EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1342-1358 ISSN 1307-5543 – ejpam.com Published by New York Business Global Intuitionistic fuzzy ordered subalgebras in ordered BCI-algebras Eun Hwan Roh1,∗, Eunsuk Yang2, Young Bae Jun3 1 Department of Mathematics Education, Chinju National University of Education, Jinju 52673, Korea 2 Department of Philosophy, Jeonbuk National University, Jeonju 54896, Korea 3 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea Abstract. In this paper, we apply the concept of an intuitionistic fuzzy set to ordered subalge- bras in ordered BCI-algebras in the sense of intuitionistic fuzzy point. We introduce the notion of an intuitionistic fuzzy (ordered) subalgebra in ordered BCI-algebras, and investigate some re- lated properties. We provide relations between an intuitionistic fuzzy ordered subalgebra and an intuitionistic fuzzy subalgebra. We give characterizations of an intuitionistic fuzzy (ordered) sub- algebra. Finally, we provide relations between a q(t,s)-level set of intuitionistic fuzzy set and an intuitionistic fuzzy ordered subalgebra. 2020 Mathematics Subject Classifications: 03G25, 06F35, 08A72 Key Words and Phrases: Intuitionistic fuzzy point, intuitionistic fuzzy (ordered) subalgebra, q(t,s)-level set 1. Introduction The speed of development of mathematics cannot be said to be fast, but it is clear that it is changing and developing through our efforts. There are many examples showing that progress is being made, but so is the appearance of BCI-algebra, which generalizes groups, and ordered BCI-algebra, which generalizes BCI-algebra. However, progress is not fast. BCI-algebra was introduced by Y. Imai and K. Iséki [11] in 1996 (see [10]), and ordered BCI-algebra was introduced by E. Yang, E. H. Roh and Y. B. Jun [8] in 2023. In [8], they introduced the notions of ordered BCI-algebras and (ordered) subalgebras and (ordered) filters of ordered BCI-algebras, and related properties are investigated. Moreover, some specific filters are introduced and their relations are discussed. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4832 Email addresses: ehroh9988@gmail.com (E. H. Roh), eunsyang@jbnu.ac.kr (E. Yang), skywine@gmail.com (Y. B. Jun) https://www.ejpam.com 1342 © 2023 EJPAM All rights reserved. E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1343 L. A. Zadeh [19] introduced the degrees of membership and truth (t) in 1965 and defined the fuzzy set. As is so well known, a fuzzy set is a mathematical concept in the field of fuzzy logic that represents a set where elements have degrees of membership. Many mathematicians have conducted research to connect the algebraic structure with the fuzzy concept and obtained meaningful results(see [6, 12–18]). The concepts of the fuzzification of ordered subalgebras in ordered BCI-algebras were introduced, and related properties were investigated in [7]. After introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets (IFS) introduced by K. Atanassov [1–4] is one among them. IFS are a mathematical concept in the field of fuzzy set theory. IFS are a generalization of traditional fuzzy sets, allowing for partial membership and uncertainty. In an IFS, an element may have a degree of membership, but also a degree of non-membership, which represents the uncertainty or the lack of information about its membership. These two degrees of membership and non-membership are used to represent the degree of belief and disbelief, respectively, in the membership of an element in the set. Many mathematicians have conducted research to connect the algebraic structure with the concept of intuitionistic fuzzy sets and obtained meaningful results (see [1–5, 9]). In this paper, we apply the concept of an IFS to ordered subalgebras in ordered BCI-algebras. We introduce the notion of an intuitionistic fuzzy (ordered) subalgebra in ordered BCI-algebras, and investigate some related properties. We provide relations between an intuitionistic fuzzy ordered subalgebra and an intuitionistic fuzzy subalgebra. We give characterizations of an intuitionistic fuzzy (ordered) subalgebra. 