Interval min-plus algebraic structure and matrices over interval min-plus algebra

Max-plus algebra is the set ℝ max or ℝ ε = ℝ ∪ { ε } where ℝ is the set of all real number and ε = −∞ which is equipped with maximum (⊕) and plus (⊗) operations. The structure of max-plus algebra is semifield. Another semifield that can be learned is min-plus algebra. Min-plus algebra is the set ℝ min or ℝ ε ′ = ℝ ∪ { ε ′ } where ε ′ = ∞ which is equipped with minimum (⊕ ′) and plus (⊗) operations. Max-plus algebra has been generalized into interval max-plus algebra, so that min-plus algebra can be developed into an interval min-plus algebra. Interval min-plus algebra is defined as a set I ( ℝ ) ε ′ = { x = [ x _ , x ¯ ] | x _ , x ¯ ∈ ℝ , x _ ≤ x ¯ < ε ′ } which have minimum ( ⊕ ¯ ′ ) and addition ( ⊗ ¯ ) operations. A matrix in which its components are the element of ℝ ε is called matrix over max-plus algebra. Matrices over max-plus algebra has been generalized into interval matrices in which its components are the element of I ( ℝ ) ε . This research will discusses the interval min-plus algebraic structure and matrices over interval min-plus algebra.