set, Ortho-convex refined fuzzy set.

The fuzzy set notion, which forms the foundation of the theory of possibility, was first presented by Zadeh [1], [2]. This fuzzy set notion was expanded upon by Dubois and Prade [3], and it was used in the analysis of differential gene expression data by Liang et al. [4]. Similarity metrics for fuzzy sets were later established by Beg and Ashraf [5]. Set difference and symmetric difference of fuzzy sets were described by Vemuri et al. [6]. The work was expanded to include the complement of an extended fuzzy set by Neog and Sut [7]. Yager [8] created Pythagorean fuzzy subsets, talked about their characteristics and uses, and then applied them to multi-criteria decision making. Biswas [9], McBratney and Odeh [10] and Mamdani [11] applied fuzzy sets in students' evaluation, soil science, and control of simple dynamic plants.

Chaudhuri [3], [4] presented the idea of a concave fuzzy set, talked about some of its practical characteristics, and explained some related ideas and their computing methods. The development of fuzzy geometry and fuzzy structures benefits from this idea. This idea was expanded to include convex and concave fuzzy mappings by Syau [14]. Concavo-convex fuzzy sets were introduced by Sarkar [15], who also established some intriguing characteristics of this unique kind of fuzzy set. Concave type-2 fuzzy sets were created by Tahayori et al. [16], who also constructed and validated its operations and attributes. Measures of fuzziness for concave functions and measures of fuzzy sets were described by Weber [17]. Fuzzy uncertainty measurement for monotone rising concave functions was described by Pal and Bezdek [18].

The concept of fuzzy sets was improved by Smarandache [19], who created refined ternary fuzzy sets, refined picture fuzzy sets, refined intuitionistic fuzzy sets, and refined intuitionistic fuzzy sets. More refinements in fuzzy sets, including refined Pythagorean fuzzy sets, refined Atanassov's intuitionistic fuzzy sets of type 2, refined spherical fuzzy sets, refined n-hyper spherical fuzzy sets, refined q-rung orthopair fuzzy sets, etc., were made possible by this idea of refinement.

This study develops a new concept of convexity cum concavity on refined fuzzy sets and extends the concept of convexity cum concavity on fuzzy sets. Furthermore, in this context, certain significant traits and outcomes are highlighted.

The remainder of the paper is structured as follows. Section 2 reviews the fundamental definitions and terminology used in the literature in relation to the main finding; Section 3 introduces the novel idea of convexity cum concavity on refined fuzzy sets along with various properties and results; and Section 4 wraps up the paper.

In this section, some basic definitions and terms are recalled from the literature to support the main study of this work. Here, Z, G, and I will play the role of the universal set, R and [0, 1], respectively.

Definition 1 ([1]). Set A is a fuzzy set if A is characterized by a membership function ΓA and defined A = {(ΓA(α)), α ∈ Z}, ΓA: Z → P(I), where A ⊆ Z.

The union of two fuzzy sets A and B with respective membership functions fA(α) and fB(α) is a fuzzy set C, written as C = A ∪ B, whose membership function is related to those of A and B by fC(α) = Max(fA(α), fB(α)), α ∈ Z, or in abbreviated form fC = fA ∨ fB.

The intersection of two fuzzy sets A and B with respective membership functions fA(α) and fB(α) is a fuzzy set C, written as C = A ∩ B, whose membership function is related to those of A and B by fC(α) = Min(fA(α), fB(α)), α ∈ Z, or in abbreviated form fC = fA ∧ fB.

Definition 2 ([1]). A fuzzy set A is convex iff fA(λα1 + (1 − λ)α2) ≥ Min(fA(α1), fA(α2)), for all α1, α2 ∈ Z and λ ∈ I.

Definition 3 ([1]). A fuzzy set A is concave iff fA(λα1 + (1 − λ)α2) ≤ Max(fA(α1), fA(α2)), for all α1, α2 ∈ Z and λ ∈ I.

