The first chapter deals with idempotent analysis per se . To make
the pres- tation self-contained, in the first two sections we
define idempotent semirings, give a concise exposition of
idempotent linear algebra, and survey some of its applications.
Idempotent linear algebra studies the properties of the semirn-
ules An , n E N , over a semiring A with idempotent addition; in
other words, it studies systems of equations that are linear in an
idempotent semiring. Pr- ably the first interesting and nontrivial
idempotent semiring , namely, that of all languages over a finite
alphabet, as well as linear equations in this sern- ing, was
examined by S. Kleene [107] in 1956 . This noncommutative semiring
was used in applications to compiling and parsing (see also [1]) .
Presently, the literature on idempotent algebra and its
applications to theoretical computer science (linguistic problems,
finite automata, discrete event systems, and Petri nets),
biomathematics, logic , mathematical physics , mathematical
economics, and optimizat ion, is immense; e. g. , see [9, 10, 11,
12, 13, 15, 16 , 17, 22, 31 , 32, 35,36,37,38,39 ,40,41,52,53
,54,55,61,62 ,63,64,68, 71, 72, 73,74,77,78,
79,80,81,82,83,84,85,86,88,114,125 ,128,135,136,
138,139,141,159,160,
167,170,173,174,175,176,177,178,179,180,185,186 , 187, 188, 189].
In 1. 2 we present the most important facts of the idempotent
algebra formalism . The semimodules An are idempotent analogs of
the finite-dimensional v- n, tor spaces lR and hence endomorphisms
of these semi modules can naturally be called (idempotent) linear
operators on An .
General
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