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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.

A simple graph G = (V(G), E(G)) admits a H-covering, where H is subgraph of G, if every edge in E(G) belongs to a subgraph of G that is isomorphic to H. An (a, d)-H-antimagic total labeling of G is a bijective function ξ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , | V ( G ) | + | E ( G ) | } , such that for all subgraphs H’ isomorphic to H, the H’ weights w(H’) = ∑ v∈V(H’) ξ(v) + ∑ e∈E(H’) ξ(e) constitute an arithmetic progression a, a + d, a + 2d, …, a + (k – 1)d where a and d are positive integers and k is the number of subgraphs of G isomorphic to H. Such a labeling is called super if the smallest possible labels appear on the vertices. This research has found super (a, d)-H-antimagic total labeling of edge corona product of cycle and path denoted by Cm ◊ Pn with H is P 2 ◊ Pn and super (a, d)-P 2 ◊ Cn -antimagic total labeling of Cm ◊ Cn .

In classical linear algebra, a basis is a vector set that generates all elements in the vector space and that vector set is a linear independence set. However, the definitions of the linear dependence and independence in min-plus algebra are little more complex given that the min-plus algebra is the linear algebra over the commutative idempotent semiring. The definition of the linear dependence (independence) is used in this paper is Gondran-Minoux linear dependence (independence). A finite set is Gondran-Minoux linearly dependent if the set can be divided into two sets that form a linear space with an intersection which is not a zero vector. We will define the concept of the bases in min-plus algebra. In this paper also defined the concept of a weak bases and will be shown that the linear dependence (independence) is not needed to form a weak basis. In the last part of the research result's are proven that every basis in a semimodules in min-plus algebra is a weak basis.

The annihilator graph of a semiring S, denoted by AG(S), is the graph whose vertex set is the set of all nonzero zero-divisors of S. In commutative semiring S, two distinct vertices are adjacent if and only if ann(xy) ≠ ann(x) ∪ ann(y), where ann(x) = {s ∈ S|sx = 0}. Similarly in noncommutative semiring S, two distinct vertices are connected by an edge if and only if either l. ann(xy) ≠ l. ann(x) ∪ l. ann(y), l. ann(yx) ≠ l. ann(x) ∪ l. ann(y), r. ann(xy) ≠ r. ann(x) ∪ r. ann(y), or r. ann(yx) ≠ r. ann(x) ∪ r. ann(y) where l. ann(x) = {s ∈ S|sx = 0} and r. ann(x) = {s ∈ S|xs = 0}. In this paper we study the properties of the right annihilator and the left annihilator of semiring of matrices over Boolean semiring Mn (ℬ) and then use these results to determine the diameter of the graph AG(Mn (ℬ)).