Definition 4 ([19]). A refined fuzzy set A is defined as

1 2 p

ARFS = {(ΓA(α), ΓA (α), . . . , Γ (α), p ≥ 2, α ∈ A)},

j th

where Γ is sub-membership of degree j -type of elements of Z w.r.t. A, and is subset of I for 1 ≤ j ≤ p and

∑ sup Γα ≤ 1, for all α ∈ A.

In this section, convex and concave refined fuzzy sets are defined. Some important results are discussed. Definition 5. A refined fuzzy set ARFS in G is Convex if for all u, v ∈ G and all w on the line segment uv ΓA (w) ≥ min(ΓA (u), ΓA (v)) , 1 ≤ k ≤ n,

k th

where ΓA is sub-membership of degree k -type of the elements w.r.t. A and is subset of I for 1 ≤ k ≤ p

RFS

and ∑ sup Γ ≤ 1.

Definition 6. A refined fuzzy set ARFS in G is ortho-convex if for all u, v ∈ G and all w on the line segmentuv lie on a line that is parallel to the co-ordinate axis

j ′ j ′ j ′

Γ (w ) ≥ min (Γ (u ), Γ (v )) , 1 ≤ j ≤ n.

ARFS ARFS ARFS

Remark 1. An ortho-convex RFS is convex RFS, but its converse may or may not be true.

Definition 7. A refined fuzzy set ARFS in G is concave if for all u, v ∈ G and all w on the line segment uv ΓA (w) ≤ max (ΓA (u), ΓA (v)) , 1 ≤ k ≤ n.

Definition 8. A refined fuzzy set ARFS in G is ortho-concave if for all u, v ∈ G and all w on the line segmentuv lie on line which is parallel to co-ordinate axis

j ′ j ′ j ′

Γ (w ) ≤ max(Γ (u ), Γ (v )) , 1 ≤ j ≤ n.

ARFS ARFS ARFS

Remark 2. An ortho-concave RFS is a concave RFS, but its converse may or may not be true.

Theorem 1. The complement of convex A RFS is concave RFS.

Proof: if A is convex refined fuzzy then for any two points u and v and another point w which lies on uv ΓA (w) ≥ min(ΓA (u), ΓA (v)) , 1 ≤ k ≤ n, so it becomes

ΓA (w) ≤ 1 − min (1 − ΓA (u), 1 − ΓA (v)) , 1 ≤ k ≤ n. (1)

Now if 1 − ΓA (u) ≤ 1 − ΓA (v).

Then, min(1 − ΓA (u), 1 − ΓA (v)) = 1 − ΓA (u),

and Eq. (1) takes the form

Γ (w) ≤ Γ (u). (2)

Similarly if 1 − ΓA (v) ≤ 1 − ΓA (u), then min(1 − ΓA (u), 1 − ΓA (v)) = 1 − ΓA (v), so Eq. (1) reduces to

Γ (w) ≤ Γ (v). (3)

Using Eqs. (2) and (3), it becomes ΓA (w) ≤ max (ΓA (u), ΓA (v)) , 1 ≤ k ≤ n, which means compliment of ARFS is concave RFS.

Remark 3. The complement of ortho-convex ARFS is ortho-concave and hence concave RFS.

Theorem 2. The union of two convex refined fuzzy sets is a convex refined fuzzy set.

Proof: let ARFS and BRFS be two convex refined fuzzy sets and H = ARFS ∪ BRFS.

Consider two points u and v and another point w which lies on uv.

ΓH (u) = min (Γ (u), ΓB (u)) , 1 ≤ k ≤ n, (4) ΓH (v) = min(ΓA (v), ΓB (v)) , 1 ≤ k ≤ n, (5) ΓH (w) = min(ΓA (w), ΓB (w)) , 1 ≤ k ≤ n. (6)

min(ΓH (u), ΓH (v))

k k k k

= min(min(ΓA (u), ΓB (u)) , min(ΓA (v), ΓB (v))) (7)

k k k k

= min(ΓA (u), ΓB (u), ΓA (v), ΓB (v)).

