pair (F, A), where F is a map from A to ℘(U) is called a S.S over U. Here, the S.S is simply represented by fA. Example 2.1. Let say that there are 6 cars in the whole world U = {w1, w2, w3, w4, w5, w6} is the set of cars under regard and that A = {ρ1, ρ2, ρ3, ρ4, ρ5} is a set of parameters denoted as colors. The ra, (a = 1, 2, 3, 4, 5) it means the parameters ‘Red’, ‘Blue’, ‘Black’, ‘White’, and ’Ash,’ respectively. Consider the mapping fA given by ‘cars’ (.), where (.) is to be complete in by one of the parameters ra ∈ E. For instance, fA(ρ1) means ‘ Cars (Colors)’. Suppose that B = {ρ1, ρ2, ρ5} ⊆ A and fA(ρ1) = {w1, w4}, fA(ρ2) = U, and fA(ρ5) = {w2, w4, w5} Then, we can view the S.S FA as consisting of the following collection of approximations: FA = {(ρ1, {w1, w4}), (ρ2, U), (ρ5, {w2, w4, w5})}. Definition 2.2 (2,7). In two S.S fA, gA over U, we say that (i) fA is a soft subset of gA if (a) A ⊆ B , and (b) ∀ρ ∈ A , λ (ρ)= µ (ρ) are equal to estimations. (ii) fA is soft equal set to gA denoted by fA = gA if fA ⊆ gA and gA ⊆ fA Definition 2.3 (7). Let A = {ρ1, .... ρn} be a parameters. The ‘Not set of A’, denoted by ΓA is defined as ΓA = { Γρ1, ..., Γρn} , Γρi means not ρi ∀ i = 1, 2, 3. . .n. Definition 2.4 (7,9). Complement of a S.S fA over U, represented by f ′A is defined as f ′A = (F ′, ΓA), F ′ : ΓA −→ ℘ (U) such that(on short s.t) F ′(Γρ) = U – F(ρ), ∀ Γρ ∈ ΓA. Definition 2.5 (9). The relative complement of a S.S fA over U, stand for fCA is defined as (fA)C = (F C, A), F C : A −→ ℘ (U) s.t F C(ρ) = U–F(ρ), ∀ ρ ∈ A. Definition 2.6 (7,9). Let fA be a S.S over U , then fA is Null S.S if ∀ ρ ∈ A, F(ρ) = ϕ and is denoted by ϕA. Let fA be a S.S over U ,then fA is absolute S.S represented by UA, if ∀ ρ ∈ A, F(ρ) = U. Also, UCA = ϕA and ϕ C A = UA. 208 G. Hari Siva Annam and T. Abinaya Definition 2.7 (2,7). Union of two S.S fA, gB over U is the S.S hC , C = A ⋃ B and ∀ ρ ∈ C, κ(ρ) =   λ(ρ), if ρ ∈ A − B µ(ρ), if ρ ∈ B − A λ(ρ) ⋃ µ(ρ), if ρ ∈ A ⋂ B We write fA ⋃ gB = hC . Definition 2.8 (2,7). The intersection of two S.S fA, gB over a whole set U is the S.S hC , here C = A ⋂ B and ∀ e ∈ C, κ(ρ) = λ(ρ) or µ(ρ). We mark done fA ⋂ gB = hc. Definition 2.9 (1). Consider FA, GA ∈ S.S (U, A). The soft symmetric difference of these sets is the S.S. HA ∈ to S.S.