FUZZY ORDERED STRUCTURES AND FUZZY

FUZZY ORDERED STRUCTURES AND FUZZY LATTICE ORDERED GROUPS ˇ SELJA, ˇ ˇ ´ AND MIRNA BRANIMIR SE ANDREJA TEPAVCEVI C ˇ ´ UDOVICIC

Abstract. Fuzzy (lattice valued) posets are investigated with the order being a fuzzy relation on a fuzzy set. In this framework, fuzzy chains and generally fuzzy lattices are investigated. These are applied to fuzzy lattice ordered subgroups of lattice ordered groups. Lattice ordered subgroups are characterized in terms of cuts, and a special construction of fuzzy lattice valued subgroups is presented. In addition, classical totally ordered groups are characterized by their lattice valued subgroups. An application to ordered groups of real numbers is given. AMS Mathematics Subject Classification (2000): 03E72, 06D72, 20N25. Key words and phrases: fuzzy poset, fuzzy ordering, fuzzy lattice, fuzzy chain, fuzzy lattice ordered group.

1. Introduction This paper is a part of investigation of fuzzy ordered structures and their application to ordered groups. In the first part we deal with fuzzy posets and lattices and in the second we investigate fuzzy lattice ordered subgroups of lattice ordered groups. In our investigation the structure of membership values is a complete lattice (approach introduced firstly by Goguen, [16]), without additional operations (like residuation or multiplication). The reason is our intention to use the cutworthy approach: under the classical lattice operations (meet and join) many classical properties fulfilled by cuts are also satisfied in the fuzzy framework by the corresponding fuzzy structure. We deal with fuzzy orderings. These have been investigated in the fuzzy context at the very beginning by Zadeh, [32], and then by others. The motivation for such investigations originates in theoretical reasons, but also in applications: orderings can model various situations in which different kinds of comparison appear; in rule based systems, in control problems, in solving relational equations. From the earlier period, orderings were introduced by Ovchinnikov (e.g., [21]). In the last decade, there were papers by Bˇelohlávek (many results are collected in his book [2]). Recent important results concerning fuzzy orders and their representations are related to De Baets, Bodenhofer at al. ([6, 7, 8] and the overview about weak fuzzy orders, 1

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ˇ SELJA, ˇ ˇ ´ AND MIRNA UDOVICI ˇ C ´ BRANIMIR SE ANDREJA TEPAVCEVI C

[9], see also [15]). In most of these investigations, what is fuzzified is the ordering relation, while the underlying set is crisp. In addition, the authors use T -norms, or more generally, residuated lattices and corresponding operations. Fuzzy posets, lattices and similar structures were investigated by Demirci, ([12, 13, 14], the last one with Eken). In the mentioned papers, fuzzy orderings are defined with respect to fuzzy equality instead of the crisp one (this approach was introduced by Höhle, [17]). There are also some recent papers in this field e.g., [10]. Our approach to fuzzy structures is based on papers [24, 25, 26, 27, 28] and in particular, concerning fuzzy posets, on [29] and [31]. Fuzzy up and down–sets have been investigated in [18, 19]. Concerning fuzzy groups, we follow the well known approach initiated by Rosenfeld in [22]. Though, our topic are ordered algebraic structures, mainly ordered groups (for the classical approach see the books [4, 5]), which in fuzzy context have not been considerably investigated. Namely, Bhakat and Das (see their paper [3]) have introduced fuzzy ordered subgroups using fuzzy order and the membership values belonging to the unit interval; they have also investigated Archimedean ordered subgroups in fuzzy context. Recently Saibaba in [23] defined fuzzy lattice ordered fuzzy groups. The most extensive recent material in the topic of fuzzy ordered groups is the paper [1] by Bakhshi; in this paper the co-domain of fuzzy structures is the unit interval, while we use the complete lattice. In addition, the corresponding orderings differ. 1.1. The structure of the paper. Here we use some notions about fuzzy order on fuzzy sets which are developed in our paper [30]; these are repeated here in Preliminaries, together with relevant properties. We also advance some definitions and properties (from the same paper) concerning fuzzy ordered groups. Our results consist in two parts. In the first (Section 3), we present properties of some fuzzy ordered sets and their characterization by cuts: fuzzy chains and fuzzy lattices in general. In Section 4 we introduce fuzzy lattice ordered subgroups of lattice ordered groups. We characterize these by their crisp subgroups and we present a construction which gives fuzzy ordered subgroups in an intrinsic way: we use a particular Heyting lattice of subgroups of the given ordered group. In addition, we characterize crisp lattice ordered groups by their fuzzy lattice ordered subgroups. Finally, in Section 5, we give an example in which we apply the previous results and constructions to ordered group of the square of real numbers under addition. Finally, in Conclusion, we mention some directions for future investigations. Let us mention that our approach to fuzzy ordered groups differs from existing ones by Bakhshi, [1], Bhakat and Das, [3], and Saibaba, [23]. Namely, we use a complete lattice and a particular fuzzy order which is, by reflexivity, essentially different from others, not only in order structures but also in applications.