Let ΓA (w) ≤ ΓB (w) so Eq. (7) becomes ΓH (w) = ΓA (w), as A is refined convex fuzzy set so k k k k k k k

ΓA (w) ≥ min(ΓA (u), ΓA (v)) ≥ min(ΓA (u), ΓB (u), ΓA (v), ΓB (v)) ,

RFS RFS RFS RFS RFS RFS RFS

k k k k

ΓA (w) = ΓH (w) ≥ min(ΓH (u)ΓH (v)).

Similarly for ΓB (w) ≤ ΓA (w), Eq. (6) takes the form

ΓH (w) = ΓB (w), (8)

as B is a refined convex fuzzy set, so Eq. (8) becomes

k k k k k k k

ΓB (w) ≥ min(ΓB (u), ΓB (v)) ≥ min (ΓA (u), ΓB (u), ΓA (v), ΓB (v)),

RFS RFS RFS RFS RFS RFS RFS

ΓH (w) ≥ min(ΓH (u)ΓH (v)).

Hence, the proof is complete.

Theorem 3. The union of two ortho-convex RFS is an ortho-convex RFS and hence convex RFS.

Proof: let ARFS and BRFS be two convex refined fuzzy sets and H = ARFS ∪ BRFS.

Consider two points u and v′ and another point w′ which lies on u′v′, which is parallel to coordinate axis.

k ′ k ′ k ′

ΓH (u ) = min(ΓA (u ), ΓB (u )) , 1 ≤ k ≤ n, (9)

k ′ k ′ k ′

ΓH (v ) = min (Γ (v ), ΓB (v )) , 1 ≤ k ≤ n, (10)

k ′ k ′ k ′

ΓH (w ) = min (ΓA (w ), ΓB (w )) , 1 ≤ k ≤ n. (11)

Take

k ′ k ′

min(ΓH (u ), ΓH (v ))

k ′ k ′ k ′ k ′

= min(min(ΓA (u ), ΓB (u )) , min(ΓA (v ), ΓB (v ))) (12)

k ′ k ′ k ′ k ′

= min(ΓA (u ), ΓB (u ), ΓA (v ), ΓB (v )) .

k ′ k ′

Let ΓA (w ) ≤ ΓB (w ) in Eq. (12) so that

k ′ k ′

ΓH (w ) = ΓA (w ), as A is ortho-refined convex fuzzy set so

k ′ k ′ k ′ k ′ k ′ k ′ k ′

ΓA (w ) ≥ min (ΓA (u ), ΓA (v )) ≥ min(ΓA (u ), ΓB (u ), ΓA (v ), ΓB (v )) ,

RFS RFS RFS RFS RFS RFS RFS k ′ k ′ k ′ k ′

ΓA (w ) = ΓH (w ) ≥ min (ΓH (u )ΓH (v )).

k ′ k ′

Similarly for ΓB (w ) ≤ ΓA (w ), Eq. (11) takes the form

k ′ k ′

ΓH (w ) = ΓB (w ), as B is an ortho-refined convex fuzzy set, so

k ′ k ′ k ′ k ′ k ′ k ′ k ′

ΓB (w ) ≥ min (ΓB (u ), ΓB (v )) ≥ min (ΓA (u ), ΓB (u ), ΓA (v ), ΓB (v )) ,

RFS RFS RFS RFS RFS RFS RFS

k ′ k ′ k ′ k ′

ΓB (w ) = ΓH (w ) ≥ min(ΓH (u )ΓH (v )).

Since every ortho-convex refined fuzzy set is also a convex refined fuzzy set which leads to the completion of proof.

Remark 4. The union of a family of convex refined fuzzy sets is a convex refined fuzzy set.

Remark 5. The union of the family of ortho-convex refined fuzzy sets is ortho-convex refined fuzzy, and hence convex refined fuzzy set.

Theorem 4. The complement of concave ARFS is convex RFS.

