Semirings: Algebraic Theory and Applications in Computer Science
This 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.
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absorbing zero additively cancellative algebra assume assumption bijection binary operation called commutative semigroup complete lattice congruence contains context corresponding defined Definition denote E-semimodule equivalence relation equivalent Exercise 2.1 exists family aż)że finite follows free semigroup Hence homomorphic image idempotent identity implies indeterminate index set infimum isomorphism latter lattice Lemma mapping matrix monotony law Moreover multiplicatively cancellative neutral element notation obtain obviously oversemiring p. o. semigroup p. o. set partial order particular partition polynomial semiring positive cone power series Proof relation Remark respect ring ideal ring of differences semialgebra semifield of quotients semigroup of denominators semigroup of quotients semigroup semiring semimodule semiring ideal semiring of differences semiring of quotients smallest statements subsemigroup subsemiring subset supremum surjective Theorem Theorem 3.1 total order uniquely determined weakly p. o. semiring yields zero-divisor free zero-preserving zero-sum free