2. Preliminaries Definition 1 ([8]). Let X be a set with a binary operation “ → ”, a constant “e” and a binary relation “ ≤X ”. Then X := (X, →, e, ≤X) is called an ordered BCI-algebra (briefly, OBCI-algebra) if it satisfies the following conditions: (∀x, y, z ∈ X)(e ≤X (x → y) → ((y → z) → (x → z))), (1) (∀x, y ∈ X)(e ≤X x → ((x → y) → y)), (2) (∀x ∈ X)(e ≤X x → x), (3) (∀x, y ∈ X)(e ≤X x → y, e ≤X y → x ⇒ x = y), (4) (∀x, y ∈ X)(x ≤X y ⇔ e ≤X x → y), (5) (∀x, y ∈ X)(e ≤X x, x ≤X y ⇒ e ≤X y). (6) Proposition 1 ([8]). Every OBCI-algebra X := (X, →, e, ≤X) satisfies: (∀x ∈ X)(e → x = x). (7) (∀x, y, z ∈ X)(z → (y → x) = y → (z → x)). (8) (∀x, y, z ∈ X)(e ≤X x → y ⇒ e ≤X (y → z) → (x → z)). (9) (∀x, y, z ∈ X)(e ≤X x → y, e ≤X y → z ⇒ e ≤X x → z). (10) E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1344 (∀x, y, z ∈ X)(e ≤X (z → (y → x)) → (y → (z → x))). (11) (∀x, y, z ∈ X)(e ≤X z → (y → x) ⇒ e ≤X y → (z → x)). (12) (∀x, y ∈ X)(((x → y) → y) → y = x → y). (13) (∀x ∈ X)((x → x) → x = x). (14) (∀x, y, z ∈ X)(e ≤X (y → z) → ((x → y) → (x → z))). (15) (∀x, y, z ∈ X)(e ≤X x → y ⇒ e ≤X (z → x) → (z → y)). (16) Definition 2 ([8]). A subset A of X is called • a subalgebra of an OBCI-algebra X := (X, →, e, ≤X) if it satisfies: (∀x, y ∈ X)(x, y ∈ A ⇒ x → y ∈ A). (17) • an ordered subalgebra of an OBCI-algebra X := (X, →, e, ≤X) if it satisfies: (∀x, y ∈ X)(x, y ∈ A, e ≤X x, e ≤X y ⇒ x → y ∈ A). (18) A function f : X → [0, 1] is called a fuzzy set in a set X, and the complement of f is denoted by ¬f , and is given as follows: ¬f : X → [0, 1], x 7→ 1− fI(x). For every fuzzy sets f and g in X, we say f ≤ g if f(x) ≤ g(x) fo all x ∈ X. A fuzzy set f in a set X of the form f(b) := { t ∈ (0, 1] if b = a, 0 if b ̸= a, is said to be a fuzzy point with support a and value t and is denoted by at. Definition 3 ([7]). A fuzzy set f in X is called • a fuzzy subalgebra of an OBCI-algebra X := (X, →, e, ≤X) if it satisfies: (∀x, y ∈ X)(∀t, s ∈ (0, 1]) ( xt ∈ f, ys ∈ f ⇒ ⟨(x → y)min{t,s}⟩ ∈ f. ) . (19) • a fuzzy ordered subalgebra of an OBCI-algebra X := (X, →, e, ≤X) if it satisfies: (∀x, y ∈ X)(e ≤X x, e ≤X y ⇒ f(x → y) ≥ min{f(x), f(y)}). (20) The concept of intuitionistic fuzzy set was introduced by Atanassov (see [1, 2, 4]) as follows: An intuitionistic fuzzy set on a set X is an expression I given by I := {⟨x, fI , gI⟩ | x ∈ X} E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1345 where fI and gI are fuzzy sets in X such that 0 ≤ fI(x) + gI(x) ≤ 1 for all x ∈ X. Every fuzzy set f in a set X is obviously an intuitionistic fuzzy set having the form {⟨x, f,¬f⟩ | x ∈ X} (see [2]). The notion of intuitionistic fuzzy point is considered in the paper [5] as follows: Given elements b ∈ X and (t, s) ∈ (0, 1]× [0, 1) satisfying t+ s ≤ 1, the intuitionistic fuzzy set b(t,s) := {⟨x, bt,¬b1−s⟩ | x ∈ X} (21) is called an intuitionistic fuzzy point in X. Let I := {⟨x, fI , gI⟩ | x ∈ X} be an intuitionistic fuzzy set in X. An intuitionistic fuzzy point b(t,s) := {⟨x, bt,¬b1−s⟩ | x ∈ X} is said to be • contained in I := {⟨x, fI , gI⟩ | x ∈ X}, denoted by b(t,s) ∈ I, if bt ≤ fI and ¬b1−s ≥ gI , or equivalently, fI(b) ≥ t and gI(b) ≤ s. • quasi-coincident with I := {⟨x, fI , gI⟩ | x ∈ X}, denoted by b(t,s) q I, if fI(b)+ t > 1 and gI(b) + s < 1. If b(t,s) β I is not established for β ∈ {∈, q}, it is denoted by b(t,s) β I. The set I∈ (t,s) := {b ∈ X | b(t,s) ∈ I} is called the ∈(t,s)-level set of I. It is clear that I∈ (t,s) = U(fI , t) ∩ L(gI , s) where U(fI , t) := {a ∈ X | fI(a) ≥ t} and L(gI , s) := {a ∈ X | gI(a) ≤ s}, which are called the upper t-level set and the lower s-level set of I := {⟨x, fI , gI⟩ | x ∈ X}. 