(U, A), here the map H : A−→℘(U) defined as follows: h(ρ) = ((f(ρ) \ g(ρ)) ⋃ ((g(ρ) \ (f(ρ)) for each ρ ∈ A. We mark down HA = FA ∆ GA. Definition 2.10 (3,6,10). A sublatice of a lattice L is a non-void subset of L that is a lattice with the same meet and join operation as L, ie., α, β ∈ L implies α ∧ β, α ∨ β ∀ α, β ∈ L. Definition 2.11 (3,6,10). A Complete lattice L and A is the parameters of the S.L over L. The triplet M = (f, A, L), f : A−→℘(L) is S.L if f(ρ) is the sublattice of L for each ρ ∈ A. Then the S.L is represented by fLA. Definition 2.12 (10). Two S.L. fLA and g L A over L its difference is denoted by hLA = f L A \ g L A, is stated as h(ρ) = ((f(ρ) \ g(ρ)) ∀ρ ∈ A. Definition 2.13 (10). Let us consider L be any complete lattice and A be the non void set of parameters. Let θ contains complete members, uniquely complemented S.L. over L, then θ is S.L.T, then the condition hold: (i) ϕA, LA ∈ θ. (ii) ⋃ a ∈ n ηa ∈ θ, ∀ { ηa : a ∈ n} ⊆ θ (iii) η1 ⋂ η2 ∈ θ, ∀ η1, η2 ∈ θ. Then the triplet (L, θ, A) is called a S.L.T.S. (soft L – space or soft L – topological space) over L. The members of θ are called soft lattice open sets in L. Also, a soft lattice (fLA) is called soft lattice closed if the relative complement (fLA) C belongs to θ. 209 Elongation of Sets in Soft Lattice Topological Spaces 3 Extension of S.L - sets Definition 3.1. In S.L.T.S, the S.L - I of (fLA) is the union of all S.L - O sets contained in fLA denoted by (f L A) ◦. i.e., (fLA) ◦ = ⋃ {gLA : g L A ∈ θ and g L A ⊆ f L A}. Theorem 3.1. Let ( L, θ, A) be a S.L.T.S over L and fLA , g L A are S.L. over L. Then, (i) ϕ◦A = ϕA and LA = L ◦ A (ii) (fLA) ◦ ⊆ (fLA) (iii) fLA is a S.L − O set ⇐⇒ (f L A) ◦ = fLA (iv) ((fLA) ◦)◦ = (fLA) ◦ (v) fLA ⊆ g L A ⇒ (f L A) ◦ ⊆ (gLA) ◦ (vi) (fLA) ◦ ⋂ (gLA)◦ = (fLA ⋂ gLA)◦ (vii) (fLA) ◦ ⋃ (gLA)◦ ⊆ (fLA ⋃ gLA)◦ Proof Results (i), (ii) are trival. (iii) If (fLA) is S.L−O set, (f L A) is itself a S.L−O set contained in (f L A). Since, (fLA) ◦ is the largest S.L − O set contained in (fLA) , (f L A) = (f L A) ◦. Conversely, Suppose that (fLA) = (f L A) ◦. Since (fLA) ◦ is a S.L − O set, so (fLA) is S.L − O set over L. (iv) since (fLA) ◦ is S.L − O set, by (iii) ((fLA) ◦)◦ = (fLA) ◦. (v) suppose that (fLA) ⊆ (g L A). Since,(f L A) ◦ ⊆ (fLA) ⊆ (g L A). (f L A) ◦ is a S.L − O subset of (gLA), so by the definition of (g L A) ◦, (fLA) ◦ ⊆ (gLA) ◦. (vi) we have (fLA ⋂ gLA) ⊆ f L A and (f L A ⋂ gLA) ⊆ g L A. This implies (by v) (fLA ⋂ gLA) ◦ ⊆ (fLA) ◦and (fLA ⋂ gLA) ◦ ⊆ (gLA) ◦ so that, (fLA ⋂ gLA) ◦ ⊆ (fLA) ◦ ⋂ (gLA)◦. Also, since (fLA) ◦ ⊆ fLA and (g L A) ◦ ⊆ gLA implies (fLA) ◦ ⋂ (gLA)◦ ⊆ (fLA ⋂ gLA) so that, (fLA ⋂ gLA)◦ is the largest S.L − O subsets of (fLA ⋂ gLA). Hence, (f L A) ◦ ⋂ (gLA)◦ ⊆ (fLA ⋂ gLA)◦. Thus, (fLA) ◦ ⋂ (gLA)◦ = (fLA ⋂ gLA)◦. 