On Soft Ternary Γ-Semirings-I

In this paper with the help of soft set theory the notion of soft ternary Γ-semiring and soft ternary Γ-sub semiring are introduced. Some of the properties of soft ternary Γ-semiring are discussed and some illustrations are also given. It is proved that (1) a soft set ( q , P1, Γ) over a ternary Γ-semiring T is a soft ternary Γ-semiring over T if and only if for all x ∈ P 1 ; q ( x ) ≠ Ø a ternary Γ-subsemiring of T. (2) A soft set ( r , P1, Γ) over T is a soft L(R, M) Γ-ideal over T iff ∀ p 1 ∈ P 1 ; r ( p 1 ) ≠ Ø is a L(R, M) Γ -ideal of T. (3) Let ( q , P1, Γ ) and ( r , P2, Γ ) be two soft ternary Γ -semirings over T, such that P 1 ∩ P 2 ≠ Ø Then ( q , P1, Γ ) ∩ r ( r , P2, Γ ) is a soft ternary Γ -semiring over T. (4) Let ( q , P1, Γ ) and ( r , P2, Γ ) are two soft ternary Γ -semirings over T, such that P 1 ∩ P 2 ≠ Ø Then ( q , P1, Γ ) ∪ E ( r , P2, Γ ) is a soft ternary Γ -semiring over T. (5) Let ( q , P1, Γ ) and ( r , P2, Γ ) are two soft ternary Γ -semirings over T. Then ( q , P1, Γ ) Λ( r , P2, Γ ) is a soft ternary Γ -semiring over T. (6) Let ( q , P1, Γ ) and ( r , P2, Γ ) are two soft ternary Γ -semirings over T. Then ( q , P1, Γ ) Λ( r , P2, Γ ) is a soft ternary Γ -semiring over T. (7) Let ( q , P1, Γ ), ( r , P2, Γ ) and ( s , P3, Γ) are any three soft ternary Γ -semirings over a commutative ternary Γ -semiring T. Then ( q , P1, Γ ) ⊚( r , P2, Γ ) ⊚( s , P3, Γ)is a soft ternary Γ -semiring over T. (8) Let ( q , P 1 , Γ), ( r , P 2 , Γ) are any two soft Γ -ideals over a ternary Γ -semiring T with P 1 ∩ P 2 ≠ Ø Then ( q , P 1 , Γ) ∪R ( r , P 2 , Γ) is a soft Γ -ideal over T contained in both ( q , P 1 , Γ), ( r , P 2 , Γ). (9) Let ( q , P 1 , Γ), ( r , P 2 , Γ) are any two soft Γ -ideals over a ternary Γ -semiring T. Then ( q , P 1 , Γ). E ( r , P 2 , Γ) is also soft Γ -ideal over T contained in both ( q , P 1 , Γ) and( r , P 2 , Γ). In section 5, the term soft ternary Γ-sub semiring is introduced and some examples are given and characterized the soft ternary Γ-semirings.