Applied Science and Computer Mathematics, 2(1) (2021), 1-8 www.furthersci.com Further Science Rough interval valued Pythagorean fuzzy ideals in ternary semigroups V. Chinnadurai *† , A. Arulselvam ‡ Received: 12 January 2021; Accepted: 30 January 2021; Published: 5 February 2021 Abstract In this paper, we discuss Rough interval valued Pythagorean fuzzy sets in ternary semigroups, Rough interval valued Pythagorean fuzzy ideals in ternary semigroups (RIVPFITS) and investigate some inter- esting properties. 1 Introduction Zadeh[16, 17] introduced the concept of fuzzy set and several of researchers have studied the generalizations of the fuzzy set. In all such generalizations the membership/nonmembership values lie in the closed interval [0,1]. Atanassov et al.[2, 3] introduced the notion of interval-valued intuitionistic fuzzy sets. Many researchers applied the concept of interval-valued intuitionistic fuzzy sets to algebraic structures. In 1932, Lehmer[9] discussed the existence of ternary operations that originated from the study of a ternary analog of the Abelian group. In 1982, Pawlak[10] initiated the fundamental concept of rough set. Ansari and Yaqoob[1] discussed the concept of T-rough ternary subsemigroups, T-rough ideals, T-rough bi-ideals, and T-rough interior ideals in ternary semigroups. Lekkoksung[8] introduced the notion of intuitionistic fuzzy bi-ideals in a ternary semigroup. The rough intuitionistic fuzzy ideals in semigroups were introduced by Jayanta Ghosh et al.[6]. In 1997 the idea of rough ideals in semigroup was presented by Kuroki[7]. Thillaigovindan et al.[12] discussed rough ideals in -semigroup. Thillaigovindan and Chinnadurai[13] discussed on interval- valued fuzzy quasi- ideals of semigroups. In 2013, Yager[14, 15] initiated the notion of Pythagorean fuzzy set the sum of their squares of membership, and non-membership belongs to the unit interval [0,1]. Hussain et al.[5] presented the idea of rough Pythagorean fuzzy ideals in semigroups (RPFIS) and discussed lower and upper approximations of Pythagorean fuzzy ideals, bi-ideals, and interior ideals in semigroups. Peng et al.[11] presented some results on Pythagorean fuzzy sets. In this paper, we introduce the concept of rough interval-valued Pythagorean fuzzy ideals in ternary semigroups(RIVPFITS) and investigate some of its properties. 2 Preliminaries Throughout this paper S denotes a semigroup and TS the ternary semigroup associated with S. Definition 1 [9] A TS is a nonempty set S togeather with a ternary operation (p,q,r)= ⇒ [pqr] satisfying the associative law [[pqr]xy]=[p[qrx]y]=[pq[rxy]] ∀ p,q,r,x,y ∈ S. Definition 2 [9] Let S and TS be an above. A transformation m : T → [0, 1] is called fuzzy subset of S. Let m and n be fuzzy subsets of TS. Then (i) m ⊆ n if m(w) ≤ n(w) ∀ w ∈ S. * Department of Mathematics, Annamalai University, Chidambaram 608002, India † Corresponding author email: chinnaduraiau@gmail.com ‡ Department of Mathematics, Annamalai University, Chidambaram, India 1