210 G. Hari Siva Annam and T. Abinaya (vii) Since, fLA ⊆ (f L A ⋃ gLA) and, g L A ⊆ (f L A ⋃ gLA). So, by (v) (fLA) ◦ ⊆ (fLA ⋃ gLA) ◦ and (gLA) ◦ ⊆ (fLA ⋃ gLA) ◦. So that (fLA) ◦ ⋃ (gLA)◦ ⊆ (fLA ⋃ gLA)◦. Example 3.1. Now the given example to show that the statement of theorem 1(v) may be strict or equal, Let L = {Sl1, Sl2, Sl3, Sl4, Sl5, Sl6, Sl7, Sl8} ;A = {ρ1, ρ2}; θ = {fL1A, f L 2A, f L 3A, f L 4A, f L 5A, LA, ϕA} Figure 1: Complete lattice fL1A = {(ρ1, {Sl4, Sl7, Sl8}), ( ρ2, {Sl3, Sl6})}, fL2A = {(ρ1, {Sl6, Sl8}), (ρ2, {Sl1, Sl4})} fL3A = {(ρ1, {Sl4, Sl6, Sl7, Sl8}), (ρ2, {Sl1, Sl3, Sl4, Sl6})}, fL4A = {(ρ1, {Sl8}), (ρ2, ϕ)} and fL5A = {(ρ1, {Sl4, Sl7}), (ρ2, {Sl3, Sl6})} For Equal Condition, We choose any two S.L from figure:1, fLC = {(ρ1, {Sl6, Sl8}), (ρ2, {Sl1, Sl4})} and gLC = {(ρ1, {Sl1, Sl6, Sl7, Sl8}), (ρ2, {Sl1, Sl3, Sl4, Sl6})} (fLC ) ◦ = fL2A and (g L C) ◦ = fL2A. Hence, fLA ⊂ g L A implies(f L A) ◦ = (gLA) ◦. 211 Elongation of Sets in Soft Lattice Topological Spaces For inclusion condition, We choose any two S.L from figure:1, fLD = {(ρ1, {Sl8}), (ρ2, {Sl3, Sl6})} and gLD = {(ρ1, {Sl4, Sl7, Sl8}), (ρ2, {Sl3, Sl6})} (fLD) ◦ = fL4A and (g L D) ◦ = fL1A. Hence, fLA ⊂ g L A implies(f L A) ◦ ⊂ (gLA) ◦. Example 3.2. Now the given example to show that the statement of theorem 1(vii) may be strict or equal, Let us consider the lattice and S.L.T given in Example: 3.1 For inclusion Condition, We choose any two S.L from figure:1, fLC = {(ρ1, {Sl6, Sl8}), (ρ2, {Sl1, Sl3, Sl6})} and gLC = {(ρ1, {Sl4, Sl7, Sl8}), (ρ2, {Sl1, Sl3, Sl4, Sl6})} (fLC ) ◦ = fL4A and (g L C) ◦ = fL1A, which implies (f L C ) ◦ ⋃ (gLC)◦ = fL1A. (fLC ⋃ gLC) ◦ is fL3A. Hence, (fLA) ◦ ⋃ (gLA) ◦ ⊂ (fLA ⋃ gLA) ◦. For equal condition, We choose any two S.L from figure:1, fLC = {(ρ1, {Sl6, Sl8}), (ρ2, {Sl1, Sl4})} and gLC = {(ρ1, {Sl1, Sl6, Sl7, Sl8}), (ρ2, {Sl1, Sl3, Sl4, Sl6})} (fLC ) ◦ = fL2A and (g L C) ◦ = fL2A, which implies (f L C ) ◦ ⋃ (gLC)◦ = fL2A. Hence, (fLA) ◦ ⋃ (gLA) ◦ = (fLA ⋃ gLA) ◦. Definition 3.2. Let (L, θ, A) be a S.L.T.S over L, then the S.L - E. of S.L fLA is denoted by (fLA)◦ and is defined as (f L A)◦ = ((f L A) C)◦. 