FUZZY ORDERED STRUCTURES ...

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2. Preliminaries In this preliminary section we list relevant notions from the classical order and lattice theory and ordered groups. In addition we give some notions from fuzzy (lattice valued) structures which we use throughout the present paper. 2.1. Order, lattice. Here we present some notions concerning order and lattices, and we list relevant properties of these. For more details about given notions, see e.g., books [4, 11]. A poset is a nonempty set X together with an ordering relation 6 . Hence, a poset is denoted as an ordered pair (X, 6 ), or by the underlying set X. A sub–poset of (X, 6 ) is a poset on a subset Y of X in which the order is the one restricted from X, and usually denoted in the same way ( 6 ). For an x ∈ X, a sub–poset ↑x := {y ∈ X | x 6 y} is a principal filter generated by x. The dual notion is a principal ideal generated by x, denoted by ↓x. If x, y ∈ X, and x 6 y, then [x, y] := ↑x ∩ ↓y is an interval in the poset X. A lattice is a poset L in which for each pair of elements x, y there is a greatest lower bound (glb, infimum, meet) and a least upper bound (lub, supremum, join), denoted respectively by x ∧ y and x ∨ y. These are binary operations on L. A non-empty poset L is said to be a complete lattice if infima and suprema exist for each subset of L. A complete lattice possesses the top (1) and the bottom element (0). A lattice L is distributive if the following identities hold in L: x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) and x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). In a lattice L with a bottom element 0, x∗ is a pseudocomplement of x, if x∗ = max{y ∈ L | x ∧ y = 0}. L is pseudocomplemented if every x ∈ L has a pseudocomplement. A Heyting lattice is a distributive lattice in which every interval is pseudocomplemented. 2.2. Ordered groups. A structure (G, · ) with a binary operation · is a groupoid, which is a group if the operation is associative, there is a neutral element e (e · x = x · e = x for every x ∈ G) and for each x ∈ G there is an inverse x−1 ∈ G (x · x−1 = x−1 · x = e). For groups we use the language with a binary operation, unary and a constant, denoting it by (G, · , −1 , e). A groupoid (G, · ) may be equipped with an ordering relation 6 which is compatible with the binary operation in the following sense: for all x, y, z ∈ G (1)

x 6 y implies z · x 6 z · y and x · z 6 y · z.

If there is such an order on G, then the structure (G, · , 6 ) is an ordered groupoid. An ordered group is a structure (G, · , −1 , e, 6 ) where (G, · , −1 , e) is a group which is ordered by 6 in the sense of (1). An example of an ordered group is the set of integers under the addition and the classical order 6 . A subgroup of an ordered group G is convex, if it is a convex subposet of G under the order of this group. If (G, · , −1 , e, 6 ) is an

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ordered group and the poset (G, 6 ) is a lattice, then this structure is said to be a lattice ordered group. The following are known definitions and properties of lattice ordered groups, see e.g., books [4, 5]. A subgroup H of a lattice ordered group G is an ℓ-subgroup if it is also a sublattice of G. Proposition 1. A subgroup H of a lattice ordered group G is an ℓ-subgroup of G if and only if for every x ∈ H, x ∨ e ∈ H. Proposition 2. The collection of all convex ℓ-subgroups of a lattice ordered group form a complete Heyting sub-lattice of the lattice of all subgroups of G. Proposition 3. A lattice ordered group G is totally ordered if and only if each subgroup of G is an ℓ-subgroup. If H is a convex normal subgroup of an ordered group G, then an order 6H on the quotient group G/H is given by (2)

x + H 6H y + H ⇔ (∃h ∈ H) h + x 6 y.

