Power Algebras over Semirings (eBook)

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 "e;natural"e; 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 "e;idempotent analysis"e; 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 [185]: "e;. . . to what extent basic fields of mathematics like algebra and topology are dependent on the underlying set theory?"e; 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.