212 G. Hari Siva Annam and T. Abinaya Theorem 3.2. Let fLA and g L A be S.L of a S.L.T.S (L, θ, A). Then, (i) (fLA ⋃ gLA)◦ = (f L A)◦ ⋂ (gLA)◦. (ii) (fLA)◦ ⋃ (gLA)◦ ⊆ (f L A ⋂ gLA)◦. (iii) fLA ⊆ g L A implies (f L A)◦ ⊇ (g L A)◦. Proof (i) (fLA ⋃ gLA)◦ = ((f L A ⋃ gLA) C)◦ = ((fLA) C ⋂ (gLA) C)◦ = ((fLA) C)◦ ⋂ ((gLA) C)◦ = (fLA)◦ ⋂ (gLA)◦ (ii) (fLA)◦ ⋃ (gLA)◦ = ((f L A) C)◦ ⋃ ((gLA) C)◦ ⊆ ((fLA) C ⋃ (gLA) C)◦ = ((fLA ⋂ gLA) C)◦ = (fLA ⋂ gLA)◦ (iii) (gLA)◦ = ((g L A) C)◦ ⊆ ((fLA) C)◦ = (fLA)◦ Example 3.3. Now the given example to show that the statement of theorem 2(ii) may be strict or equal, Let LA = {Sl1, Sl2, Sl3, Sl4, Sl5, Sl6, Sl7}; A = {ρ1, ρ2}; θ = {fL1A, f L 2A, f L 3A, f L 4A, LA, ϕA} fL1A = {(ρ1, {Sl3, Sl6}), (ρ2, {Sl4, Sl5})}, f L 2A = {(ρ1, {Sl6}), (ρ2, ϕ)} fL3A = {(ρ1, {Sl2, Sl3, Sl5, Sl6}), (ρ2, {Sl4, Sl5, Sl6})} and fL4A = {(ρ1, {Sl2, Sl5, Sl6}), (ρ2, {Sl6})} Figure 2: Complete lattice 213 Elongation of Sets in Soft Lattice Topological Spaces For inclusion condition, Now we take any two S.L from the figure:2, fLC = {(ρ1, {Sl2, Sl6}), (ρ2, {Sl6})} and gLC = {(ρ1, {Sl2, Sl3, Sl5}), (ρ2, {Sl4, Sl5})} Then, (fLC ⋂ gLC) = {(ρ1, {Sl2}), (ρ2, ϕ)}. (f L C )◦ = ϕA and (g L C)◦ = f L 2A, which implies (fLC )◦ ⋃ (gLC)◦ = f L 2A. (f L C ⋂ gLC)◦ is f L 1A. Hence, (fLA)◦ ⋃ (gLA)◦ ⊂ (f L A ⋂ gLA)◦. For Equal condition, Now we take any two S.L from the figure:2, fLC = {(ρ1, {Sl5, Sl7}), (ρ2, {Sl4, Sl7})} and gLC = {(ρ1, {Sl3, Sl5, Sl7}), (ρ2, {Sl4, Sl5, Sl7})} Then, (fLC ⋂ gLC) = {(ρ1, {Sl5, Sl7}), ((ρ2, {Sl4, Sl7})}. (fLC )◦ = f L 2A and (g L C)◦ = f L 2A, which implies (f L C )◦ ⋃ (gLC)◦ = f L 2A. (fLC ⋂ gLC)◦ is f L 2A. Hence, (fLA)◦ ⋃ (gLA)◦ = (f L A ⋂ gLA)◦. Example 3.4. Now the given example to show that the statement of theorem 2(iii) may be strict or equal, Let us consider the lattice and S.L.T given in Example: 3.3 For Equal condition, Now we take any two S.L from the figure:2, fLD = {(ρ1, {Sl3, Sl6}), (ρ2, {Sl4, Sl5, Sl6})} and gLD = {(ρ1, {Sl1, Sl3, Sl5, Sl6}), (ρ2, {Sl4, Sl5, Sl6})} (fLD)◦ = ϕA and (g L D)◦ = ϕA. Hence, fLA ⊆ g L A implies (f L A)◦ = (g L A)◦. 