This work is an updated and expanded version of the author's book "The Theory of Semirings, with Applications to Mathematics and Theoretical Science", which has been recognized as a notable reference work in this area. This edition includes many of the new results in this area, as well as further applications of semiring theory in such areas as idempotent analysis, discrete dynamical systems, formal language theory, fuzzy set theory, and optimization. The book contains an extensive bibliography and a large number of examples.
1. Hemirings and semirings: definitions and examples.- 2. Sets and relations with values in a semiring.- 3. Building new semirings from old.- 4. Some conditions on semirings.- 5. Complemented elements in semirings.- 6. Ideals in semirings.- 7. Prime and semiprime ideals in semirings.- 8. Factor semirings.- 9. Morphisms of semirings.- 10. Kernels of morphisms.- 11. Semirings of fractions.- 12. Euclidean semirings.- 13. Additively-regular semirings.- 14. Semimodules over semirings.- 15. Factor semimodules.- 16. Some constructions for semimodules.- 17. Free, projective, and injective semimodules.- 18. Localization of semimodules.- 19. Linear algebra over a semiring.- 20. Partially-ordered semirings.- 21. Lattice-ordered semirings.- 22. Complete semirings.- 23. Complete semimodules.- 24. CLO-semirings.- 25. Fixed points of affine maps.- References.- Index of applications.- Index of terminology.