This order is compatible with the group operation, hence G/H is an ordered group under 6H . 2.3. Fuzzy structures. We present some notions from the theory of fuzzy structures with the membership values belonging to a complete lattice. More about our approach as well as about the relevant properties can be found e.g., in [26, 27]. Fuzzy or lattice–valued sets are mappings from a non-empty set X (domain) into a complete lattice L (co-domain) in which the top and bottom elements are 1 and 0, respectively. If α : X → L is a fuzzy set on X then, for p ∈ L, the set αp := {x ∈ X | α(x) ≥ p} is said to be the p-cut, a cut set or simply a cut of α. Obviously, a p-cut of α is the inverse image (under α) of the principal filter in L generated by p : αp = α−1 (↑p). If (G, · , −1 , e) is a group and (L, ∧, ∨, ≤) a complete lattice, then the mapping µ : G → L is a fuzzy subgroup of G if the following holds: for all x, y ∈ G (i) µ (x) ∧ µ (y) ≤ µ (x · y) ( ) (ii) µ (x) ≤ µ x−1 (iii) µ (e) = 1. From the above definition it follows that


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( ) µ (x) = µ x−1 . 2.4. Fuzzy sub–poset; fuzzy order on fuzzy set. The majority of material presented in preliminary parts 2.4 and 2.5 is developed in [30]. Here we repeat necessary notions and their properties, omitting proofs. Let (P, 6) be a poset and (L, ∧, ∨, ≤), as indicated, a complete lattice. By k6 we denote the characteristic function of the order in P : for any x, y ∈ P { 1 if x 6 y k6 (x, y) := 0 otherwise . A mapping α : P → L is a fuzzy sub–poset of (P, 6). Lemma 1. Every cut of a fuzzy sub–poset α of a poset P is a crisp sub– poset of P . 2 A fuzzy set µ : P → L is said to be a fuzzy convex sub–poset in P if for all x, y, z ∈ P the following holds: (3)

µ (x) ∧ µ (z) ∧ k6 (x, y) ∧ k6 (y, z) ≤ µ (y) .

Some characterizations of fuzzy convex sub–posets are given in the mentioned paper [30]. In the present investigation we use fuzzy relations with the domain being a fuzzy set. Therefore, we accordingly adopt the known relational properties, listed in the sequel. A fuzzy relation ρ : P 2 → L on a set P is a fuzzy relation on a fuzzy subset µ : P → L of P if for all x, y ∈ P ρ(x, y) ≤ µ(x) ∧ µ(y).

(4)

A fuzzy relation ρ : P 2 → L on µ : P → L is - reflexive if for all x ∈ P (5)

ρ(x, x) = µ(x);

- antisymmetric if for all x, y ∈ P x ̸= y implies ρ(x, y) ∧ ρ(y, x) = 0;

(6) - transitive if (7)

for all x, y, z ∈ P, ρ(x, y) ∧ ρ(y, z) ≤ ρ(x, z).

We say that a fuzzy relation ρ on a fuzzy subset µ is a fuzzy ordering (fuzzy order) on µ, if it is reflexive in the sense of (5), antisymmetric as defined by (6) and transitive in the sense of (7). Proposition 4. Let (P, 6) be a poset and µ : P → L its fuzzy sub–poset. Further, let ρ : P → L be a fuzzy relation on P defined as follows: (8)

ρ(x, y) := µ(x) ∧ µ(y) ∧ k6 (x, y),

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where k6 is the characteristic function of the crisp order 6 on P . Then, the function ρ defined by (8) is a fuzzy order on fuzzy set µ on P . Now, if (P, 6) is a poset, then a pair (µ, ρ) is a fuzzy poset with fuzzy ordering if µ : P → L is a fuzzy subset of P and ρ : P 2 → L is the fuzzy ordering on P defined by (8). Theorem 1. Let (P, 6) be a poset, µ : P → L a fuzzy subset of P , and ρ : P 2 → L a fuzzy relation on µ. Then (µ, ρ) is a fuzzy poset with fuzzy order on (P, 6), if and only if for every p ∈ L, p ̸= 0, pair (µp , ρp ) is a crisp sub–poset of (P, 6). 2.5. Fuzzy–ordered structures. Let (G, ·) be a groupoid and ρ : G2 → L a fuzzy relation on G. We say that ρ is compatible with operation ”·” on G, if for all x, y, z ∈ G the following holds: (9)

ρ(x, y) ≤ ρ(x · z, y · z) ∧ ρ(z · x, z · y).