214 G. Hari Siva Annam and T. Abinaya For inclusion condition, Now we take any two S.L from the figure:2, fLB = {(ρ1, {Sl4, Sl5}), (ρ2, {Sl2, Sl3, Sl6, Sl7})} and gLB = {(ρ1, {Sl2, S l4, Sl5, Sl7}), (ρ2, {Sl1, Sl2, Sl3, Sl5, Sl6, Sl7})} (fLB)◦ = f L 1A and (g L B)◦ = f L 2A. Hence,fLA ⊆ g L A implies (f L A)◦ ⊃ (g L A)◦. Definition 3.3. In S.L.T.S, then the S.L - B of S.L fLA is denoted by (f L A) B and is defined as (fLA) B = fLA ⋂ (fLA) C. Theorem 3.3. Let (L, θ, A) be a S.L.T.S: (i) (fLA) B ⋂ (fLA) ◦ = fLϕ (ii) (fLA) B ⋂ (fLA)◦ = f L ϕ Proof (i) (fLA) B ⋂ (fLA) ◦ = ( fLA ⋂ (fLA) C ) ⋂ (fLA) ◦ = fLA ⋂ (fLA) C ⋂ (fLA) ◦ = fLϕ (ii) (fLA) B ⋂ (fLA)◦ = f L A ⋂ (fLA) C ⋂ (fLA) C)◦ = fLA ⋂ (fLA) C ⋂ ((fLA)) C = fLϕ Example 3.5. Now the given example for find the boundary, Let us consider the lattice and S.L.T given in Example: 3.1 Now we take any S.L from the figure:1, fLC = {(ρ1, {Sl2, Sl8}), (ρ2, {Sl1, Sl3})} Then, (f L C ) B = (fL4A) C Definition 3.4. Let (L, θ, A) be a S.L.T.S over L, then the S.L - Bor of S.L fLA is denoted by (f L A) • and is defined as (fLA) • = fLA − (f L A) ◦. Theorem 3.4. Let (L, θ, A) be a S.L.T.S. Then the following hold: 215 Elongation of Sets in Soft Lattice Topological Spaces (i) (fLA) • = A ⋂ (LA − fLA) (ii) (ϕLA) • = ϕLA (iii) (fLA) • ⊆ ((fLA) ◦)C (iv) (fLA) • ⊆ fLA ⊆ fLA Proof (i) fLA ⋂ ((fLA) ◦)C = fLA ⋂ (fLA) C = fLA ⋂ (LA − fLA) (ii) ϕLA ⋂ ((ϕLA) ◦)C = (ϕLA) ⋂ (ϕLA) C = ϕLA (iii) fLA − (f L A) ◦ = fLA ⋂ ((fLA) ◦)C ⊆ ((fLA) ◦)C (iv) By definition of (fLA) •, (fLA) • ⊆ fLA . we know that,fLA ⊂ fLA Therefore,(f L A) • ⊆ fLA ⊆ fLA Example 3.6. Now the given example to show that the statement of theorem 4(iv) may be strict or equal, Let us consider the lattice and S.L.T given in Example: 3.3 We choose any two S.L from the figure:2, gLB = {(ρ1, {Sl2, Sl3, Sl5, Sl6}), (ρ2, {Sl4, Sl5, Sl6})} Now, the Closure of gLB is LA , Then border of g L B, is ϕA Hence,(gLB) • ⊂ gLB ⊂ gLB. 4 Conclusions In the present work, we defined and discussed some S.L – sets of S.L.T.S. We extended some basic results relating to S.L - I, S.L - E, S.L - B, and S.L - Bor of S.L.T.S. In the interior section, idempotent and monotonicity results are held. Formerly the intersection of the boundary and interior soft lattice gives the null set and the intersection of the boundary and exterior soft lattice should not give the non-empty soft sets. In end, this paper is the inception of a novel 216 G. Hari Siva Annam and T. Abinaya structure. Further, we learned a few viewpoints, it will be needed to carry out a new seeking work to build future applications. 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