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The 1st bankruptcy bargains with idempotent research in keeping with se . To make the pres- tation self-contained, within the first sections we outline idempotent semirings, provide a concise exposition of idempotent linear algebra, and survey a few of its purposes. Idempotent linear algebra reviews the homes of the semirn- ules An , n E N , over a semiring A with idempotent addition; in different phrases, it reports structures of equations which are linear in an idempotent semiring. Pr- ably the 1st attention-grabbing and nontrivial idempotent semiring , specifically, that of all languages over a finite alphabet, in addition to linear equations during this sern- ing, used to be tested via S. Kleene [107] in 1956 . This noncommutative semiring used to be utilized in functions to compiling and parsing (see additionally [1]) . almost immediately, the literature on idempotent algebra and its functions to theoretical laptop technology (linguistic difficulties, finite automata, discrete occasion platforms, and Petri nets), biomathematics, common sense , mathematical physics , mathematical economics, and optimizat ion, is titanic; e. g. , see [9, 10, eleven, 12, thirteen, 15, sixteen , 17, 22, 31 , 32, 35,36,37,38,39 ,40,41,52,53 ,54,55,61,62 ,63,64,68, seventy one, seventy two, 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 current crucial proof of the idempotent algebra formalism . The semimodules An are idempotent analogs of the finite-dimensional v- n, tor areas lR and accordingly endomorphisms of those semi modules can obviously be known as (idempotent) linear operators on An .

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