Let (G, · ) be a groupoid and µ : G → L its fuzzy subgrupoid. We say that a fuzzy relation ρ : G2 → L on µ is compatible with operation ”·” on µ, if for all x, y, z ∈ G the following holds: µ(z) ∧ ρ(x, y) ≤ ρ(x · z, y · z) ∧ ρ(z · x, z · y). ( ) Proposition 5. Let G, ·, −1 , e, 6 be an ordered group and µ : G → L a fuzzy subgroup of G. The fuzzy relation ρ : G2 → L on µ defined by (8), i.e., by ρ(x, y) = µ(x) ∧ µ(y) ∧ k6 (x, y), is a fuzzy order on µ which is compatible with the group operation in the sense of (10). ( ) Let G, ·, −1 , e, 6 be an ordered group. Let also µ : G → L and ρ : G2 → L be a fuzzy set on G and a fuzzy relation on µ, respectively. The pair (µ, ρ) is a fuzzy ordered subgroup of G if the following hold: 1. µ is a fuzzy subgroup of G; 2. ρ is the fuzzy relation on µ defined by (10)

ρ(x, y) = µ(x) ∧ µ(y) ∧ k6 (x, y). Obviously, by Proposition 5, ρ is a fuzzy ordering relation on µ, compatible with the group operation. In addition, the fuzzy order ρ is by the definition uniquely determined by the function µ. Theorem 2. Let G be an ordered group, µ : G → L a fuzzy subset of G and ρ : G2 → L a fuzzy relation on µ. Then (µ, ρ) is a fuzzy-ordered subgroup of G if and only if for every p ∈ L, the cut µp is an ordered subgroup of G. Theorem 3. Let F be a collection of subgroups of an ordered group (G, ·, −1 , e, 6) which is closed under set intersections and contains G. Then there is a complete lattice L and an ordered fuzzy subgroup (µ, ρ) of G,


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such that for every subgroup H ∈ F , the cut µH coincides with H and it is ordered by ρH . 2 a fuzzy-ordered subgroup (µ, ρ) of an ordered group ( We−1say that ) G, ·, , e, 6 is a fuzzy-convex subgroup of G if µ is a fuzzy-convex subset on the poset (G, 6) .

3. Fuzzy order, fuzzy chain and fuzzy lattice In this section we present some properties of special fuzzy posets, like fuzzy chain and in general, fuzzy lattice. We use the notions introduced in preliminary sections. As in the previous parts, L is a complete lattice of membership values. Recall that if (P, 6) is a poset, then a pair (µ, ρ) is a fuzzy poset with fuzzy ordering if µ : P → L is a fuzzy subset of P and ρ : P 2 → L is the fuzzy ordering on P defined by (8): ρ(x, y) = µ(x) ∧ µ(y) ∧ k6 (x, y). Lemma 2. Let (P, 6) be a poset, µ : P → L a fuzzy subset of P , and ρ : P 2 → L a fuzzy relation on µ. Then (µ, ρ), a fuzzy poset with fuzzy order on (P, 6) is convex if and only if all the cuts are crisp convex subposets. Proof. Suppose that (µ, ρ) is convex, i.e., that for all x, y, z ∈ P , µ (x) ∧ µ (z) ∧ k6 (x, y) ∧ k6 (y, z) ≤ µ (y) . Let p ∈ L, and let µp be a cut. Suppose that x, z ∈ µp and x 6 y 6 z. Then, we have that µ(x) ≥ p and µ(z) ≥ p and k6 (x, y) = 1 and k6 (y, z) = 1. Therefore, p ≤ µ (x) ∧ µ (z) ∧ k6 (x, y) ∧ k6 (y, z) ≤ µ (y) and y ∈ µp . To prove the converse, suppose that all the cuts are convex and take x, y, z ∈ P Let p = µ (x) ∧ µ (z). If k6 (x, y) ∧ k6 (y, z) = 0, then the inequality is satisfied. If k6 (x, y) ∧ k6 (y, z) = 1, then x 6 y 6 z and since x ∈ µp and z ∈ µp , we have that y ∈ µp and µ(y) ≥ p, i.e., µ(y) ≥ µ (x) ∧ µ (z) ∧ k6 (x, y) ∧ k6 (y, z). 2 Let (M, 6) be a poset and (µ, ρ) a fuzzy sub-poset of M . Then (µ, ρ) is a fuzzy chain, or fuzzy linearly ordered sub-poset of M if for all x, y ∈ M (11)

ρ(x, y) ∨ ρ(y, x) = µ(x) ∧ µ(y).

