The purpose of this paper is to introduce and discuss the concept of T-rough ternary subsemigroups, T-rough ideals, T-rough bi-ideals and T-rough interior ideals in ternary semigroups. Proof. The proof is similar to the proof of Theorem 3.12. Theorem 3.16. Let S 1 and S 2 be two ternary semigroups and T : Proof. The proof is similar to the proof of Theorem 3.13. Theorem 3.17. Let S 1 and S 2 be two ternary semigroups and T : S 1 → P (S 2 ) be a strong set-valued homomorphism. If A is a

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... al of S 2 , then Apr T (A) is, if it is non-empty, a quasi-ideal of S 1 . Proof. Let A be a quasi-ideal of S 2 , we have AS 2 S 2 ∩ S 2 AS 2 ∩ S 2 S 2 A ⊆ A and AS 2 S 2 ∩ S 2 S 2 AS 2 S 2 ∩ S 2 S 2 A ⊆ A. So T (AS 2 S 2 ∩ S 2 AS 2 ∩ S 2 S 2 A) ⊆