3. Intuitionistic fuzzy (ordered) subalgebras In what follows, let X := (X, →, e, ≤X) denote an OBCI-algebra unless otherwise specified. Definition 4. An intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} is called • an intuitionistic fuzzy subalgebra of X := (X, →, e, ≤X) if it satisfies: (∀x, y ∈ X) ( fI(x → y) ≥ min{fI(x), fI(y)} gI(x → y) ≤ max{gI(x), gI(y)} ) . (22) • an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X) if it satisfies: (∀x, y ∈ X) ( e ≤X x, e ≤X y, x(t1,s1) ∈ I, y(t2,s2) ∈ I ⇒ (x → y)(min{t1,t2},max{s1,s2}) ∈ I ) (23) for all (t1, s1), (t2, s2) ∈ (0, 1]× [0, 1). E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1346 Table 1: Cayley table for the binary operation “→” → 1 e ∂ 0 1 1 0 0 0 e 1 e ∂ 0 ∂ 1 ∂ e 0 0 1 1 1 1 Example 1. Let X = {1, e, ∂, 0} be a set, where 1 and 0 are the greatest element and the least element of X, respectively. Define a binary operation “ → ” on X by Table 1 Let ≤e:= {(0, 0), (e, e), (∂, ∂), (1, 1), (0, e), (0, ∂), (e, 1), (∂, 1)}. Then X := (X, →, e, ≤X) is an OBCI-algebra (see [8]). Define an intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X as follows: fI : X → [0, 1], x 7→ { 0.68 if x ∈ {1, e, 0}, 0.24 otherwise, and gI : X → [0, 1], x 7→ { 0.31 if x ∈ {1, e, 0}, 0.59 otherwise. It is routine to verify that I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy subalgebra of X := (X, →, e, ≤X). Also, if we define an intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X by fI : X → [0, 1], x 7→ { 0.63 if x ∈ {e, 0}, 0.27 otherwise, and gI : X → [0, 1], x 7→ { 0.29 if x ∈ {e, 0}, 0.62 otherwise, then I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). It is clear that every intuitionistic fuzzy subalgebra is an intuitionistic fuzzy ordered subalgebra, but the converse is not true as seen in the example below. Example 2. Let X = {0, 1, 34 , 1 2 , 1 4} be a set with a binary operation “ → ” given by Table 2 and let ≤e be the natural order in X. Then X := (X, →, e, ≤X), where e = 3 4 , is an OBCI-algebra (see [8]). Define an intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X as follows: fI : X → [0, 1], x 7→ { 0.78 if x ∈ {3 4 , 0}, 0.23 otherwise, E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1347 Table 2: Cayley table for the binary operation “→” → 1 3 4 1 2 1 4 0 1 1 0 0 0 0 3 4 1 3 4 1 2 1 4 0 1 2 1 3 4 3 4 1 2 0 1 4 1 3 4 3 4 3 4 0 0 1 1 1 1 1 and gI : X → [0, 1], x 7→ { 0.12 if x ∈ {3 4 , 0}, 0.61 otherwise. It is routine to verify that f is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). But it is not an intuitionistic fuzzy subalgebra of X := (X, →, e, ≤X) since fI(0 → 3 4) = fI(1) = 0.23 ≱ 0.78 = min{fI(0), fI(34)} and/or gI(0 → 3 4) = fI(1) = 0.61 ≰ 0.12 = max{gI(0), gI(34)}. We provide a condition in which the intuitionistic fuzzy ordered subalgebra becomes the intuitionistic fuzzy subalgebra. Theorem 1. Let I := {⟨x, fI , gI⟩ | x ∈ X} be an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). If its ∈(t,s)-level set I∈ (t,s) satisfies e ≤X x for all x ∈ I∈ (t,s), then I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy subalgebra of X := (X, →, e, ≤X). Proof. Straightforward. Theorem 2. An intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X is an intuitionistic fuzzy subalgebra of X := (X, →, e, ≤X) if and only if it satisfies: x(t1,s1) ∈ I, y(t2,s2) ∈ I ⇒ (x → y)(min{t1,t2},max{s1,s2}) ∈ I (24) for all x, y ∈ X and (ti, si) ∈ (0, 1]× [0, 1) for i = 1, 2. Proof. Assume that I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy subalgebra of X := (X, →, e, ≤X). Let x, y ∈ X be such that x(t1,s1) ∈ I and y(t2,s2) ∈ I for all (ti, si) ∈ (0, 1]×[0, 1) for i = 1, 2. Then fI(x) ≥ t1, fI(y) ≥ t2, gI(x) ≤ s1, and gI(y) ≤ s2. It follows from (22) that fI(x → y) ≥ min{fI(x), fI(y)} ≥ min{t1, t2} and gI(x → y) ≤ max{gI(x), gI(y)} ≤ max{s1, s2}. Hence (x → y)(min{t1,t2},max{s1,s2}) ∈ I. E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1348 Conversely, suppose that I := {⟨x, fI , gI⟩ | x ∈ X} satisfies (24) for all x, y ∈ X and (ti, si) ∈ (0, 1] × [0, 1) for i = 1, 2. then fI(a → b) < min{fI(a), fI(b)} or gI(a → b) > max{gI(a), gI(b)} for some a, b ∈ X. Taking t := min{fI(a), fI(b)} and s := max{gI(a), gI(b)} induces a(t,s) ∈ I, and b(t,s) ∈ I. It follows from (24) that (a → b)(t,s) = (a → b)(min{t,t},max{s,s}) ∈ I. But fI(a → b) < t or gI(a → b) > s imply that (a → b)(t,s) ∈I, a contradiction. Therefore I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy subalgebra of X := (X, →, e, ≤X). Theorem 3. An intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X) if and only if it satisfies: (∀x, y ∈ X)  e ≤X x, e ≤X y ⇒ { fI(x → y) ≥ min{fI(x), fI(y)} gI(x → y) ≤ max{gI(x), gI(y)}  . (25) Proof. Assume that I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). If the assertion (25) is not valid, then fI(a → b) < t < min{fI(a), fI(b)} or gI(a → b) > s > max{gI(a), gI(b)} for some (t, s) ∈ (0, 1)×(0, 1) and a, b ∈ X with e ≤X a, e ≤X b. Then t + s ≤ 1, at ≤ fI , bt ≤ fI , ¬a1−s ≥ gI , and ¬b1−s ≥ gI . Hence a(t,s) ∈ I and b(t,s) ∈ I. It follows from (23) that (a → b)(t,s) = (a → b)(min{t,t},max{s,s}) ∈ I. Thus (a → b)t ≤ fI and ¬(a → b)s ≥ gI , that is, fI(a → b) ≥ t and gI(a → b) ≤ s. This is a contradiction, and so (25) is valid. Conversely, suppose that I := {⟨x, fI , gI⟩ | x ∈ X} satisfies (25). Let x, y ∈ X be such that e ≤X x, e ≤X y, x(t1,s1) ∈ I and y(t2,s2) ∈ I for every (t1, s1), (t2, s2) ∈ (0, 1]× [0, 1). Then fI(x) ≥ t1, gI(x) ≤ s1, fI(y) ≥ t2, and gI(y) ≤ s2. It follows from (25) that fI(x → y) ≥ min{fI(x), fI(y)} ≥ min{t1, t2} and gI(x → y) ≤ max{gI(x), gI(y)} ≤ max{s1, s2}. Hence (x → y)(min{t1,t2},max{s1,s2}) ∈ I. Therefore I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). Lemma 1. An intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X) if and only if fI and gcI are fuzzy ordered subalgebras of X := (X, →, e, ≤X), where gcI is defined by gcI(x) = 1−gI(x) for all x ∈ X. Proof. Let I := {⟨x, fI , gI⟩ | x ∈ X} be an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). Obviously, fI is a fuzzy ordered subalgebra of X := (X, →, e, ≤X) by Theorem 3. Let x, y ∈ X be such that e ≤X x and e ≤X y. Using Theorem 3 induces gcI(x → y) = 1− gI(x → y) ≥ 1−max{gI(x), gI(y)} = min{1− gI(x), 1− gI(y)} = min{gcI(x), gcI(y)}. E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1349 Hence gcI is a fuzzy ordered subalgebra of X := (X, →, e, ≤X). Conversely, suppose that fI and gcI are fuzzy ordered subalgebras of X := (X, →, e, ≤X). For every x, y ∈ X with e ≤X x and e ≤X y, we have fI(x → y) ≥ min{fI(x), fI(y)} and 1− gI(x → y) = gcI(x → y) ≥ min{gcI(x), gcI(y)} = min{1− gI(x), 1− gI(y)} = 1−max{gI(x), gI(y)}, that is, gI(x → y) ≤ max{gI(x), gI(y)}. Therefore I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X) by Theorem 3. Theorem 4. An intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X) if and only if □I := {⟨x, fI , f c I⟩ | x ∈ X} and ♢I := {⟨x, gcI , gI⟩ | x ∈ X} are intuitionistic fuzzy ordered subalgebras of X := (X, →, e, ≤X) Proof. It is straightforward by Lemma 1. Theorem 5. If I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X), then the set I(0,1) := {x ∈ X | fI(x) > 0, gI(x) < 1}, which is called the intuitionistic support of I, is an ordered subalgebra of X := (X, →, e, ≤X). Proof. Let x, y ∈ X be such that e ≤X x, e ≤X y and x, y ∈ I(0,1). Then fI(x) > 0, gI(x) < 1, fI(y) > 0, and gI(y) < 1. Using Theorem 3, we have fI(x → y) ≥ min{fI(x), fI(y)} > 0 and gI(x → y) ≤ max{gI(x), gI(y)} < 1. Hence x → y ∈ I(0,1), and so I(0,1) is an ordered subalgebra of X := (X, →, e, ≤X). Theorem 6. If an intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X satisfies: (∀x, y ∈ X) ( e ≤X x, e ≤X y, x(t1,s1) ∈ I, y(t2,s2) ∈ I ⇒ (x → y)(min{t1,t2},max{s1,s2}) q I ) (26) where (ti, si) ∈ (0, 1]× [0, 1) for i = 1, 2, then its intuitionistic support I(0,1) is an ordered subalgebra of X := (X, →, e, ≤X). E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1350 Proof. Let x, y ∈ X be such that e ≤X x, e ≤X y and x, y ∈ I(0,1). Then (fI(x), gI(x)), (fI(y), gI(y)) ∈ (0, 1]× [0, 1). Note that x(fI(x),gI(x)) ∈ I and y(fI(y),gI(y)) ∈ I. Using (26), we get (x → y)(min{fI(x),fI(y)},max{gI(x),gI(y)}) q I. Hence fI(x → y) > 1−min{fI(x), fI(y)} ≥ 0 and gI(x → y) < 1−max{gI(x), gI(y)} ≤ 1. This shows that x → y ∈ I(0,1), and therefore I(0,1) is an ordered subalgebra of X := (X, →, e, ≤X). Theorem 7. If an intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X satisfies: (∀x, y ∈ X) ( e ≤X x, e ≤X y, x(t1,s1) q I, y(t2,s2) q I ⇒ (x → y)(min{t1,t2},max{s1,s2}) ∈ I ) (27) for all (ti, si) ∈ (0, 1]× [0, 1) for i = 1, 2, then its intuitionistic support I(0,1) is an ordered subalgebra of X := (X, →, e, ≤X). Proof. Let x, y ∈ X be such that e ≤X x, e ≤X y and x, y ∈ I(0,1). Then (fI(x), gI(x)), (fI(y), gI(y)) ∈ (0, 1] × [0, 1), and so fI(x) + 1 > 1, gI(x) + 0 < 1, fI(y) + 1 > 1, and gI(y) + 0 < 1. This shows that x(1,0) q I and y(1,0) q I. It follows from (27) that (x → y)(1,0) ∈ I. Hence fI(x → y) = 1 > 0 and gI(x → y) = 0 < 1, which imply that x → y ∈ I(0,1). Hence I(0,1) is an ordered subalgebra of X := (X, →, e, ≤X). Theorem 8. If an intuitionistic fuzzy set If I := {⟨x, fI , gI⟩ | x ∈ X} in X satisfies: (∀x, y ∈ X) ( e ≤X x, e ≤X y, x(t1,s1) q I, y(t2,s2) q I ⇒ (x → y)(min{t1,t2},max{s1,s2}) q I ) (28) for all (ti, si) ∈ (0, 1]× [0, 1) for i = 1, 2, then its intuitionistic support I(0,1) is an ordered subalgebra of X := (X, →, e, ≤X). Proof. Let x, y ∈ X be such that e ≤X x, e ≤X y and x, y ∈ I(0,1). Then (fI(x), gI(x)), (fI(y), gI(y)) ∈ (0, 1] × [0, 1), and so fI(x) + 1 > 1, gI(x) + 0 < 1, fI(y) + 1 > 1, and gI(y) + 0 < 1. This shows that x(1,0) q I and y(1,0) q I. Using (28), we have (x → y)(1,0) q I. If fI(x → y) = 0 or gI(x → y) = 1, then fI(x → y) + 1 = 1 or gI(x → y) + 1 = 2, i.e., (x → y)(1,0) q I, a contradiction. Hence fI(x → y) > 0 and gI(x → y) < 1, that is, x → y ∈ I(0,1). Therefore I(0,1) is an ordered subalgebra of X := (X, →, e, ≤X). E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1351 Theorem 9. Given a nonempty subset B of X, let IB := {⟨x, fB I , gBI ⟩ | x ∈ X} be an intuitionistic fuzzy set in X in which fB I and gBI are given as follows: fB I : X → [0, 1], x 7→ { t1 if x ∈ B, t2 otherwise, and gBI : X → [0, 1], x 7→ { s1 if x ∈ B, s2 otherwise, where t1 > t2 in (0, 1] and s1 < s2 in [0, 1). Then IB := {⟨x, fB I , gBI ⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X) if and only if B is an ordered subalgebra of X := (X, →, e, ≤X). Proof. Assume that IB := {⟨x, fB I , gBI ⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). Let x, y ∈ X be such that x, y ∈ B, e ≤e x and e ≤e y. Using Theorem 3, we have fB I (x → y) ≥ min{fB I (x), fB I (y)} = t1 and gBI (x → y) ≤ max{gBI (x), gBI (y)} = s1. Hence fB I (x → y) = t1 and gBI (x → y) = s1, and so x → y ∈ B. Therefore B is an ordered subalgebra of X := (X, →, e, ≤X). Conversely, suppose that B is an ordered subalgebra of X := (X, →, e, ≤X). Let x, y ∈ X be such that e ≤e x and e ≤e y. If x /∈ B (or y /∈ B), then fB I (x) = t2 (or fB I (y) = t2) and gBI (x) = s2 (or gBI (y) = s2). Thus fB I (x → y) ≥ t2 = min{fB I (x), fB I (y)} and gBI (x → y) ≤ s2 = max{gBI (x), gBI (y)}. If x ∈ B and y ∈ B, then x → y ∈ B since B is an ordered subalgebra of X := (X, →, e, ≤X). Thus fB I (x → y) = t1 = min{fB I (x), fB I (y)} and gBI (x → y) = s1 = max{gBI (x), gBI (y)}. It follows from Theorem 3 that IB := {⟨x, fB I , gBI ⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). Theorem 10. An intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X is an intuition- istic fuzzy ordered subalgebra of X := (X, →, e, ≤X) if and only if its upper t-level set and lower s-level set are ordered subalgebras of X := (X, →, e, ≤X) for all (t, s) ∈ (0, 1]×[0, 1). Proof. Suppose that I := {⟨x, fI , gI⟩ | x ∈ X} in X is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). Let x, y, a, b ∈ X be such that x, y ∈ U(fI , t) and a, b ∈ E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1352 L(gI , s) whenever e ≤e x, e ≤e y, e ≤e a, and e ≤e b. Then fI(x) ≥ t, fI(y) ≥ t, gI(a) ≤ s and gI(b) ≤ s. It follows from Theorem 3 that fI(x → y) ≥ min{fI(x), fI(y)} ≥ t and gI(x → y) ≤ max{gI(x), gI(y)} ≤ s. Hence x → y ∈ U(fI , t) and a → b ∈ L(gI , s). Therefore U(fI , t) and L(gI , s) are ordered subalgebras of X := (X, →, e, ≤X). Conversely, assume that the upper t-level set and lower s-level set are ordered sub- algebras of X := (X, →, e, ≤X) for all (t, s) ∈ (0, 1] × [0, 1). Let x, y ∈ X and (t1, s1), (t2, s2) ∈ (0, 1]× [0, 1) be such that e ≤X x, e ≤X y, x(t1,s1) ∈ I, and y(t2,s2) ∈ I. Then fI(x) ≥ t1, gI(x) ≤ s1, fI(y) ≥ t2, and gI(y) ≤ s2. Hence x ∈ U(fI , t1) ⊆ U(fI ,min{t1, t2}), y ∈ U(fI , t2) ⊆ U(fI ,min{t1, t2}), x ∈ L(gI , s1) ⊆ L(gI ,max{s1, s2}), and y ∈ L(gI , s2) ⊆ L(gI ,max{s1, s2}). Since U(fI ,min{t1, t2}) and L(gI ,max{s1, s2}) are ordered subalgebras of X := (X, →, e, ≤X) by hypothesis, it follows that x → y ∈ U(fI ,min{t1, t2}) ∩ L(gI ,max{s1, s2}) = I∈ (min{t1,t2},max{s1,s2}). Hence (x → y)(min{t1,t2},max{s1,s2}) ∈ I, and therefore I := {⟨x, fI , gI⟩ | x ∈ X} in X is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). Corollary 1. If an intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X is an intuition- istic fuzzy ordered subalgebra of X := (X, →, e, ≤X), then its ∈(t,s)-level set is an ordered subalgebra of X := (X, →, e, ≤X) for all (t, s) ∈ (0, 1]× [0, 1). Proof. Straightforward. We make an intuitionistic fuzzy ordered subalgebra using a collection of ordered sub- algebras. Theorem 11. Let {Bt | t ∈ Λ ⊆ [0, 1]} be a collection of ordered subalgebras of X := (X, →, e, ≤X) such that X is represented as the union of Bt, i.e., X = ⋃ t∈Λ Bt, and (∀t, s ∈ Λ)(t > s ⇔ Bt ⊂ Bs). (29) Then an intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X defined by fI : X → [0, 1], x 7→ sup{t ∈ Λ | x ∈ Bt}, gI : X → [0, 1], x 7→ inf{t ∈ Λ | x ∈ Bt} (30) is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). Proof. According to Theorem 10, it is sufficient to show that U(fI , t) and L(gI , s) are ordered subalgebras of X := (X, →, e, ≤X) for every (t, s) ∈ (0, 1]× [0, 1). We first show that U(fI , t) is an ordered subalgebra of X := (X, →, e, ≤X). To do that, we consider the following two cases: E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1353 (i) t = sup{k ∈ Λ | k < t}, (ii) t ̸= sup{k ∈ Λ | k < t}. The first case induces (∀x ∈ X) ( x ∈ U(fI , t) ⇔ (∀k < t)(x ∈ Bk) ⇔ x ∈ ∩ k 0. So x /∈ Bk for all k > t − δ, which means that if x ∈ Bk, then k ≤ t − δ. Thus fI(x) ≤ t − δ < t, i.e., x /∈ U(fI , t). Therefore U(fI , t) = ⋃ k≥t Bk and it is an ordered subalgebra of X := (X, →, e, ≤X). By the similarly, we can verify that L(gI , s) is an ordered subalgebra of X := (X, →, e, ≤X). Consequently, I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X) by Theorem 10. Theorem 12. Given an intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X, its nonempty ∈(t,s)-level set is an ordered subalgebra of X := (X, →, e, ≤X) for all (t, s) ∈ (0.5, 1]× [0, 0.5) if and only if I := {⟨x, fI , gI⟩ | x ∈ X} satisfies: (∀x, y ∈ X)  e ≤X x, e ≤X y ⇒ { max{fI(x → y), 0.5} ≥ min{fI(x), fI(y)} min{gI(x → y), 0.5} ≤ max{gI(x), gI(y)}  . (31) Proof. Assume that the ∈(t,s)-level set I∈ (t,s) is a nonempty ordered subalgebra of X := (X, →, e, ≤X) for all (t, s) ∈ (0.5, 1]× [0, 0.5). If I does not satisfy (31), then max{fI(a → b), 0.5} < min{fI(a), fI(b)} or min{gI(a → b), 0.5} > max{gI(a), gI(b)} for some a, b ∈ X with e ≤e a and e ≤e b. If we put t := min{fI(a), fI(b)} and s := max{gI(a), gI(b)}, then t ∈ (0.5, 1] and s ∈ [0, 0.5), a, b ∈ I∈ (t,s) but a → b /∈ I∈ (t,s). This is a contradiction, and so max{fI(x → y), 0.5} ≥ min{fI(x), fI(y)} and min{gI(x → y), 0.5} ≤ max{gI(x), gI(y)} for all x, y ∈ X with e ≤e x and e ≤e y. Conversely, suppose that I satisfies (31). Let x, y ∈ X, t ∈ (0.5, 1] and s ∈ [0, 0.5) be such that e ≤e x, e ≤e y and x, y ∈ I∈ (t,s). Then max{fI(x → y), 0.5} ≥ min{fI(x), fI(y)} ≥ t > 0.5 E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1354 and min{gI(x → y), 0.5} ≤ max{gI(x), gI(y)} ≤ s < 0.5, and so fI(x → y) ≥ t and gI(x → y) ≤ s. Hence x → y ∈ I∈ (t,s), and therefore I∈ (t,s) is an ordered subalgebra of X := (X, →, e, ≤X) for all t ∈ (0.5, 1] and s ∈ [0, 0.5). Given an intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} and (t, s) ∈ (0, 1] × [0, 1), the set Iq (t,s) := {y ∈ X | y(t,s) q I} (32) is called the q(t,s)-level set of I. It is clear that I q (t1,s1) ⊆ Iq (t2,s2) for all (ti, si) ∈ (0, 1]×[0, 1), i = 1, 2, satisfying (t1, s1) ≪ (t2, s2), i.e., t1 ≤ t2 and s1 ≥ s2. We know that Iq (t,s) := {y ∈ X | y(t,s) q I} = Q(fI , t) ∩Q(gI , s) where Q(fI , t) := {y ∈ X | f(y) > 1− t} and Q(gI , s) := {y ∈ X | g(y) < 1− s} which are called the upper q-level set and the lower q-level set of I related to t and s, respectively. Theorem 13. If I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X), then its q(t,s)-level set is an ordered subalgebra of X := (X, →, e, ≤X) for all (t, s) ∈ (0, 1]× [0, 1). Proof. Assume that I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X). Let x, y ∈ X be such that e ≤e x, e ≤e y and x, y ∈ Iq (t,s) for all (t, s) ∈ (0, 1]× [0, 1). Then x(t,s) q I and y(t,s) q I, that is, fI(x)+ t > 1, gI(x) + s < 1, fI(y) + t > 1 and gI(y) + s < 1. Hence fI(x → y) + t ≥ min{fI(x), fI(y)}+ t = min{fI(x) + t, fI(y) + t} > 1 and gI(x → y) + s ≤ max{gI(x), gI(y)}+ s = max{gI(x) + s, gI(y) + s} < 1, and so (x → y)(t,s) q I, i.e., x → y ∈ Iq (t,s). Consequently, Iq (t,s) is an ordered subalgebra of X := (X, →, e, ≤X) for all (t, s) ∈ (0, 1]× [0, 1). The example below describes Theorem 13. Example 3. Consider the OBCI-algebra X := (X, →, e, ≤X) in Example 2. Define an intuitionistic fuzzy set I := {⟨x, fI , gI⟩ | x ∈ X} in X as follows: fI : X → [0, 1], x 7→  0.74 if x = 1, 0.53 if x = 3 4 , 0.48 if x = 1 2 , 0.36 if x = 1 4 , 0.67 if x = 0, E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1355 and gI : X → [0, 1], x 7→  0.24 if x = 1, 0.13 if x = 3 4 , 0.21 if x = 1 2 , 0.46 if x = 1 4 , 0.24 if x = 0. The upper t-level set U(fI , t) and the lower s-level set L(gI , s) of I are calculated as follows: U(fI , t) =  ∅ if t ∈ (0.74, 1], {1} if t ∈ (0.67, 0.74], {1, 0} if t ∈ (0.53, 0.67], {1, 0, 34} if t ∈ (0.48, 0.53], {1, 0, 34 , 1 2} if t ∈ (0.36, 0.48], X if t ∈ (0, 0.36], and L(gI , s) =  ∅ if s ∈ [0, 0.13), {3 4} if s ∈ [0.13, 0.21), {3 4 , 1 2} if s ∈ [0.21, 0.24), {1, 0, 34 , 1 2} if s ∈ [0.24, 0.46), X if s ∈ [0.46, 1). It is routine to verify that U(fI , t) and L(gI , s) are ordered subalgebras of X := (X, →, e, ≤X) for all (t, s) ∈ (0, 1] × [0, 1). Hence I := {⟨x, fI , gI⟩ | x ∈ X} is an intuitionistic fuzzy ordered subalgebra of X := (X, →, e, ≤X) by Theorem 10. The upper q-level sets Q(fI , t) related to t are provided by Tables 3. Table 3: Calculation of Q(fI , t) t 1− t Q(fI , t) (0.74, 1] [0, 0.26) X (0.67, 0.74] [0.26, 0.33) X (0.53, 0.67] [0.33, 0.47) X or {1, 0, 34 , 1 2} (0.48, 0.53] [0.47, 0.52) {1, 0, 34 , 1 2} or {1, 0, 34} (0.36, 0.48] [0.52, 0.64) {1, 0, 34} or {1, 0} (0, 0.36] [0.64, 1) {1, 0}, {1} or ∅ The lower q-level sets Q(gI , s) related to s are provided by Tables 4. We can observe that Q(fI , t) and Q(gI , s) are ordered subalgebras of X := (X, →, e, ≤X) for all (t, s) ∈ (0, 1]× [0, 1). Hence Iq (t,s) = Q(fI , t) ∩Q(gI , s) is an ordered subalgebra of X := (X, →, e, ≤X) for all (t, s) ∈ (0, 1] × [0, 1), and they are displayed as follows: {1}, {3 4}, {1, 0}, { 3 4 , 1 2}, {1, 0, 3 4}, {1, 0, 3 4 , 1 2} and X. E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (3) (2023), 1342-1358 1356 Table 4: Calculation of Q(gI , s) s 1− s Q(gI , s) [0, 0.13) (0.87, 1] X [0.13, 0.21) (0.79, 0.87] X [0.21, 0.24) (0.76, 0.79] X [0.24, 0.46) (0.54, 0.76] X [0.46, 1) (0, 0.54] {3 4}, { 3 4 , 1 2}, {1, 0, 3 4 , 1 2} or X Proposition 2. Let I := {⟨x, fI , gI⟩ | x ∈ X} be an intuitionistic fuzzy set in X. For every (ti, si) ∈ (0, 0.5]× [0.5, 1), i = 1, 2, if the q(ti,si)-level set of I is an ordered subalgebra of X := (X, →, e, ≤X), then I := {⟨x, fI , gI⟩ | x ∈ X} satisfies: (∀x, y ∈ X)  { x ∈ Iq (t1,s1) , e ≤e x y ∈ Iq (t2,s2) , e ≤e y } ⇒ x → y ∈ I∈ (max{t1,t2},min{s1,s2})  . (33) Proof. Assume that the q(ti,si)-level set I q (ti,si) is an ordered subalgebra of X := (X, →, e, ≤X) for all (ti, si) ∈ (0, 0.5]× [0.5, 1), i = 1, 