The largest solution of linear equation over the complete Heyting algebra

Let (L, ≤, ∨, ∧) be a complete Heyting algebra. In this article, the linear system Ax = b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by ∨ and ∧ respectively, is studied. We obtain: (i) the necessary and sufficient conditions for S(A,b) ≠ θ; (ii) the necessary conditions for |S(A,b)| = 1. We also obtain the vector and prove that it is the largest element of S(A,b) if S(A,b) ≠ θ.