International Journal of Analysis and Applications Volume 18, Number 5 (2020), 819-837 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-819 UNION SOFT SET THEORY APPLIED TO ORDERED SEMIGROUPS RAEES KHAN1,∗, ASGHAR KHAN2, MUHAMMAD UZAIR KHAN2,3, NASIR KHAN1 1Department of Mathematics, FATA University, Darra Adam Khel, N.M.D. Kohat, KP, Pakistan 2Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan 3Department of Mathematics & Statistics, Bacha Khan University Charsadda, KP, Pakistan ∗Corresponding author: raeeskhan@fu.edu.pk Abstract. The uni-soft type of bi-ideals in ordered semigroup is considered. The notion of a uni-soft bi-ideal is introduced and the related properties are investigated. The concept of δ−exclusive set is given and the relations between uni-soft bi-ideals and δ−exclusive set are discussed. The concepts of two types of prime uni-soft bi-ideals of an ordered semigroup S are given and it is proved that, a non-constant uni-soft bi-ideal of S is prime in the second sense if and only if each of its proper δ−exclusive set is a prime bi-ideal of S. The characterizations of left and right simple ordered semigroups are considered. Using the notion of uni-soft bi-ideals, some semilattices of left and right simple semigroups are provided. By using the properties of uni-soft bi-ideals, the characterization of a regular ordered semigroup is provided. In the last section of this paper, the characterizations of both regular and intra-regular ordered semigroups are provided. 1. Introduction The notion of soft set was introduced in 1999 by Molodtsov [20] as a new mathematical tool for dealing with uncertainties. Due to its importance, it has received much attention in the mean of algebraic structures such as groups [9], semirings [11], rings [1], ordered semigroups [15] and hemirings [19, 22] and so on. Feng et al. discussed soft relations in semigroups (see [12, 13]) and explored decomposition of fuzzy soft sets with finite value spaces. Also, Feng and Li [14] considered soft product operations. Jun et al., [15] applied Received December 12th, 2017; accepted February 5th, 2018; published July 23rd, 2020. 2010 Mathematics Subject Classification. 06D72, 20M99, 20M12. Key words and phrases. ordered semigroup; left/right regular and completely regular ordered semigroup; regular and intra- regular ordered semigroup; left and right simple subsemigroup; uni-soft bi-ideal; uni-soft left/right ideal; uni-soft quasi-ideal. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 819 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-819 Int. J. Anal. Appl. 18 (5) (2020) 820 the concept of soft set theory to ordered semigroups. They applied the notion of soft sets by Molodtsov to ordered semigroups and introduced the notions of (trivial, whole) soft ordered semigroups, soft ordered subsemigroups, soft r-ideals, soft l-ideals, and r-idealistic and l-idealistic soft ordered semigroups [16]. They investigated various related properties by using these notions. In [6–8] Khan et al., characterized different classes of ordered semigroups by using uni-soft quasi-ideals and uni-soft ideals. In this paper, we introduce the notion of a uni-soft bi-ideal in ordered semigroups. The concept of δ−exclusive set is given and the relations between uni-soft bi-ideals and δ−exclusive set are discussed. We also define two types of prime uni-soft bi-ideals of an ordered semigroup S and prove that, a non-constant uni-soft bi-ideal of S is prime in the second sense if and only if each of its proper δ−exclusive set is a prime bi-ideal of S. The characterizations of some classes of ordered semigroups are given. By using the notion of uni-soft bi-ideals, some semilattices of left and right simple subsemigroups are discussed. Using the concept of an δ−exclusive set, it is proved that a soft set of S over U is a uni-soft bi-ideal if and only if the non-empty δ−exclusive set is a bi-ideal. Regular ordered semigroups are chracterized by the properties of uni-soft bi-ideals and it is shown that every uni-soft bi-ideal of S over U is idempotent if and only if the ordered semigroup is regular. In the last section of this paper, the characterizations of both regular and intra-regular ordered semigroups are discussed. 