Proposition 6. Every fuzzy sub-poset (µ, ρ) of a linearly ordered poset (M, 6) is a fuzzy chain. Proof. By the definition of the order ρ on µ, for all x, y ∈ M (12)

ρ(x, y) = µ(x) ∧ µ(y) ∧ k6 (x, y).

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Since the order 6 on M is linear by assumption, either k6 (x, y) = 1 and k6 (y, x) = 0, or vice versa. Therefore, for all x, y ∈ M ρ(x, y) ∨ ρ(y, x) = (µ(x) ∧ µ(y) ∧ k6 (x, y)) ∨ (µ(x) ∧ µ(y) ∧ k6 (y, x)) = µ(x) ∧ µ(y). 2 Next we deal with fuzzy chains (i.e., linearly ordered sub-posets) in posets which might not be linearly ordered. Proposition 7. Let (µ, ρ) be a fuzzy sub-poset of a poset (M, 6). Then (µ, ρ) is a fuzzy chain if and only if for all x, y ∈ M such that x is not comparable with y, we have µ(x) ∧ µ(y) = 0.

(13)

Proof. Let (µ, ρ) be a fuzzy sub-poset of (M, 6) which is a fuzzy chain i.e., such that (11) holds. Let x, y ∈ M and suppose that they are not comparable, i.e., that k6 (x, y) = k6 (y, x) = 0. Then by (12), ρ(x, y) ∨ ρ(y, x) = (µ(x) ∧ µ(y) ∧ k6 (x, y)) ∨ (µ(x) ∧ µ(y) ∧ k6 (y, x)) = 0, hence, by (11), µ(x) ∧ µ(y) = 0. Conversely, suppose that x, y are incomparable. Then (13) holds, and as above, k6 (x, y) = k6 (y, x) = 0, finally by (12) ρ(x, y) ∨ ρ(y, x) = (µ(x) ∧ µ(y) ∧ k6 (x, y)) ∨ (µ(x) ∧ µ(y) ∧ k6 (y, x)) = 0 = µ(x) ∧ µ(y). If x and y are comparable, e.g., if x 6 y, then k6 (x, y) = 1 and k6 (y, x) = 0. Thereby, similarly as above, ρ(x, y) ∨ ρ(y, x) = (µ(x) ∧ µ(y) ∧ 1) ∨ (µ(x) ∧ µ(y) ∧ 0) = µ(x) ∧ µ(y), and (µ, ρ) is a fuzzy chain.

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Proposition 8. Let (µ, ρ) be a fuzzy sub-poset of a poset (M, 6). Then (µ, ρ) is a fuzzy chain if and only if every its non-zero cut µp is a chain in (M, 6), with respect to ρp . Proof. Suppose that (µ, ρ) is a fuzzy chain. Let µp be a cut of µ, for some p ∈ L, p ̸= 0. We prove that for all x, y ∈ M , x, y ∈ µp implies (x, y) ∈ ρp or (y, x) ∈ ρp . Indeed, if x, y ∈ µp , then µ(x) ≥ p and µ(y) ≥ p. Therefore, µ(x)∧µ(y) ≥ p, and since p ̸= 0, by Proposition 7, x and y are comparable, e.g., x 6 y. Then k6 (x, y) = 1 and k6 (y, x) = 0, implying ρ(x, y) = µ(x) ∧ µ(y) ∧ 1 = µ(x) ∧ µ(y) ≥ p. Therefore, (x, y) ∈ ρp . To prove the converse, suppose that x and y are not comparable (if they are, then ρ(x, y)∨ρ(y, x) = µ(x)∧µ(y), as in the previous proposition.) Now,


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k6 (x, y) = 0 and k6 (y, x) = 0 and ρ(x, y) = 0 and ρ(y, x) = 0. Suppose that µ(x) ∧ µ(y) = p ̸= 0. Then, x ∈ µp and y ∈ µp . Since for every p ̸= 0, x, y ∈ µp implies (x, y) ∈ ρp or (y, x) ∈ ρp , and we would have that (x, y) ∈ ρp or (y, x) ∈ ρp , i.e., ρ(x, y) ∨ ρ(y, x) ≥ p, a contradiction! Therefore, (µ, ρ) is a fuzzy chain. 2 The following definition and properties given in Proposition 9 can also be found in [31]. Let (M, 6) be a poset which is a lattice, and (µ, ρ) a fuzzy sub-poset of M . Then, (µ, ρ) is a fuzzy sublattice of M , if µ is a fuzzy sublattice as a fuzzy algebra, i.e., if for all x, y ∈ M , (14)

µ(x ∧M y) ≥ µ(x) ∧L µ(y) and µ(x ∨M y) ≥ µ(x) ∧L µ(y).

