Semirings and their applications pdf
Semirings and their Applications | SpringerLinkIn abstract algebra , a semiring is an algebraic structure similar to a ring , but without the requirement that each element must have an additive inverse. The term rig is also used occasionally  —this originated as a joke, suggesting that rigs are ri n gs without n egative elements, similar to using rng to mean a r i ng without a multiplicative i dentity. Compared to a ring , a semiring omits the requirement for inverses under addition; that is, it requires only a commutative monoid , not a commutative group. In a ring, this implies the existence of a multiplicative zero, so here it must be specified explicitly. If a semiring's multiplication is commutative , then it is called a commutative semiring. There are some authors who prefer to leave out the requirement that a semiring have a 0 or 1.
Daniel Spielman “Miracles of Algebraic Graph Theory”
Graphs and Matroids Weighted in a Bounded Incline Algebra
Dutta and S. Lehmer A ternary analogue of abelian groups American J. Lister Ternary rings Tran. Hanumanthachari and K. Venuraju The additive semigroup structure of semiring Math. Seminar Note 11 Rao and B.
Firstly, for a graph weighted in a bounded incline algebra or called a dioid , a longest path problem LPP, for short is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and the most reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximum independent set problem is studied. In graph theory, the famous shortest path problem SPP, for short is the problem of finding a path between two vertices in a weighted graph such that the sum of the weights of its constituent edges is minimized [ 1 ]. An example is finding the quickest way to get from one location to another on a road map. In this case, the vertices represent locations and the edges represent segments of road and are weighted by the time needed to travel that segment. There are many other classical problems using various semirings in graph theory [ 2 ].
Skip to main content Skip to table of contents. Advertisement Hide. Semirings and their Applications. Front Matter Pages i-xi. Hemirings and Semirings: Definitions and Examples.