Power Algebras over Semirings: With Applications in Mathematics and Computer Science
This monograph is a continuation of several themes presented in my previous books [146, 149]. In those volumes, I was concerned primarily with the properties of semirings. Here, the objects of investigation are sets of the form RA, where R is a semiring and A is a set having a certain structure. The problem is one of translating that structure to RA in some "natural" way. As such, it tries to find a unified way of dealing with diverse topics in mathematics and theoretical com puter science as formal language theory, the theory of fuzzy algebraic structures, models of optimal control, and many others. Another special case is the creation of "idempotent analysis" and similar work in optimization theory. Unlike the case of the previous work, which rested on a fairly established mathematical foundation, the approach here is much more tentative and docimastic. This is an introduction to, not a definitative presentation of, an area of mathematics still very much in the making. The basic philosphical problem lurking in the background is one stated suc cinctly by Hahle and Sostak : ". . . to what extent basic fields of mathematics like algebra and topology are dependent on the underlying set theory?" The conflicting definitions proposed by various researchers in search of a resolution to this conundrum show just how difficult this problem is to see in a proper light.
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Some Hopefully Motivating Examples
Powers of a semiring
Relations with Values in a Semiring
Change of Base Semirings
Semiringvalued Subsemigroups and Submonoids
Semiringvalued Submodules and Subspaces
Semiringvalued Ideals in Semirings and Rings
additively idempotent additively-idempotent semiring applications assume Chu spaces CLO-semiring closure operator commutative semiring complete distributive lattice complete semiring Computer Science congruence relation convolution context elements of RA equivalence relation exists an element f G RM facile fl-valued subgroup facile subgroup field F fl-valued facile fl-valued left fl-valued submonoid fl-valued subsemigroup function f fuzzy games fuzzy groups Fuzzy Sets fuzzy subgroups G RA homomorphism identity element Info ISBN left 5-module left ideal linearly independent Lr(f Mathematics monoid Moreover morphism of semirings multiplicatively-idempotent multisets necessary summation nonempty set nonempty subset numbers otherwise partial order Proof Proposition R-valued 5-submodule R-valued congruence relation R-valued equivalence relation R-valued ideal R-valued relation RAxA RAxB regular fl-valued relation h ring RMxM satisfying the condition semifield semiring and let Sets and Systems Similarly simple semiring structure submonoid subsemiring supp(f theory topology triangular norm vector space zerosumfree