Proposition 9. (µ, ρ) is a fuzzy sublattice of a lattice M , if and only if for every p ∈ L, the cut µp is a crisp sublattice of M . Proof. Let (µ, ρ) be a fuzzy sublattice of a lattice M , and let p ∈ L. Let x, y ∈ µp . Then µ(x) ≥ p and µ(y) ≥ p. By µ(x ∧M y) ≥ µ(x) ∧L µ(y) and µ(x ∨M y) ≥ µ(x) ∧L µ(y), we have that µ(x ∨M y) ≥ p and µ(x ∧M y) ≥ p, hence x ∨M y ∈ µp and x ∧M y ∈ µp . To prove the converse, for x, y ∈ M let µ(x) ∧L µ(y) = p ∈ L. Then µ(x) ≥ p and µ(y) ≥ p, hence x, y ∈ µp . Since µp is a sublattice, it follows that x ∧M y, x ∨M y ∈ µp , therefore µ(x ∧M y) ≥ p and µ(x ∨M y) ≥ p, which proves (14). 2 4. Fuzzy lattice ordered group Starting with the definition of a fuzzy ordered subgroup given in Preliminaries, now we introduce fuzzy lattice ordered subgroups. ( ) Let G, ·, −1 , e, 6 be an ordered group. Let also µ : G → L and ρ : G2 → L be a fuzzy set on G and a fuzzy relation on µ, respectively. The pair (µ, ρ) is a fuzzy ordered subgroup of G if the following hold: 1. µ is a fuzzy subgroup of G; 2. ρ is the fuzzy relation on µ defined by ρ(x, y) = µ(x) ∧ µ(y) ∧ k6 (x, y). Under the assumptions that G is a lattice under the above order 6, we say that (µ, ρ) is a fuzzy lattice ordered subgroup of G, or a fuzzy ℓ-subgroup of G if for every x ∈ G (15)

µ(x) ≤ µ(x ∨ e).

The following theorem shows that a fuzzy ℓ-subgroup is uniquely determined by the corresponding cuts.

ˇ SELJA, ˇ ˇ ´ AND MIRNA UDOVICI ˇ C ´ 10 BRANIMIR SE ANDREJA TEPAVCEVI C

Theorem 4. Let µ be a fuzzy subgroup of a lattice ordered group G and ρ a fuzzy relation on µ. Then, (µ, ρ) is a fuzzy ℓ-subgroup of G if and only if, for every p ∈ L, the cut µp is an ℓ-subgroup of G. Proof. Let µ : G → L be a fuzzy subgroup of G, and assume that (15) holds. For p = 0, we have that µp = G, and it is a trivial ℓ-subgroup of itself. Further, by Theorem 2, we have that µ is ordered by ρ : G2 → L, ρ(x, y) = µ(x) ∧ µ(y) ∧ k6 (x, y), and for every p ∈ L, p ̸= 0, (µp , ρp ) is an ordered subgroup of G. In addition, ρp is a suborder of 6 on G. Further, for every p ∈ L and every x ∈ G, x ∈ µp implies µ(x) ≥ p, hence by (15) µ(x ∨ e) ≥ p, i.e., x ∨ e ∈ µp . By Proposition 1, µp is an ℓ-subgroup of G. Conversely, let for every p ∈ L, the cut µp be an ℓ-subgroup of G. Then, for every x ∈ G, ∨ ∨ µ(x) = {p ∈ L | x ∈ µp } ≤ {p ∈ L | x ∨ e ∈ µp } = µ(x ∨ e), 2

and (15) holds.