2. Preliminaries In this section, we give some basic definitions and results, which are necessary for the subsequent sections. By an ordered semigroup we mean a structure (S, ·,≤) such that: (OS1) (S, ·) is a semigroup. (OS2) (S,≤) is a poset. (OS3) (∀a,b,x ∈ S) (a ≤ b =⇒ ax ≤ bx and xa ≤ xb). For A ⊆ S, we denote (A] := {t ∈ S : t ≤ h for some h ∈ A}. For A,B ⊆ S, we have AB := {ab : a ∈ A,b ∈ B}. A nonempty subset A of an ordered semigroup S is called a subsemigroup of S if A2 ⊆ A. A nonempty subset A of S is called a left (resp. right ) ideal of S if: (1) SA ⊆ A(resp. AS ⊆ A) and (2) a ∈ A, S 3 b ≤ a, implies b ∈ A. By a two-sided ideal or simply an ideal of S we mean a non-empty subset of S which is both a left and a right ideal of S. A nonempty subset A of an ordered semigroup S is called a bi-ideal of S if: (1) A2 ⊆ A, (2) ASA ⊆ A and (3) a ∈ A,S 3 b ≤ a, implies b ∈ A. Let S be an ordered semigroup and ∅ 6= A ⊆ S. Then the set (A ∪ A2 ∪ ASA] is the bi-ideal of S generated by A. In particular, if A = {x} (x ∈ S), then we write (x ∪ x2 ∪ xSx], instead of ({x}∪{x2}∪{x}S{x}] (see [2]). Int. J. Anal. Appl. 18 (5) (2020) 821 3. Basic operations of soft sets From now on, U is an initial universe, E is a set of parameters, P(U) is the power set of U and A,B,C... ⊆ E. Definition 3.1. A soft set FA over U is defined as FA : E −→ P(U) such that FA(x) = ∅ if x /∈ A. Hence FA is also called an approximation function. A soft set FA over U can be represented by the set of ordered pairs FA = {(x,FA(x))|x ∈ E,FA(x) ∈ P(U)} . It is clear from Definition 3.1, that a soft set is a parameterized family of subsets of U. Note that the set of all soft sets over U will be denoted S(U). Definition 3.2. (i) Let FA,FB ∈ S(U). Then FA is called a soft subset of FB, denoted by FA⊆̃FB if FA(x) ⊆ FB(x) for all x ∈ E. (ii) Let FA,FB ∈ S(U). Then the soft union of FA and FB, denoted by FA∪̃FB = FA∪B, is defined by( FA∪̃FB ) (x) = FA(x) ∪FB(x) for all x ∈ E. (iii) Let FA,FB ∈ S(U). Then the soft intersection of FA and FB, denoted by FA∩̃FB = FA∩B, is defined by ( FA∩̃FB ) (x) = FA(x) ∩FB(x) for all x ∈ E. For x ∈ S, we define Ax = {(y,z) ∈ S ×S|x ≤ yz}. Definition 3.3. Let (FS,S) and (GS,S) be two soft sets over U. Then, the soft intersection-union product, denoted by fS♦gS, is defined by FS♦GS : S −→ P(U),x 7−→   ⋂ (y,z)∈Ax {FS(y) ∪GS(z)} if Ax 6= ∅, U if Ax = ∅, for all x ∈ S. One can easily prove that ”♦” on S(U) is well defined and the set ( S(U),♦,⊆̃ ) forms an ordered semigroup (see [7]). 4. Uni-soft bi-ideals For an ordered semigroup, the soft sets ”∅S” and ”>S” of S over U are defined as follows: ∅S : S −→ P(U),x 7−→∅S(x) = ∅, >S : S −→ P(U),x 7−→>S(x) = U for all x ∈ S. Clearly, the soft set ”∅S” (resp. ”>S”) of an ordered semigroup S over U is the least (resp., the greatest ) element of the ordered semigroup (S(U),♦,⊆̃). The soft set ”∅S” is the null element of (S(U),♦,⊆̃) (that Int. J. Anal. Appl. 18 (5) (2020) 822 is FS♦∅S = ∅S♦FS = ∅S and ∅S⊆̃FS for every FS ∈ S(U). The soft set (>S,S) is called the whole soft set over U, where >S(x) = U for all x ∈ S. For a non-empty subset A of S, the characteristic soft set (χA,A) over U is a soft set defined as follows: χA : S −→ P(U),x 7−→   U if x ∈ A,∅ if x ∈ S\A. For the characteristic soft set (χA,A) over U, the soft set (χ c A,A) over U is given as follows: χcA : S −→ P(U),x 7−→   ∅ if x ∈ A,U if x ∈ S\A. Definition 4.1. (cf. [15]). Let S be an ordered semigroup. A soft set (FS,S) of S over U is called a union-soft semigroup (briefly, uni-soft semigroup) of S over U if: FS(xy) ⊆ FS(x) ∪FS(y) ∀x,y ∈ S. Definition 4.2. (cf. [15]). Let S be an ordered semigroup. A soft set (FS,S) of S over U is called a union-soft left (resp. right) ideal (briefly, uni-soft left (resp. right) ideal) of S over U if (SU1) x ≤ y =⇒ FS(x) ⊆ FS(y), (SU2) FS(xy) ⊆ FS(y) (resp. FS(xy) ⊆ FS(x) ) ∀x,y ∈ S. If (FS,S) is both a uni-soft left ideal and a uni-soft right ideal of S over U, then (FS,S) is called a uni-soft ideal of S over U. Definition 4.3. (cf. [6]). Let S be an ordered semigroup. A uni-soft semigroup (FS,S) of S over U is called a union-soft bi-ideal (briefly, uni-soft bi-ideal) of S over U if (SU5) x ≤ y =⇒ FS(x) ⊆ FS(y). (SU6) FS(xyz) ⊆ FS(x) ∪FS(z) ∀ x,y,z ∈ S. Definition 4.4. A soft set (FS,S) of an ordered semigroup (S, ·,≤) over U is called idempotent if (FS♦FS,S) = (FS,S). For a soft set (FA,S) over U and a subset δ of U, the δ-exclusive set of (FA,S) denoted by eA(FA; δ) is defined by eA(FA; δ) := {x ∈ A|FA(x) ⊆ δ} . Int. J. Anal. Appl. 18 (5) (2020) 823 Theorem 4.1. A soft set (FS,S) over U is a uni-soft bi-ideal over U if and only if the nonempty δ-exclusive set of (FS,S) is a bi-ideal of S for all δ ∈ P(U). Proof. Assume that (FS,S) is a uni-soft bi-ideal over U. Let δ ∈ P(U) be such that eS(FS; δ) 6= ∅. Let x,y ∈ S with x ≤ y be such that y ∈ eS(FS; δ). Then FS(y) ⊆ δ. By (SU5) we have FS(x) ⊆ FS(y) ⊆ δ and that x ∈ eS(FS; δ). Let x,y ∈ eS(FS; δ). Then FS(x) ⊆ δ and FS(y) ⊆ δ. By (SU6) we have FS(xy) ⊆ FS(x) ∪FS(y) ⊆ δ and so xy ∈ eS(FS; δ). For x,z ∈ eS(FS; δ). Then FS(x) ⊆ δ and FS(z) ⊆ δ. By (SU6) we have FS(xyz) ⊆ FS(x) ∪FS(z) ⊆ δ hence xyz ∈ eS(FS; δ). Therefore eS(FS; δ) is a bi-ideal of S. Conversely, suppose that the nonempty δ-exclusive set of (FS,S) is a bi-ideal of S for all δ ∈ P(U). Let x,y ∈ S with x ≤ y be such that FS(x) ⊃ FS(y) = δy then y ∈ eS(FS; δy) but x /∈ eS(FS; δy). This is a contradiction. Hence FS(x) ⊆ FS(y) for all x ≤ y. If there exist x,y ∈ S such that FS(xy) ⊃ FS(x) ∪FS(y) = δx ∪ δy = δz then x ∈ eS(FS; δz) and y ∈ eS(FS; δz) but xy /∈ eS(FS; δz). This is a contradiction. Hence FS(xy) ⊆ FS(x) ∪FS(y) for all x,y ∈ S. If there exist x,y,z ∈ S such that FS(xyz) ⊃ FS(x) ∪FS(z) = δx ∪δz = δs then x ∈ eS(FS; δs) and z ∈ eS(FS; δs) but xyz /∈ eS(FS; δs). This is a contradiction. Hence FS(xyz) ⊆ FS(x) ∪FS(z) for all x,y ∈ S. � Corollary 4.1. (cf. [8]). For any nonempty subset B of S, the following are equivalent. (1) B is a bi-ideal of S. (2) The soft set (χcB,S) over U is a uni-soft bi-ideal over U. A bi-ideal P of an ordered semigroup S is called prime if P 6= S and for any bi-ideals A,B of S from AB ⊆ P it follows that A ⊆ P or B ⊆ P. By analogy a non-constant uni-soft bi-ideal (FS,S) of S over U is called prime (in the first sense) if for any uni-soft bi-ideals (GS,S), (HS,S) of S over U from (GS♦HS,S)⊇̃(FS,S) it follows that (GS,S)⊇̃(FS,S) or (HS,S)⊇̃(FS,S) . Theorem 4.2. A bi-ideal P of an ordered semigroup S is prime if and only if for all a,b ∈ S from (aSb] ⊆ P it follows that a ∈ P or b ∈ P . Int. J. Anal. Appl. 18 (5) (2020) 824 Proof. Assume that P is a prime bi-ideal of S and (aSb] ⊆ P for some a,b ∈ S. Then obviously, the sets A = (aSa] and B = (bSb] are bi-ideals of S, because (aSa]S(aSa] = (aSa](S](aSa] ⊆ (aSaSaSa] ⊆ (aSa] and if x ∈ S and x ≤ y ∈ (aSa], then x ∈ ((aSa]] = (aSa]. Similarly, (bSb] is a bi-ideal of S. So, AB ⊆ (AB] = ((aSa](bSb]] ⊆ (aSabSb] ⊆ (aSb] ⊆ P, and consequently A ⊆ P or B ⊆ P . Let 〈x〉 be the bi-ideal of S generated by x ∈ S. If A ⊆ P, then 〈a〉 ⊆ (aSa] = A ⊆ P, whence a ∈ P . If B ⊆ P , then 〈b〉⊆ (bSb] = B ⊆ P, whence b ∈ P . The converse part is obvious. � Corollary 4.2. A bi-ideal P of a commutative ordered semigroup S with identity is prime if and only if for all a,b ∈ S from ab ∈ P it follows a ∈ P or b ∈ P . The result expressed by Corollary 4.2, suggests the following definition of prime uni-soft bi-ideals. Definition 4.5. A non-constant uni-soft bi-ideal (FS,S) of S over U is called prime (in the second sense) if for all δ ∈ P(U) and a,b ∈ S, the following condition is satisfied: if FS(axb) ⊆ δ for every x ∈ S then FS(a) ⊆ δ or FS(b) ⊆ δ. In other words, a non-constant uni-soft bi-ideal is prime if from the fact that axb ∈ eS(FS; δ) for every x ∈ S it follows a ∈ eS(FS; δ) or b ∈ eS(FS; δ). It is clear that any uni-soft bi-ideal which is prime in the first sense is prime in the second