Proof: if A is concave refined fuzzy, then for any two points u and v and another point w which lies on uv. ΓA (w) ≤ max (ΓA (u), ΓA (v)) , 1 ≤ k ≤ n, so it becomes

ΓA (w) ≥ 1 − max (1 − ΓA (u), 1 − ΓA (v)) , 1 ≤ k ≤ n. (13)

Now if 1 − ΓA (u) ≤ 1 − ΓA (v), then max (1 − ΓA (u), 1 − ΓA (v)) = 1 − ΓA (v), Eq. (13) takes the form

Γ (w) ≥ Γ (v). (14)

Similarly if 1 − ΓA (v) ≤ 1 − ΓA (u), then max (1 − ΓA (u), 1 − ΓA (v)) = 1 − ΓA (u), so Eq. (13) reduces to Γ (w) ≥ Γ (u). (15)

Eqs. (14) and (15) results in ΓA (w) ≥ min (ΓA (u), ΓA (v)) , 1 ≤ k ≤ n,

which means compliment of ARFS is convex RFS.

Remark 6. The complement of ortho-concave ARFS is ortho-convex and hence convex RFS.

Theorem 5. The union of two concave refined fuzzy sets is a concave refined fuzzy set.

Proof: let ARFS and BRFS be two convex refined fuzzy sets and H = ARFS ∪ BRFS.

Consider two points u and v and another point w which lies on uv.

ΓH (u) = max(Γ (u), ΓB (u)) , 1 ≤ k ≤ n, (16) ΓH (v) = max (ΓA (v), ΓB (v)) , 1 ≤ k ≤ n, (17) ΓH (w) = max (ΓA (w), ΓB (w)) , 1 ≤ k ≤ n. (18)

Consider max (ΓH (u), ΓH (v))

k k k k

= max (max(ΓA (u), ΓB (u)) , max(ΓA (v), ΓB (v))) (19)

k k k k

= max (ΓA (u), ΓB (u), ΓA (v), ΓB (v)).

Using ΓA (w) ≥ ΓB (w) in Eq. (18) reduces it to

ΓH (w) = ΓA (w), (20)

as A is refined concave fuzzy so, Eq. (20) becomes

k k k k

ΓH (w) = ΓA (w) ≤ max (ΓA (u), ΓA (v)) ,

k k k k k

ΓH (w) ≤ max (ΓA (u), ΓB (u), ΓA (v)ΓB (v)).

RFS RFS RFS RFS RFS

ΓH (w) ≤ max (ΓH (u)ΓH (v)) .

Similarly, if ΓB (w) ≥ ΓA (w), then Eq. (18) results in

ΓH (w) = ΓB (w), (21)

as B is a refined concave fuzzy set so, Eq. (21) becomes

k k k k

ΓH (w) = ΓB (w) ≤ max (ΓB (u), ΓB (v)) , k k k k k

ΓH (w) ≤ max (ΓA (u), ΓB (u), ΓA (v)ΓB (v)),

RFS RFS RFS RFS RFS

ΓH (w) ≤ max (ΓH (u)ΓH (v)).

Hence, the required result is proved.

Theorem 6. The union of two ortho-concave RFS is an ortho-concave RFS and hence concave RFS. Proof: let ARFS and BRFS be two convex refined fuzzy sets and H = ARFS ∪ BRFS. Consider two points u′ and v′ and another point w′ on u′v′ with condition that u′v′ is parallel to coordinate axis.

as

k ′ k ′ k ′

ΓH (u ) = max (Γ (u ), ΓB (u )) , 1 ≤ k ≤ n, (22)

k ′ k ′ k ′

ΓH (v ) = max (ΓA (v ), ΓB (v )) , 1 ≤ k ≤ n, (23)

k ′ k ′ k ′

ΓH (w ) = max(ΓA (w ), ΓB (w )) , 1 ≤ k ≤ n. (24)

k ′ k ′

max (ΓH (u ), ΓH (v ))

k ′ k ′ k ′ k ′

= max (max(ΓA (u ), ΓB (u )) , max(ΓA (v ), ΓB (v ))) (25)

k ′ k ′ k ′ k ′

= max (ΓA (u ), ΓB (u ), ΓA (v ), ΓB (v )) .

