Annihilator in ternary semiring

Introduction

In 1932, Lehmer [1] explored ternary algebraic system theory. He studied "triplexes", which are ternary algebraic systems that finally turn out to be ternary groups. A generalization of ternary rings are called ternary semirings, as presented by Dutta et al [2] . Ternary semiring is a superset of semiring. Negative integers form a ternary semiring but it does not form a semiring. In 1920, E.H. Moore established the concept of the generalized inverse of a matrix, defining the unique generalized inverse through the use of matrices projectors. Penrose demonstrated in 1955 that Moore's inverse is the unique matrix that can satisfy four equations [3] . The notion of regular element was first introduced by Von Neumann in 1936 [4] . We present some properties of generalized inverses, elements in Ternary semirings that we use throughout the article and particular generalized inverses are defined or studied by means of their associated principal or annihilator ideals. The several characterizations of these inverses are given. Some new properties are obtained and some known results are generalized.

Preliminaries

In this work, T represents the Ternary Semiring; it is abbreviated as T.S.R. throughout this paper. Remark: z -is an natural example of a ternary semiring, which has ternary multiplication and regular addition. Definition 2.2. If an element 𝑎 exists in T such that 𝑎𝑏𝑎𝑏𝑎 = 𝑎, then 𝑎 is considered regular. The element 𝑏 is represented as 𝑎 -and is also known as the generalized inverse or 1-inverse of the element ′𝑎′and the set of all 1-inverses is denoted by 𝑎{1}.If every element in 𝑎 T.S.R is regular, then we say T.S.R is regular. Definition 2.3. If there is an element 𝑎 in 𝑇 ∃ 𝑏𝑎𝑏𝑎𝑏 = 𝑏, then 𝑎 is considered anti-regular. The name for element 𝑏 is 2-inverse of 𝑎 and set of all 2-inverses of 𝑎 is denoted by 𝑎{2}.

The both conditions in Definition (2.2) and Definition (2.3) satisfied by a single element 𝑏 is called 1-2 inverse and is denoted by 𝑎 ̂. In Definition (2.2), if the ′𝑏′ is unit then ′𝑎′ is called unit regular. Definition 2.4. An element 𝑎 ∈ 𝑇 has Moore penrose inverse if there exists an 𝑏 ∈ 𝑇 such that the following equations hold: (1) 𝑎𝑏𝑎𝑏𝑎 = 𝑎 (2) 𝑏𝑎𝑏𝑎𝑏 = 𝑏 (3) (𝑎𝑏𝑒) * = 𝑎𝑏𝑒 (4)(𝑒𝑏𝑎) * = 𝑒𝑏𝑎. Where 𝑒 is identity element, here 𝑏 is unique and denoted by 𝑎 † . In Definition (2.4), the element 𝑏 is satisfy the equations ( 1) and (3), then we say that element is {1,3}-inverse of ′𝑎 ′ and 𝑏 satisfy the equation ( 1) and (4) then we say that 𝑏 is {1,4}-inverse of ′𝑎′. Definition 2.7. For any 𝑡 1 , 𝑡 2 ∈ 𝑇 and 𝑖 ∈ 𝐼, an additive subsemigroup𝐼 of 𝑇 is considered a left (right, lateral) ideal of 𝑇 if 𝑡 1 𝑡 2 𝑖 (similarly 𝑖𝑡 1 𝑡 2 , 𝑡 1 𝑖𝑡 2 ) ∈ 𝐼. 𝐼 is said to as an ideal 𝑇 if𝐼 is both a left, right, and lateral ideal of 𝑇.

Results in ternary semiring Proposition 3.1. [6] Let𝑚, 𝑚 -∈ 𝑇, Then the following holds:

∘ Proposition.3.2 [6] Let 𝑚 ∈ 𝑇. From Equations ( 5), ( 6), ( 7) and ( 8 Proposition 3.3. [7] Let 𝑚 ∈ 𝑇. Then we have the following results: 1. 𝑚 is {1,3} invertible with {1,3} inverse 𝑦 iff 𝑦 * 𝑚 * 𝑚𝑦𝑚 = 𝑚 2. 𝑚 is {1,4} invertible with {1,4} inverse 𝑦 iff 𝑚𝑥𝑚𝑚 * 𝑦 * = 𝑚.

Theorem 3.1. [8] Let 𝑚, 𝑛 ∈ 𝑇and 𝑛 -∈ 𝑛{1} Then Similarly, 𝑇𝑒𝑦 = 𝑇𝑚 -𝑚 Theorem 3.4. [8] Let 𝑚, 𝑦 ∈ 𝑇, The following statements are equivalent 1. 𝑦 is Moore-Penrose inverse of 𝑚 2. 𝑦𝑒𝑇 = 𝑚 * 𝑦 * 𝑇 and 𝑇𝑒𝑚 * = 𝑇𝑚𝑦 From the equations ( 5),( 6), ( 7) and ( 8

Conclusion

In this paper, we characterize annihilators in ternary semirings by means of appropriate ideals and Moore-Penrose inverses.

References

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