Semirings: Algebraic Theory And Applications In Computer ScienceThis book provides an introduction to the algebraic theory of semirings and, in this context, to basic algebraic concepts as e.g. semigroups, lattices and rings. It includes an algebraic theory of infinite sums as well as a detailed treatment of several applications in theoretical computer science. Complete proofs, various examples and exercises (some of them with solutions) make the book suitable for self-study. On the other hand, a more experienced reader who looks for information about the most common concepts and results on semirings will find cross-references throughout the book, a comprehensive bibliography and various hints to it. |
Contents
| 1 | |
CHAPTER II EXTENSIONS OF SEMIRINGS | 90 |
CHAPTER III PARTIALLY ORDERED SEMIRINGS | 144 |
CHAPTER IV SEMIRINGS WITH INFINITE SUMS | 193 |
CHAPTER V SEMIALGEBRAS SEMIGROUP SEMIRINGS AND SEMIRINGS OF FORMAL POWER SERIES | 265 |
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Common terms and phrases
a₁ absorbing zero additively cancellative ai)iel algebra assume axiom b₁ bijection binary operation called commutative semigroup complete lattice congruence considered contains corresponding defined Definition denote E-algebra E-semiring equivalence relation equivalent Example Exercise exists finite follows free semigroup Hence holds idempotent identity implies index set isomorphism k-ideal latter lattice left cancellative Lemma mapping Math matrix monotony law Moreover multiplicatively cancellative multiplicatively left neutral element notation obtain oversemiring p. o. semigroup p. o. set partial order particular polynomial semiring positive cone power series Proof Remark ring ideal ring of differences S-semialgebra S-semimodule satisfies semifield semifield of quotients Semigroup Forum semigroup of denominators semigroup of quotients semilattice semimodule semiring ideal semiring of differences semiring of quotients smallest statements subsemigroup subsemiring subset surjective T₁ Theorem total order uniquely determined weakly p. o. semiring yields zero-divisor free zero-sum free