sense. The converse is not true. Example 4.1. Let S = {a,b,c} be an ordered semigroup with the following Cayley table and order relation (see [3]). · a b c a a a a b a b b c a c c ≤:= {(a,a), (b,b), (c,c), (a,b)}. Let (FS,S) be a soft set over U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} in which FS is given by FS : S −→ P(U),x 7−→   {1, 3, 5} if x = c {1, 2, 3, 5, 6} if x = b {1, 2, 3, 5, 6, 8} if x = a is a uni-soft bi-ideal over U. It is prime in the second sense but it is not prime in the first sense. Int. J. Anal. Appl. 18 (5) (2020) 825 Theorem 4.3. A non-constant soft set (FS,S) over U is a prime uni-soft bi-ideal over U in the second sense if and only if the nonempty δ-exclusive set of (FS,S) is a prime bi-ideal of S for all δ ∈ P(U). Proof. Let a uni-soft bi-ideal (FS,S) over U be prime in the second sense and let eS(FS; δ) be its arbitrary proper δ−exclusive set, i.e., ∅ 6= eS(FS; δ) 6= S (obviously eS(FS; δ) is a bi-ideal of S (Theorem 4.1)) . If (aSb] ⊆ eS(FS; δ), then FS(axb) ⊆ δ for every x ∈ S. Hence FS(a) ⊆ δ or FS(b) ⊆ δ, i.e., a ∈ eS(FS; δ) or b ∈ eS(FS; δ) which means that eS(FS; δ) is prime bi-ideal of S (Corollary 4.2). For the converse part consider a non-constant uni-soft bi-ideal (FS,S) over U. If it is not prime in the second sense, then there exist a,b ∈ S such that FS(axb) ⊆ δ for all x ∈ S, but FS(a) ⊃ δ and FS(b) ⊃ δ. Thus (aSb] ⊆ eS(FS; δ) but a /∈ eS(FS; δ) and b /∈ eS(FS; δ). Therefore eS(FS; δ) is not prime. This is a contradiction, which proves that (FS,S) is prime in the second sense. � Corollary 4.3. A soft set (χcB,S) over U is a prime uni-soft bi-ideal over U if and only if B is a prime bi-ideal of S. An ordered semigroup (S, ·,≤) is called regular, if for every a ∈ S, there exists x ∈ S such that a ≤ axa or equivalently: (1) (∀a ∈ S) (a ∈ (aSa]) and (2) (∀A ⊆ S) (A ⊆ (ASA]) . An ordered semigroup S is called left (resp., right ) regular, if for every a ∈ S, there exists x ∈ S such that a ≤ xa2 (resp., a ≤ a2x) or equivalently: (1) (∀a ∈ S) ( a ∈ (Sa2] )( resp., a ∈ (a2S] ) and (2) (∀A ⊆ S) ( A ⊆ (SA2] )( resp., A ⊆ (A2S] ) . An ordered semigroup is called completely regular if it is regular, left regular and right regular. Lemma 4.1. (cf. [4]). An ordered semigroup S is completely regular if and only if A ⊆ (A2SA2] for every A ⊆ S. Equivalently, if a ∈ (a2Sa2] for every a ∈ S. Theorem 4.4. An ordered semigroup (S, ·,≤) is completely regular if and only if for every uni-soft bi-ideal (FS,S) of S over U, we have (FS(a),S) = ( FS(a 2),S ) for every a ∈ S. Proof. (=⇒) . Assume that (FS,S) is a uni-soft bi-ideal of S over U. Let a ∈ S. Since S is completely regular, then a ∈ (a2Sa2]. That is, a ≤ a2xa2 for some x ∈ S. Since (FS,S) is a uni-soft bi-ideal of S over U, we have FS(a) ⊆ FS(a2xa2) ⊆ FS(a2) ∪FS(a2) = FS(a 2) ⊆ FS(a) ∪FS(a) = FS(a) (since (FS,S) is a uni-soft semigroup) . Thus, (FS(a),S) = ( FS(a 2),S ) . Int. J. Anal. Appl. 18 (5) (2020) 826 (⇐=) . Let A(a2) be a bi-ideal of S generated by a (a ∈ S), i.e., the set A(a2) = (a2 ∪a4 ∪a2Sa2]. By Corollary 4.1, the soft set (χcA,S) defined by χcA : S −→ P(U),x 7−→   ∅ if x ∈ A(a 2), U if x ∈ A(a2), is a uni-soft bi-ideal of S over U. By hypothesis, we have (χcA(a),S) = ( χcA(a 2),S ) . Since a2 ∈ A(a2), we have χcA(a 2) = ∅. Then χcA(a) = ∅ and hence a ∈ A(a 2) = (a2 ∪ a4 ∪ a2Sa2]. Thus a ≤ a2 or a ≤ a4 or a ≤ a2xa2 for some x ∈ S. If a ≤ a2, then a ≤ a2 = a.a ≤ a2.a2 = a.a2.a ≤ a2.a2.a2 ∈ a2Sa2. Similarly, in other cases we get a ≤ a2va2 for some v ∈ S. Consequently, a ∈ (a2Sa2] and by Lemma 4.1, S is completely regular. � 5. Semilattices of left and right simple semigroups A subsemigroup F of S is called a filter (see [5]) of S if: (1) (∀a,b ∈ S) (ab ∈ F =⇒ a ∈ F and b ∈ F) and (2) (∀a ∈ S)(∀b ∈ F)(a ≥ b =⇒ a ∈ F). For x ∈ S, we denote by N(x), the least filter of S generated x (x ∈ S). By N we mean the equivalence relation on S defined by N := {(x,y) ∈ S × S|N(x) = N(y)} (see [4]). An equivalence relation σ on S is called congruence if (a,b) ∈ σ implies (ac,bc) ∈ σ and (ca,cb) ∈ σ for every c ∈ S. A congruence σ on S is called semilattice congruence on S, if (a,a2) ∈ σ and (ab,ba) ∈ σ for each a,b ∈ S (see [5]). If σ is a semilattice congruence on S then the σ-class (x)σ of S containing x is a subsemigroup of S for every x ∈ S (see [4]). An ordered semigroup S is called a semilattice of left and right simple semigroups if there exists a semilattice congruence σ on S such that the σ-class (x)σ of S containing x is a left and right simple subsemigroup of S for every x ∈ S. Equivalently, there exists a semilattice Y and a family {Sα}α∈Y of left and right simple subsemigroups of S such that: (i) Sα ∩Sβ = ∅, ∀α,β ∈ Y, α 6= β, (ii) S = ⋃ α∈Y Sα, (iii) SαSβ ⊆ Sαβ, α,β ∈ Y. The semilattice congruences in ordered semigroups are defined exactly as in semigroups without ordered−so the two definitions are equivalent (see [4, 5]). Lemma 5.1. (cf. [2]). An ordered semigroup (S, ·,≤) is a semilattice of left and right simple semigroups if and only if for all bi-ideals A,B of S, we have (A2] = A and (AB] = (BA]. Int. J. Anal. Appl. 18 (5) (2020) 827 Theorem 5.1. Let (S, ·,≤) be an ordered semigroup. Then the following are equivalent: (1) S is regular, left and right simple. (2) Every uni-soft bi-ideal (FS,S) of S over U is a constant mapping. Proof. (1) =⇒ (2). Assume that S is regular and left and right simple. Let (FS,S) be a uni-soft bi-ideal over U. We consider the set ES := {e ∈ S|e2 ≥ e}. Since S is regular, so for every a ∈ S, there exists x ∈ S such that a ≤ axa and we have (ax)2 = (axa)x ≥ ax, hence ax ∈ ES. Thus ES 6= ∅. (A) Let t ∈ ES, we prove that (FS,S) is constant mapping on ES. That is, for every e ∈ ES, we have FS(e) = FS(t). Since t ∈ S and S is left and right simple, we have S = (tS] and S = (St]. Since e ∈ S, we have e ∈ (tS] and e ∈ (St], then e ≤ ts and e ≤ zt for some s,z ∈ S. Hence e2 ≤ t(sz)t. Since (FS,S) is a uni-soft bi-ideal over U, we have FS(e 2) ⊆ FS(t(sz)t) ⊆ FS(t) ∪FS(t) = FS(t). On the other hand, since e ∈ ES, we have e2 ≥ e and hence FS(e) ⊆ FS(e2). Thus, FS(e) ⊆ FS(e2) ⊆ FS(t). In a similar way, we can prove that FS(t) ⊆ FS(e). Thus, FS(t) = FS(e). (B) Now we prove that (FS,S) is constant mapping on S. That is, FS(t) = FS(e) for every a ∈ S. Since S is regular, so for every a ∈ S, there exists x ∈ S such that a ≤ axa. Then (ax)2 = (axa)x ≥ xa and (xa)2 = x(axa) ≥ xa. Hence ax,xa ∈ ES. Thus by (A) we have FS(ax) = FS(t) and FS(xa) = FS(t). Since (ax)a(xa) = (axa)xa ≥ axa ≥ a, we have FS(a) ⊆ FS((ax)a(xa)) ⊆ FS(ax) ∪FS(xa) = FS(xa) = FS(t). Since S is left and right simple and a ∈ S, we have S = (aS] and S = (aS]. Since t ∈ S, we have t ∈ (aS] and t ∈ (Sa], then t ≤ as1 and t ≤ z1a for some s1,z1 ∈ S. Now, t2 ≤ a(s1z1)a and (FS,S) is a uni-soft bi-ideal of S over U, we have FS(t 2) ⊆ FS(a(s1z1)a) ⊆ FS(a) ∪FS(a) = FS(a). Since t ∈ ES, we have t2 ≥ t, then FS(t) ⊆ FS(t2) ⊆ FS(a). Therefore, FS(t) = FS(a). Int. J. Anal. Appl. 18 (5) (2020) 828 (2) =⇒ (1). Let a ∈ S. Then (i) (aS]S(aS] = (aS](S](aS] ⊆ (aSSaS] ⊆ (aS], (ii) (aS](aS] ⊆ (aSaS] ⊆ (aS] and (iii) If x ≤ y such that S 3 x ≤ y ∈ (aS], then x ∈ ((aS]] = (aS]. Therefore, (aS] is a bi-ideal of S. By Corollary 4.1, the soft set ( χc (aS] ,S ) defined by χc(aS] : S −→ P(U),x 7−→   ∅ if x ∈ (aS],U if x /∈ (aS], is a uni-soft bi-ideal of S over U. By hypothesis, ( χc (aS] ,S ) is a constant mapping, that is, for every x ∈ S, there exists a subset δ ⊆ U such that χc(aS](x) = δ Let (aS] ⊂ S and t ∈ S be such that t /∈ (aS]. Then χc (aS] (t) = U. Since a2 ∈ (aS], we have χc (aS] (a2) = ∅. This is a contradiction. Thus, S = (aS]. Similarly, we can prove that S = (Sa] and therefore, S is left and right simple. Since S = (aS] = (Sa], we have a ∈ S = (aS] = (a(Sa]] = (aSa] and hence S is regular. This completes the proof. � Lemma 5.2. (cf. [2]). Let (S, ·,≤) be an ordered semigroup and B(x) and B(y) be the bi-ideals of S generated by x and y, respectively. Then B(x)SB(y) ⊆ (xSy]. Theorem 5.2. An ordered semigroup (S, ·,≤) is a semilattice of left and right simple semigroups if and only if for every uni-soft