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Introduction
Published in John N. Mordeson, Davender S. Malik, Fuzzy Automata and Languages, 2002
John N. Mordeson, Davender S. Malik
Recall that a set R with the operations of addition + and multiplication × is called a semiring if the following three conditions are satisfied: (1) + is associative and commutative; (2)× is associative; (3)× distributes over +, i.e., a×(b+c)=a×b+a×c,(b+c)×a=b×a+c×a,
Boolean Algebra
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
A semiring is a nonempty set S provided with two binary operations “+” and “·” such that S is closed and associative under both operations, and commutative under +, satisfying also the distributive law.
Scheduling and control of high throughput screening systems with uncertainties and disturbances
Published in Production & Manufacturing Research, 2022
Adetola Oke, Laurent Hardouin, Xin Chen, Ying Shang
Definition 1. A semiring is a set , equipped with two operations and , such that is a commutative monoid (the zero element will be denoted ), is a monoid (the unit element will be denoted ), operation is right and left distributive over , and is absorbing for the product (i.e. ).
Analysis of P-time event graphs in (max,+) and (min,+) semirings
Published in International Journal of Systems Science, 2021
Pavel Špaček, Jan Komenda, Sébastien Lahaye
Finally we point out that the two extreme cycle means coincide with those obtained using linear programming in Becha et al. (2013). There is a small difference between the computed initializations that is due to the fact that our approach is purely algebraic, while using linear programming some rounding errors can occur. We have preferred algebraic computation in and semirings with additional benefits such as polynomial time complexity (not always guaranteed when using linear programming). The negative numbers occur in the state component that is fictive, i.e. stemming from extension of state vector due to two tokens of initial marking in one of the places. In practice there are two possibilities how to avoid the negative components: either we multiply all components by (add to them in the conventional algebra) the inverse of this negative number (while remaining in the same space) or we can also ignore the fictive component.
Ideals of an EMV-semiring
Published in International Journal of General Systems, 2020
R. A. Borzooei, M. Shenavaei, A. Di Nola, O. Zahiri
Idempotent semirings have many applications in linguistic problems, discrete mathematics, computer science, computer languages, automata, optimization problems (see Katsov 1997, 2004; Cohen, Gaubert, and Quadrat 1999; Golan 1999; Litvinov 2007). On the other hand, the theory arising from the substitution of the fields of real and complex numbers with idempotent semirings and/or semifields is often referred to as idempotent or tropical mathematics. Consider the above approach, it can be an important application of MV-algebras.