By Theorem 4, all cuts of a fuzzy ℓ-subgroup of G should be ℓ-subgroups of the same group. However, in the crisp case, not every subgroup of a lattice ordered group G is an ℓ-subgroup of G, unless the group is linearly ordered. Therefore, it is reasonable to investigate the existence of fuzzy ℓ-subgroups of a lattice ordered group which is not necessarily linearly ordered. Our answer to this existence problem is affirmative for fuzzy convex ℓ-subgroups, as proven in the sequel. Observe that (µ, ρ) is a fuzzy convex ℓ-subgroup of a lattice ordered group G, if and only if by Lemma 2, every cut of µp is a convex ℓ-subgroup of G. In the following, for an element x in a group G, and a collection F of ℓ-subgroups of G closed under set intersection and containing G, we denote by ⟨x⟩F the smallest subgroup in F, containing x. Since the intersection of any family from F belongs to F, and G is also in F, the smallest subgroup in F containing x always exists and is equal to the intersection of all subgroups in F containing x. Theorem 5. Let (G, ·, −1 , e, 6) be a lattice ordered group, and L =SubG, i.e., L is the lattice of all subgroups of G, ordered dually to the set inclusion. Further, let H ⊆ L consist of all convex ℓ-subgroups of G. Let also µ : G → L be such that for every x ∈ G, µ(x) := ⟨x⟩H , and let ρ : G2 → L be the corresponding order defined by (8). Then, (µ, ρ) is a fuzzy ℓ-subgroup of G. Proof. First we prove that µ is closed under the binary operation · in G, i.e., that for x, y ∈ G, (16)

µ(x · y) ≥ µ(x) ∧ µ(y).


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The latter is, by the definition of order in L (which is dual to inclusion) and by the definition of µ, equivalent with ⟨x · y⟩H ⊆ ⟨x⟩H ∨SubG ⟨y⟩H , which, by the definition of join in SubG, holds if and only if ⟨x · y⟩H ⊆ ⟨⟨x⟩H ∪ ⟨y⟩H ⟩H . Here we also use the fact that the collection of all convex ℓ-subgroups of a lattice ordered group form a complete sub-lattice of the lattice of all subgroups of G (Proposition 2). Since the last formula holds as a property of subgroups, formula (16) is proved. Next, we show that for every x ∈ G, (17)

µ(x−1 ) ≤ µ(x).

Similarly as above, (17) is equivalent with ⟨x⟩H ⊆ ⟨x−1 ⟩H . Since obviously ⟨x⟩H = ⟨x−1 ⟩H , (17) holds. In addition, µ(e) = {e}, which is the top element in L, since the order is dual to inclusion. Thus we have proved that µ is a fuzzy subgroup of G. As a fuzzy sub-poset of (G, 6), it is fuzzy ordered by ρ : G2 → L, ρ(x, y) = µ(x) ∧ µ(y) ∧ k6 (x, y) (by Proposition 5). To prove that µ is fuzzy ℓ-ordered, we use Theorem 4, i.e., we show that every cut of µ is an ℓ-subgroup of G. Indeed, if p is a cut of µ then it is an ordered subgroup of G, i.e., p = K ∈ SubG, and µp = {x ∈ G | µ(x) ≥ p} = {x ∈ G | ⟨x⟩H ⊆ K} ∨ = {⟨x⟩H | x ∈ K and ⟨x⟩H ⊆ K} ∈ H, since by Proposition 2, H is a complete sublattice of L = SubG. Hence, µp is an ℓ-subgroup of G. 2 Next we characterize crisp totally ordered groups in terms of their Lvalued subgroups. Theorem 6. Let G be an ordered group and L a complete lattice. Then G is totally ordered if and only if every L-valued subgroup µ of G is a fuzzy ℓ-subgroup of G under the order ρ : G → L, ρ(x, y) = µ(x) ∧ µ(y) ∧ k6 (x, y). Proof. Let G be totally ordered. Then by Proposition 6, µ is also linearly ordered. Its cuts are subposets of M , which are linearly ordered i.e., they are sublattices of M . Therefore, µ is a fuzzy ℓ-subgroup of G. Conversely, suppose that every L-valued subgroup of an ordered group G is a fuzzy ℓ-subgroup. Then it also holds for all crisp subgroups of G, represented by the corresponding characteristic functions. By Proposition


2

3, G is totally ordered.

A condition under which a fuzzy ordered subgroup of an ordered group (which is not necessarily linearly ordered) is a fuzzy chain is given in the sequel. It is a straightforward consequence of Proposition 7. Proposition 10. A fuzzy subgroup (µ, ρ) of a lattice ordered group G is a fuzzy chain under ρ if for every pair of non-comparable elements x, y ∈ G, µ(x) ∧ µ(y) = 0. 5. Application: fuzzy ordered subgroups of quotient group of real numbers In this part we deal with ordered quotient groups and we present a procedure to obtain fuzzy subgroups of these, which are correspondingly ordered by a suitable fuzzy order. We give an example constructed on the ordered groups of real numbers. Let (G, · , −1 , e, 6) be an ordered group, H a convex normal subgroup of G, and G/H the corresponding quotient group, ordered by 6H , (2). Let also K be a fuzzy subgroup of G/H. Now, we define a fuzzy relation ρK : G/H × G/H → L as follows: (18)

ρK (x + H, y + H) = K(x + H) ∧ K(y + H) ∧ k6H (x + H, y + H).