bi-ideal (FS,S) of S over U, we have (FS(a),S) = ( FS(a 2),S ) and (FS(ab),S) = (FS(ba),S) for all a,b ∈ S. Proof. (=⇒) (A) Suppose that (FS,S) be a uni-soft bi-ideal of S over U and let S be a semilattice of left and right simple semigroups. Then by hypothesis, there exists a semilattice Y and a family {Si : i ∈ Y} of left and right simple subsemigroups of S such that (i) Si ∩Sj = ∅ ∀i,j ∈ Y and i 6= j, (ii) S = ⋃ i∈Y Si, (iii) SiSj ⊆ Sij. Let a ∈ S. Since S = ⋃ i∈Y Si, then there exists i ∈ Y such that a ∈ Si. Since Si is left and right simple for every i ∈ Y . Then Si = (aSia] = {t ∈ S : t ≤ axa for some t ∈ Si}, Then a ≤ axa for some x ∈ Si. Since Si is left and right simple, we have x ≤ aya for some y ∈ Si. Then we have a ≤ axa ≤ a(aya)a = a2ya2 ∈ a2Sa2 and so a ∈ (a2Sa2]. Thus S is completely regular. Since (FS,S) is a uni-soft bi-ideal of S over U. By Theorem 4.4, we have (FS(a),S) = ( FS(a 2),S ) . Int. J. Anal. Appl. 18 (5) (2020) 829 (B) Let a,b ∈ S. Then by (A), we have FS(ab) = FS((ab)2) = FS((ab)4). On the other hand, (ab)4 = (aba)(babab) ∈ B(aba)B(babab) ⊆ (B(aba)B(babab)] = (B(babab)B(aba)] (Lemma 5.1) = (B(babab)(B(aba)2]] (Lemma 5.1) = ((B(babab)](B(aba)B(aba)]] ⊆ ((B(babab)B(aba)B(aba)]] (since (A](B] ⊆ (AB]) = (B(babab)B(aba)B(aba)] ( since ((A]] = (A]) ⊆ (B(babab)SB(aba)] ⊆ (((babab) S (aba)]] (Lemma 5.2) = ((babab) S (aba)] (since ((A]] = (A]) . Then (ab)4 ≤ (babab) z (aba) for some z ∈ S. Since (FS,S) is a uni-soft bi-ideal of S over U, we have FS ( (ab)4 ) ⊆ FS ((babab) z (aba)) = FS(ba(babza)ba) ⊆ FS(ba) ∪FS(ba) = FS(ba). Thus, FS (ab) ⊆ FS(ba). By symmetry we can prove that FS(ba) ⊆ FS (ab) . Therefore, (FS (ab) ,S) = (FS(ba),S) . (⇐=) Assume that (FS(a2),S) = (FS(a),S) and (FS (ab) ,S) = (FS(ba),S) hold for every uni-soft bi-ideal (FS,S) of S over U. Since (FS(a 2),S) = (FS(a),S) so by condition (1) and Theorem 4.4, it follows that S is completely. Let A be a bi-ideal of S and let a ∈ A. Since a ∈ S and S is completely regular, by Lemma 4.1, we have a ≤ a2xa2 = a(axa)a ∈ A(ASA)A ⊆ AAA ⊆ AA = A2. Then A ⊆ A2 and hence (A] ⊆ (A2] =⇒ A ⊆ (A2] (since A is a bi-ideal) . On the other hand, since A is a subsemigroup of S, we have A2 ⊆ A then (A2] ⊆ (A] = A. Therefore, A = (A2]. Let A and B be bi-ideals of S and let x ∈ (AB]. Then x ≤ ab for some a ∈ A and b ∈ B. We consider B(ab) = (ab∪abab∪abSab], the bi-ideal of S generated by ab (a,b ∈ S). By Corollary 4.1, the soft set ( χc (ab∪abab∪abSab],S ) defined by χc(ab∪abab∪abSab](x) =   ∅ if x ∈ (ab∪abab∪abSab]U if x /∈ (ab∪abab∪abSab] is a uni-soft bi-ideal of S over U. By hypothesis, χc (ab∪abab∪abSab](ab) = χ c (ab∪abab∪abSab](ba). Since ab ∈ (ab∪abababSab], we have χc (ab∪abab∪abSab](ab) = ∅. Thus, χ c (ab∪abab∪abSab](ba) = ∅ and ba ∈ (ab∪abab∪abSab] and we have ba ≤ ab or ba ≤ abab or ba ≤ (ab)x(ab) for some x ∈ S. If ba ≤ ab, then x ≤ ab ∈ AB and Int. J. Anal. Appl. 18 (5) (2020) 830 x ∈ (AB]. Thus (BA] ⊆ (AB]. Similarly in other cases we get (BA] ⊆ (AB]. By symmetry, we can prove that (AB] ⊆ (BA]. Therefore, (AB] = (BA] and by Lemma 5.1, S is a semilattice of left and right simple subsemigroups. This completes the proof. � 6. Regular ordered semigroups In this section, we characterize regular ordered semigroups in terms of uni-soft bi-ideals and prove that an ordered semigroup is regular if and only if for every uni-soft bi-ideal (FS,S) we have (FS♦∅S♦FS,S) = (FS,S). Lemma 6.1. Let (S, ·,≤) be an ordered semigroup and (FS,S) a soft set over U. If (FS,S) is a uni-soft semigroup over U then (FS♦FS,S)⊇̃(FS,S). Conversely, if (FS♦FS,S)⊇̃(FS,S) holds for every soft set (FS,S) over U, then (FS,S) is a uni-soft semigroup over U. Proof. (=⇒) Suppose that (FS,S) is a uni-soft semigroup over U. Let x ∈ S. If Ax = ∅, then (FS♦FS) (x) = U ⊇ FS(x). If Ax 6= ∅, then (FS♦FS) (x) = ⋂ (b,c)∈Ax {FS(b) ∪FS(c)} ⊇ ⋂ (b,c)∈Ax FS(bc) ⊇ ⋂ (b,c)∈Ax FS(x) ( since x ≤ bc) = FS(x). Hence (FS♦FS,S)⊇̃(FS,S). (⇐=) Assume that (FS♦FS,S)⊇̃(FS,S) holds for every soft set (FS,S) over U. Let x,y ∈ S, then FS(xy) ⊆ (FS♦FS)(xy) = ⋂ (b,c)∈Axy {FS(b) ∪FS(c)} ⊆ FS(x) ∪FS(y). Hence FS(xy) ⊆ FS(x) ∪FS(y) for all x,y ∈ S and (FS,S) is a uni-soft semigroup. � Proposition 6.1. Let (S, ·,≤) be an ordered semigroup and (FS,S) a uni-soft bi-ideal of S over U. Then (FS♦∅S♦FS,S)⊇̃(FS,S). Int. J. Anal. Appl. 18 (5) (2020) 831 Proof. Let (FS,S) be a uni-soft bi-ideal of S over U. Let x ∈ S. If Ax = ∅, then (FS♦∅S♦FS) (x) = U ⊇ FS(x). If Ax 6= ∅, then (FS♦∅S♦FS) (x) = ⋂ (b,c)∈Ax {(FS♦∅S) (b) ∪FS(c)} = ⋂ (b,c)∈Ax   ⋂ (b1,c1)∈Ab {FS(b1) ∪∅S(c1)}∪FS(c)   = ⋂ (b,c)∈Ax ⋂ (b1,c1)∈Ab {FS(b1) ∪∅S(c1) ∪FS(c)} = ⋂ (b,c)∈Ax ⋂ (b1,c1)∈Ab {FS(b1) ∪∅∪FS(c)} = ⋂ (b,c)∈Ax ⋂ (b1,c1)∈Ab {FS(b1) ∪FS(c)} . Since x ≤ bc and b ≤ b1c1, we have x ≤ bc ≤ (b1c1)c and (FS,S) is a uni-soft bi-ideal of S over U, we have FS(x) ⊆ FS((b1c1)c) ⊆ FS(b1) ∪FS(c). Hence, (FS♦∅S♦FS) (x) = ⋂ (b,c)∈Ax ⋂ (b1,c1)∈Ab {FS(b1) ∪FS(c)} ⊇ ⋂ (b,c)∈Ax ⋂ (b1,c1)∈Ab FS(x) = FS(x). There- fore, (FS♦∅S♦FS,S) ⊇ (FS,S) . � Lemma 6.2. (cf. [15]). Let (χcA,S) and (χ c B,S) be soft sets over U where A and B are nonempty subsets of S. Then the following properties hold: (1) (χcA,S)∪̃(χ c B,S) = (χ c A∩B,S) . (2) (χcA,S) ♦ (χ c B,S) = ( χc (AB] ,S ) . Lemma 6.3. (cf. [18]). Let (S, ·,≤) be an ordered semigroup. Then the following are equivalent: (1) S is regular. (2) B = (BSB] for every bi-ideal B of S. (3) B(a) = (B(a)SB(a)] for every a ∈ S. Theorem 6.1. An ordered semigroup (S, ·,≤) is regular if and only if for every uni-soft bi-ideal (FS,S) over U, we have (FS,S) = (FS♦∅S♦FS,S) . Int. J. Anal. Appl. 18 (5) (2020) 832 Proof. Suppose that S is regular. Let (FS,S) be a uni-soft bi-ideal of S over U and let a ∈ S. Since S is regular, there exists x ∈ S such that a ≤ axa = a(xa). Then (a,xa) ∈ Aa and we have (FS♦∅S♦FS) (a) = ⋂ (b,c)∈Aa {FS(b) ∪ (∅S♦FS) (c)} ⊆ FS(a) ∪ (∅S♦FS) (xa) = FS(a) ∪ ⋂ (b1,c1)∈Axa {∅S(b1) ∪FS(c1)} ⊆ FS(a) ∪∅S(x) ∪FS(a) = FS(a) ∪∅ = FS(a). Then (FS♦∅S♦FS,S)⊆̃(FS,S). On the other hand, by Proposition 6.1, we have (FS♦∅S♦FS,S)⊇̃(FS,S), therefore, (FS♦∅S♦FS,S) = (FS,S). Conversely, assume that (FS♦∅S♦FS,S) = (FS,S) holds for every uni-soft bi-ideal (FS,S) over U. To prove that S is regular, by Lemma 6.2, it is enough to prove that B(a) = (B(a)SB(a)] ∀a ∈ S. Let y ∈ B(a). Since B(a) is the bi-ideal of S generated by a (a ∈ S). By Corollary 4.1, ( χc B(a) ,S ) is a uni-soft bi-ideal of S over U. By hypothesis, we have( χcB(a)♦∅S♦χ c B(a) ) (y) = χcB(a)(y). Since y ∈ B(a), we have χc B(a) (y) = ∅ and hence ( χc B(a) ♦∅S♦χcB(a) ) (y) = ∅. But by Lemma 6.2, we have χc B(a) ♦∅S♦χcB(a) = χ c (B(a)SB(a)] . Thus, χc (B(a)SB(a)] (y) = ∅ and y ∈ (B(a)SB(a)]. Therefore, B(a) ⊆ (B(a)SB(a)]. On the other hand, since B(a) is the bi-ideal of S, we have (B(a)SB(a)] ⊆ (B(a)] = B(a). Thus B(a) = (B(a)SB(a)] and S is regular (Lemma 6.3). � Lemma 6.4. Let (S, ·,≤) be an ordered semigroup. Let (FS,S) and (GS,S) be uni-soft bi-ideals of S over U. Then the soft product (FS♦GS,S) of (FS,S) and (GS,S) is again a uni-soft bi-ideal of S over U. Proof. Let (FS,S) and (GS,S) be uni-soft bi-ideals of S over U. Let x,y,z ∈ S. Then (FS♦GS) (x) ∪ (FS♦GS) (z) =   ⋂ (p,q)∈Ax {FS(p) ∪GS(q)}∪ ⋂ (p1,q1)∈Az {FS(p1) ∪GS(q1)}   = ⋂ (p,q)∈Ax ⋂ (p1,q1)∈Az [{FS(p) ∪GS(q)}∪{FS(p1) ∪GS(q1)}] = ⋂ (p,q)∈Ax ⋂ (p1,q1)∈Az [FS(p) ∪GS(q) ∪FS(p1) ∪GS(q1)] ⊇ ⋂ (p,q)∈Ax ⋂ (p1,q1)∈Az [FS(p) ∪GS(q1)] . Int. J. Anal. Appl. 18 (5) (2020) 833 Since x ≤ pq and z ≤ p1q1, hence xyz ≤ (pq)y(p1q1) = p(qyp1)q1 and (p(qy)p1,q1) ∈ Axyz. Thus, Axyz 6= ∅ and we have (FS♦GS) (x) ∪ (FS♦GS) (z) ⊇ ⋂ (p,q)∈Ax ⋂ (p1,q1)∈Az [FS(p) ∪FS(p1) ∪GS(q1)] ⊇ ⋂ (p(qy)p1,q1)∈Axyz [FS(p(qyp1)q1) ∪GS(q1)] = ⋂ (p(qy)p1,q1)∈Axyz (FS♦GS) (xyz) = (FS♦GS) (xyz). Thus, (FS♦GS) (xyz) ⊆ (FS♦GS) (x) ∪ (FS♦GS) (z) . Similarly, we can prove that (FS♦GS) (xy) ⊆ (FS♦GS) (x) ∪ (FS♦GS) (y) . Let x,y ∈ S be such that x ≤ y. Then (FS♦GS) (x) ⊆ (FS♦GS) (y) . In- fact, if (p,q) ∈ Ay, then y ≤ pq and we have x ≤ y ≤ pq it follows that (p,q) ∈ Ax and hence Ay ⊆ Ax. If Ax = ∅, then Ay = ∅ and we have (FS♦GS) (y) = U ⊇ (FS♦GS) (x). If Ax 6= ∅, then Ay 6= ∅ and we have (FS♦GS) (y) = ⋂ (p,q)∈Ay {FS(p) ∪GS(q)} ⊇ ⋂ (p,q)∈Ax {FS(p) ∪GS(q)} = (FS♦GS) (x). Hence in both the cases, we have (FS♦GS) (x) ⊆ (FS♦GS) (y) for all x,y ∈ S with x ≤ y. This completes the proof. � 7. Regular and intra-regular ordered semigroups In this section, we characterize regular and intra-regular ordered semigroups in terms of uni-soft bi-ideals. Lemma 7.1. Let S be an ordered semigroup and (FS,S) a uni-soft bi-ideal of S over U. Then (FS♦FS,S)⊇̃(FS,S). Proof. Let (FS,S) be a uni-soft bi-ideal of S over U and let a ∈ S. If Aa = ∅, then (FS♦FS) (a) = U ⊇ FS(a). If Aa 6= ∅, then (FS♦FS) (a) = ⋂ (p,q)∈Aa {FS(p) ∪FS(q)} ⊇ ⋂ (p,q)∈Aa FS(pq) ⊇ ⋂ (p,q)∈Aa FS(a) (since a ≤ pq =⇒ FS(a) ⊆ FS(pq)) = FS(a). Therefore, (FS♦FS,S)⊇̃(FS,S). � Int. J. Anal. Appl. 18 (5) (2020) 834 7.1. Lemma. Let (FS,S) and (GS,S) be soft subsets of an ordered semigroup S over U. Then (FS♦GS,S)⊇̃(∅S♦GS,S) ( resp., (FS♦GS,S)⊇̃(FS♦∅S) ) . Proof. Straightforward. � Lemma 7.2. Let (FS,S) and (GS,S) be uni-soft bi-ideals of an ordered semigroup S over U. Then (FS∪̃GS,S) is a uni-soft bi-ideal of S over U. Proof. Straightforward. � Theorem 7.1. Let S be an ordered semigroup. Then the following are equivalent: (1) S is both regular and intra-regular. (2) (FS♦FS,S) = (FS,S) for every uni-soft bi-ideal (FS,S) over U. (3) (FS∪̃GS,S) = ((FS♦GS)∪̃(GS♦FS),S) for all uni-soft bi-ideals (FS,S) and (GS,S) of S over U. Proof. (1)=⇒(2). Let (FS,S) be a uni-soft bi-ideal of S over U and let a ∈ S. Since S is regular and intra-regular, there exist x,y,z ∈ S such that a ≤ axa ≤ axaxa and a ≤ ya2z. Then a ≤ axaxa ≤ ax(yaz)xa = (axya)(azxa), and hence (axya,azxa) ∈ Aa. Then (FS♦FS) (a) = ⋂ (p,q)∈Aa {FS(p) ∪FS(q)} ⊆ FS(axya) ∪FS(azxa) ⊆{FS(a) ∪FS(a)}∪{FS(a) ∪FS(a)} = FS(a). and hence (FS♦FS,S)⊆̃(FS,S). On the other hand, by Lemma 7.1, we have (FS♦FS,S)⊇̃(FS,S). There- fore, (FS♦FS,S)⊇̃(FS,S). (2)=⇒(3). Let (FS,S) and (GS,S) be uni-soft bi-ideals of S over U. Then ( FS∪̃GS,S ) is a uni-soft bi-ideal of S over U (Lemma 7.2). By (2), we have ( FS∪̃GS,S ) = (( FS∪̃GS,S ) ♦ ( FS∪̃GS,S ) ,S ) ⊇̃(FS♦GS,S) . Similarly, we can prove that ( FS∪̃GS,S ) ⊇̃(GS♦FS,S) . Therefore, ( FS∪̃GS,S ) ⊇̃(FS♦GS,S) ∪̃(GS♦FS,S) . Int. J. Anal. Appl. 18 (5) (2020) 835 On the other hand, since (FS♦GS,S) and (GS♦FS,S) are uni-soft bi-ideals of S over U. Again by Lemma 7.2, (FS♦GS,S) ∪̃(GS♦FS,S) is a uni-soft bi-ideal of S over U. By (2), we have ( (FS♦GS,S)∪̃(GS♦FS,S) ,S ) =   ((FS♦GS,S) ∪̃(GS♦FS,S))♦( (FS♦GS,S) ∪̃(GS♦FS,S) ) ,S   ⊇̃((FS♦GS,S) ♦ (GS♦FS,S) ,S) = (FS♦ (GS♦GS) ♦FS,S) = (FS♦GS♦FS,S) (as (GS♦GS,S) = (GS,S) by (1) above) ⊇̃(FS♦∅S♦FS,S) = (FS,S) (as (FS♦∅S♦FS,S) = (FS,S)) . (7.1) Similarly, we can prove that ( (FS♦GS,S) ∪̃(GS♦FS,S) ,S ) ⊇̃(GS,S). Therefore, ( (FS♦GS,S) ∪̃(GS♦FS,S) ,S ) ⊇̃(FS∪̃GS,S). Consequently, we have ( (FS♦GS,S) ∪̃(GS♦FS,S) ,S ) = (FS∪̃GS,S). (3)=⇒(1). To prove that S is both regular and intra-regular, it is enough to prove that P ∩Q = (PQ] ∩ (QP] for every bi-ideal P and Q of S. Let b ∈ P ∩Q. By Corollary 4.1, (χcP ,S) and ( χcQ,S ) are uni-soft bi-ideals of S over U. By (3), we have ( χcP∪̃χ c Q,S ) (b) = (( χcP ♦χ c Q ) ∪̃ ( χcQ♦χ c P ) ,S ) (b). By Lemma 6.2, ( χcP ♦χ c Q ) ∪̃ ( χcQ♦χ c P ) = χc (PQ]∩(QP] and χ c P∪̃χ c Q = χ c P∩Q, hence we have χ c (PQ]∩(QP](b) = χcP∩Q(b) = ∅. Thus, χ c (PQ]∩(QP](b) = ∅ and b ∈ (PQ] ∩ (QP] =⇒ P ∩Q ⊆ (PQ] ∩ (QP]. On the other hand, if b ∈ (PQ] ∩ (QP], then ∅ = ( χc(PQ]∩(QP],S ) (b) = ( χc(PQ]∪̃χ c (QP] ) (b) = (( χcP ♦χ c Q ) ∪̃ ( χcQ♦χ c P )) (b) = ( χcP∪̃χ c Q,S ) (b) (by (3)) = ( χcP∩Q,S ) (b) . Hence, b ∈ P ∩Q and (PQ] ∩ (QP] ⊆ PQ. Therefore, P ∩Q = (PQ] ∩ (QP] and S is both regular and intra-regular. � Int. J. Anal. Appl. 18 (5) (2020) 836 8. Conclusion In the present paper, we introduced the notion of uni-soft type of bi-ideals of ordered semigroups. Fur- thermore The notion of a uni-soft bi-ideal is introduced and their related properties is provided. The concept of δ−exclusive set is given and the relations between uni-soft bi-ideals and δ−exclusive set are discussed. 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