2, Let x, y ∈ X and (ti, si) ∈ (0, 0.5]× [0.5, 1) be such that x ∈ Iq (t1,s1) , y ∈ Iq (t2,s2) , e ≤e x and e ≤e y. Then 1 < fI(x) + t1 ≤ fI(x) + max{t1, t2}, 1 < fI(y) + t2 ≤ fI(y) + max{t1, t2}, 1 > gI(x) + s1 ≥ gI(x) + min{s1, s2}, 1 > gI(y) + s2 ≥ gI(y) + min{s1, s2}. Hence x, y ∈ Iq (max{t1,t2},min{s1,s2}), and so x → y ∈ Iq (max{t1,t2},min{s1,s2}) since max{t1, t2} ∈ (0, 0.5], min{s1, s2} ∈ [0.5, 1) and Iq (max{t1,t2},min{s1,s2}) is an ordered subalgebra of X := (X, →, e, ≤X). Thus fI(x → y) > 1−max{t1, t2} ≥ max{t1, t2} and gI(x → y) < 1−min{s1, s2} ≤ min{s1, s2} because of max{t1, t2} ≤ 0.5 and min{s1, s2} ≥ 0.5. Therefore x → y ∈ I∈ (max{t1,t2},min{s1,s2}) which shows that (33) is valid. Proposition 3. Let I := {⟨x, fI , gI⟩ | x ∈ X} be an intuitionistic fuzzy set in X. For every (ti, si) ∈ (0.5, 1]×[0, 0.5), i = 1, 2,, if the q(ti,si)-level set of I is an ordered subalgebra REFERENCES 1357 of X := (X, →, e, ≤X), then I := {⟨x, fI , gI⟩ | x ∈ X} satisfies: (∀x, y ∈ X)  { x ∈ I∈ (t1,s1) , e ≤e x y ∈ I∈ (t2,s2) , e ≤e y } ⇒ x → y ∈ Iq (max{t1,t2},min{s1,s2})  . (34) Proof. Suppose that the q(ti,si)-level set I q (ti,si) is an ordered subalgebra of X := (X, →, e, ≤X) for all (ti, si) ∈ (0.5, 1]×[0, 0.5), i = 1, 2. Let x, y ∈ X and (ti, si) ∈ (0.5, 1]×[0, 0.5) be such that x ∈ I∈ (t1,s1) , y ∈ I∈ (t2,s2) , e ≤e x and e ≤e y. Then fI(x) ≥ t1 > 1− t1, gI(x) ≤ s1 < 1− s1, fI(y) ≥ t2 > 1− t2, gI(y) ≤ s2 < 1− s2, i.e., x(t1,s1) q I and y(t2,s2) q I. Hence x ∈ Iq (t1,s1) ⊆ Iq (max{t1,t2},min{s1,s2}) and y ∈ Iq (t2,s2) ⊆ Iq (max{t1,t2},min{s1,s2}). Since max{t1, t2} ∈ (0.5, 1], min{s1, s2} ∈ [0.0.5) and max{t1, t2}+ min{s1, s2} ≤ 1, it follows from the hypothesis that Iq (max{t1,t2},min{s1,s2}) is an ordered subalgebra of X := (X, →, e, ≤X). Hence x → y ∈ Iq (max{t1,t2},min{s1,s2}). References [1] K. Atanassov. Intuitionistic fuzzy sets. VII ITKRs Session, Deposed in Central Sci.- Techn. Library of Bulg. Acd. of Sci., pages 1684–1697, 1983. [2] K. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1):87–96, 1986. [3] K. Atanassov. Two operators on intuitionistic fuzzy sets. Comptes Rendus Acaémi bulgare Sci. Tome, 41, 1988. [4] K. Atanassov. More on intuitionistic fuzzy sets. Fuzzy Sets and Systems, 33(1):37–45, 1989. [5] D. Çoker and M. Demirci. On intuitionistic fuzzy points. Notes IFS, 1(2):79–84, 1995. [6] B. Davvaz. (∈,∈∨q)-fuzzy subnear-rings and ideals. Soft Comput., 10:206–211, 2006. [7] E. H. Roh E. Yang and Y. B. Jun. Fuzzy ordered subalgebras in ordered bci- algebras. Revista de la Real Academia de Ciencias Exactas, F́ısicas y Naturales. Serie A. Matemáticas (RaCSaM), page (submitted). [8] E. H. Roh E. Yang and Y. B. Jun. Ordered bci-algebras. Journal of Algebra and its Applications, page (submitted). [9] S. Ghorbani. Intuitionistic fuzzy congruence relations on residuated lattices. Acta Universitatis Apulensis, 29:301–314, 2012. REFERENCES 1358 [10] Y. S. Huang. BCI-algebra. Science Press, Beijing, China, 2006. [11] Y. Imai and K. Iséki. On axiom systems of proposition calculi. Proc. Japan. Acad., 42:19–22, 1966. [12] Y. B. Jun. On (α, β)-fuzzy subalgebras of bck/bci-algebras. Bull. Korean Math. Soc., 42(4):703–711, 2005. [13] Y. B. Jun. Fuzzy subalgebras of type (α, β) in bck/bci-algebras. Kyungpook Math. J., 47:403–410, 2007. [14] Y. B. Jun. Generalizations of (∈,∈∨q)-fuzzy subalgebras in bck/bci-algebras. Com- put. Math. Appl., 58:1383–1390, 2009. [15] Y. B. Jun and S. Z. Song. Generalized fuzzy interior ideals in semigroups. Inform. Sci., 176:3079–3093, 2006. [16] P. M. Pu and Y. M. Liu. Fuzzy topology i, neighborhood structure of a fuzzy point and moore-smith convergence. J. Math. Anal. Appl., 76:571–599, 1980. [17] M. Shabir W. A. Dudek and M. Irfan Ali. (α, β)-fuzzy ideals of hemirings. Comput. Math. Appl., 58:310–321, 2009. [18] B. Davvaz X. Ma, J. Zhan and Y. B. Jun. Some kinds of (∈,∈∨q)-interval-valued fuzzy ideals of bci-algebras. Inform. Sci., 178:3738–3754, 2008. [19] L. A. Zadeh. Fuzzy sets. Inform. Control, 8:338–353, 1965.