By Proposition 5, it is obvious that ρK is a fuzzy order on K, hence (K, ρK ) is a fuzzy ordered subgroup of the quotient group G/H. In the following example, we present a concrete construction of a fuzzy ordered subgroup of a quotient group on R2 (R is the set of real numbers), and then we apply the above ordering (18), obtaining a fuzzy ordered subgroup of the mentioned quotient group. Example 1. Let (R2 , +, −, 0 6) be the (naturally) componentwise ordered group of the square R2 = R × R of real numbers. Let A = {(0, y) | y ∈ R}. R2 ,

A is a convex subgroup of which is straightforward. Next, let µ : R×R → L be an arbitrary fuzzy subgroup of R2 , and K : R2 /A → L defined by K((x, y) + A) := µ(x, 0). If ρK : G/H ×G/H → L is defined as in (18), then (K, ρK ) is a fuzzy ordered subgroup of the quotient group R2 /A. 6. Conclusion In this investigation we have used our approach to fuzzy ordered structures (fuzzy sets equipped with a particular fuzzy orderings) in order to deal with fuzzy lattice ordered groups. Our goal was not only to develop these fuzzy algebras, but also to present some nontrivial applications of these in


13

the ordered groups constructed on real numbers. It turned out that the new fuzzy structures are an appropriate generalization of the classical lattice ordered groups. Therefore, our next task would be to develop in more details the theory of fuzzy ordered algebras (special fuzzy algebras arising from lattices, other properties of ordered groups, but also ordered semigroups and other related structures), and also to characterize ordered and lattice ordered groups by their fuzzy substructures. 7. Acknowledgements The research of the first two authors was supported by Serbian Ministry of Science, Grant No. 174013 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, Grant “Ordered structures and applications”. References [1] M. Bakhshi, On fuzzy convex lattice–ordered subgroups, Iranian Journal of Fuzzy Systems 10 No 3 (2013) 159–172. [2] R. Bˇelohl´ avek, Fuzzy Relational Systems, Kluwer Academic Publishers, Dordrecht 2002. [3] S.K. Bhakat, P. Das, Fuzzy partially ordered fuzzy subgroups, Fuzzy Sets and Systems 67 2 (1994) 191-198. [4] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Providence R.I., 1967. [5] T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005. [6] U. Bodenhofer, M. De Cock, E.E. Kerre, Openings and closures of fuzzy preorderings: theoretical basics and applications to fuzzy rule-based systems, Int. J. General Systems. 01/2003 [32]:343–360. [7] U. Bodenhofer, A New Approach to Fuzzy Orderings, Tatra Mt. Math. Publ. Vol. 16 Part I, 1999 21-29. [8] U. Bodenhofer, Representations and constructions of similarity based fuzzy orderings, Fuzzy Sets and Systems 137 (2003) 113–136. [9] U. Bodenhofer, B. De Baets, J. Fodor, A compendium of fuzzy weak orders: Representations and constructions, Fuzzy Sets and Systems 158 (2007) 811–829. [10] I. Chon, Fuzzy partial order relations and fuzzy lattices, Korean J. Math. 17 (2009), No. 4, pp. 361–374. [11] B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1992. [12] M. Demirci, A theory of vague lattices based on many-valued equivalence relationsI: general representation results, Fuzzy Sets and Systems 151 (2005) 437-472. [13] M. Demirci, A theory of vague lattices based on many-valued equivalence relationsII: complete lattices, Fuzzy Sets and Systems 151 (2005) 473-489. [14] M. Demirci, Z. Eken, An Introduction to Vague Complemented Ordered Sets, Information Sciences 177(1) (2007) 150—160. [15] S. D´ıaz, B. De Baets and S. Montes, Additive decomposition of fuzzy pre-orders, Fuzzy Sets and Systems 158 (2007), 830–842. [16] J.A. Goguen, L-fuzzy Sets, J. Math. Anal. Appl. 18 (1967), 145–174. [17] U. H¨ ohle, Quotients with respect to similarity relations, Fuzzy Sets and Systems 27 (1988) 31-44. ˇ selja and A. Tepavˇcević, On lattice valued up-sets and [18] J. Jiménez, S. Montes, B. Seˇ down-sets, Fuzzy Sets and Systems 161 (2010) 1699–1710.


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