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Necessary and Sufficient Conditions for The Solutions of Linear Equation System

Gregoria Ariyanti
Jurnal Matematika, Statistika dan Komputasi , Volume 17, pp 82-88; doi:10.20956/jmsk.v17i1.10352

Abstract: A Semiring is an algebraic structure (S,+,x) such that (S,+) is a commutative Semigroup with identity element 0, (S,x) is a Semigroup with identity element 1, distributive property of multiplication over addition, and multiplication by 0 as an absorbent element in S. A linear equations system over a Semiring S is a pair (A,b) where A is a matrix with entries in S and b is a vector over S. This paper will be described as necessary or sufficient conditions of the solution of linear equations system over Semiring S viewed by matrix X that satisfies AXA=A, with A in S. For a matrix X that satisfies AXA=A, a linear equations system Ax=b has solution x=Xb+(I-XA)h with arbitrary h in S if and only if AXb=b.
Keywords: structure / Sufficient Conditions / Semiring / Linear Equations System / Satisfies / aXa

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