k ′ k ′

Let ΓA (w ) ≥ ΓB (w ) in Eq. (25) so that

k ′ k ′

ΓH (w ) = ΓA (w ), as A is ortho-concave refined fuzzy, so

k ′ k ′ k ′ k ′

ΓH (w ) = ΓA (w ) ≤ max (ΓA (u ), ΓA (v )),

k ′ k ′ k ′ k ′ k ′

ΓH (w ) ≤ max(ΓA (u ), ΓB (u ), ΓA (v )ΓB (v )) ,

RFS RFS RFS RFS RFS

k ′ k ′ k ′

ΓH (w ) ≤ max(ΓH (u )ΓH (v )) .

k ′ k ′

Similarly for ΓB (w ) ≥ ΓA (w ), Eq. (24) takes the form

k ′ k ′

ΓH (w ) = ΓB (w ), (26)

as B is an ortho-refined concave fuzzy set so, Eq. (26) becomes

k ′ k ′ k ′ k ′

ΓH (w ) = ΓB (w ) ≤ max (ΓB (u ), ΓA (v )) ,

k ′ k ′ k ′ k ′ k ′

ΓH (w ) ≤ max(ΓA (u ), ΓB (u ), ΓA (v )ΓB (v )) ,

RFS RFS RFS RFS RFS k ′ k ′ k ′

ΓH (w ) ≤ max(ΓH (u )ΓH (v )).

Since every ortho-concave refined fuzzy set is also a concave refined fuzzy set, which leads to the completion of the proof.

Remark 7. The union of a family of concave refined fuzzy sets is a concave refined fuzzy set.

Remark 8. The union of the family of ortho-concave refined fuzzy sets is ortho-concave refined fuzzy and hence concave refined fuzzy set.

Definition 9. For any point p ∈ l where l is a line, lp is perpendicular to l at A, the Inf-Projection Al of concave refined fuzzy set A in R2 is mapping of each point p ∈ l into inf{A(r), r ∈ l }.

Definition 10. For any point p ∈ l where l is a line, lp is perpendicular to l at A, the Sup-Projection Al of concave refined fuzzy set A in R is mapping of each point p ∈ l into sup{A(r), r ∈ lp}.

Theorem 7. If A is concave RFS, so is Al.

Proof: if u, v, w are three points of l such that w lies on uv, given any ε > 0, let u and v be points on lu and

k k ′ k k ′ ′ ′ ′

lv so that ΓA (u) > ΓA (u ) − ε and ΓA (v) > ΓA (v ) − ε. Let w be the intersection of line segment u v with lw.

′ ′ ′

Since A is concave and w ∈ u v , then we have

k ′ k ′ k ′

ΓA (w ) ≤ max (ΓA (u ), ΓA (v )) , 1 ≤ k ≤ n, < max(ΓA (u) + ϵ, ΓA (v) + ε) = max (ΓA (u), ΓA (v)) + ε.

But by the definition of inf-projection

k ′ k

ΓA (w ) ≥ ΓA (w).

Hence ΓA (w) < max (ΓA (u), ΓA (v)) + ε.

l l l

Since ε > 0 is arbitrary so ΓA (w) ≤ max (ΓA (u), ΓA (v)),

l l l

which means Al is concave.

Remark 9. If A is convex RFS, so is Al.

In this paper, convexity cum concavity is defined on a refined fuzzy set, and some useful results are established. This work can further be extended by developing convex hull, convex cone, α-cuts, and other types of convexity like graded convexity, triangular convexity, etc.

The auhors are thankful to the reviewers for their valuable comments to improve the paper.

Conceptualization, M.A. and M.S.; Methodology, M.A. and A.U.R..; Software, M.A. and A.U.R.; Validation, M.A., M.S. and A.U.R..; formal analysis, M.S.; investigation, M.A. and A.U.R..; data maintenance, M.A. and A.U.R.; writing-creating the initial design, M.A.; writing-reviewing and editing, M.S. and A.U.R..; visualization, M.S. and A.U.R.. All authors have read and agreed to the published version of the manuscript.

The author has approved the manuscript for publication.

This article does not include experiments involving humans or animals.

Not applicable.

No data is associated with this study.

The authors declare no conflict of interest.