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Index of Applications

Semiring theory is applications-driven, not an exercise in generalization for gen­eralization's sake. Mention of many applications to areas of pure and appliedmathematics, computer science, physics , etc. is sprinkled throughout the text.As a rule, these applications are not developed in detail, but references to theappropriate technicalliterature are generally provided.

4, 164, 165, 17315

60

algebraic geometry 19algorithmics 23analysis of computer programsanalysis of industrial processesasymptotic computation 16automata theory 3, 13, 15, 16, 42, 43, 103, 112, 156, 173automated theorem proving 130automatie parallelization of linear code ,. 103automatie sequences 24Banach algebras 20Banach spaces 19bottleneck problems 6codes 15, 136combinatorial matrix theorycombinatorics 155, 157command algebras 12communication networks 16, 62computationallinguistics 103computer architecture design 16computer languages 41, 159concurrency analysis 103constraint programming 14control theory 15, 152cryptography 60database systems 41, 156dimension theory 23

225

226 INDEX OF APPLICATIONS

discrete-event dynamical systems 3, 7, 15, 46, 61, 130, 155document classification systems 62dynamic programming 64economic modelling 61ftowchart schemes 103formallanguage theory 7, 11, 13, 16, 40, 42, 43, 51, 130, 136, 152, 156fuzzy logic 12, 14, 51fuzzy number theory 3fuzzy set theory 2, 32, 46, 136game theory 2, 61geometry 41graph algorithmics 16graph theory 6, 7, 15, 16, 41, 61, 62, 64, 65, 106, 107, 155, 159, 171group rings .. 17hybrid Pet ri nets 16hybrid systems 5idempotent analysis 19, 39information retrieval systems 62information theory 61integration spaces 117interval analysis 31intuit ionist ic logic 51knowledge bases 130linear logic 33, 46many-valued logic 2, 51, 52Markov processes 15modal logic 8models of memory redundancy 15models of vagueness 17monoidallogic 51natural language processing 103network theory 30non -deterministic computation .... ...... 109, 119, 156numerical analysis 31, 135operations research 62, 175optimal control 16, 116optimization 11,15, 16,30,113,130,155,176partition functions 28pattern recognition 2perturbation analysis 15Petri nets 15, 16Presburger logic 34probabiiity theory 2, 16, 113programming languages 50quantum computation 62quantum field theory 11

quantum mechanics 12, 46queueing systems 16, 62rewriting systems 136semantics of programming languages 16, 46, 156signal processing .... 102soft computing 2statistical physics 15, 16statistics 2, 155stochastic process algebras 15stochastic systems 62string rewriting 11string theory .. 19symbolic dynamics 61synchronization 16systems theory 4, 102transportation networks 16

227

Index of terminology

-A-action algebra 50- lattice .. 50additively extremal semiring 11- idempotent semiring .... ...... 6adjacency matrix .... ...... 64adjoint 180- , fuH 189affine map 129Alexandrov space 96algebra, action 50- , bottleneck 12-, convolution 40-, demonie refinement .. 173-, extended optimization 16- , extended schedule 15- , fuzzy 28- , information 7-, Kleene 165-, (max, +)- 15-, MV- 55- ,optimization 16- , schedule 15-, standard Kleene 112-, symmetrized 95algebraic assignment problem .... .... .. 176aperiodic matrix 65arborescence 186automorphism, of a semimodule 119-, of a semiring 23-, permutation 119

-B-balance topology 95

229

230

balanced 83, 109base 24basic logic 52- semiring 1basis 109- , canonical .......... 109Bellman problem, stationary 157bideterminant 181bimodule 101bisemimodule 101boolean semiring 12bottleneck algebra 12bounded-complete semimodule 112bounded funct ion ... ....... 102bounded-complete semiring 39Bourne relation 5, 124Brouwer residual 51Butkoviö semiring 11

-c-cancellable ........ .. 14, 106cancellative semimodule 106- semiring 14canonical basis 109Cauchy product 13Cayley-Hamilton Theorem 187, 190center .. ...... .. 1central element 1chain semimodule 137- semiring 28character 23characteristic polynomial .... .. .... 187- , full 190circuit 64closed, locally 161closure operation 9- , transitive 171cofactor 180commutative semiring ...... .... 1complement 8complemented element 8complete derivation 44- homomorphism 116- semimodule 112- semiring 39cone 108

INDEX OF TERMINOLOGY

-, polyhedral 108congruence relation, improper 4- , on a semimodule 123- , on a semiring 4- , proper 4continuous semimodule 148- semiring 42convolution algebra 40Conway conditions 164- semiring 164Correspondence Principle 12countably-complete deri vation 44- homomorphism 116- semimodule 112- semiring 41countably-continuous semimodule 148- semiring 42countably-finitary semiring 45covering . .... 46cyclic mean ..... 154

-D-demonie refinement algebra 173dense 27derivation 25-, complete 44- , countably-complete 44-, increasing 38- , normal 25determinant . 188difference order 29difference-chain semimodule 137- semiring 30difference-ordered semimodule 137- semiring 29digit representation .. 24Dijkstra 14diord 6discrete metric .. . 34division semiring 18

-E-edge, weighted 64eigenelement 151, 156-, principal 151eigenpair 151, 156

231

232

eigensubsemimodule 151eigenvalue 151, 156endomorphism 118entire semiring 7epigraph 27equalizer 125, 130equatorial semiring 16evaluation morphism 24, 60exotie semiring .... . 16expeetation semiring 103extended optimization algebra 16- sehedule algebra 15extension, symmetrie 82ext remal semimodule 104

- F-faetor semimodule 123- semiring 4finitary semiring 45finitely generated . 108first-order logieal formula 35fixed point 151, 156formal language 11- power series 13free monoid 11- semimodule 109fuH eharaeterist ie polynomial. ......... 190- adjoint 189fusion produet 33fuzz semiring 2fuzzy algebra 28

INDEX OF TERMINOLOGY

-G -Gel 'fand semi ring 20generalized stationary Bellman problem 157Gödel residual .......... 51Green's relation 29graph, of a funetion ... .. 118- , preeedent 64group, lattiee-ordered .......... 52

- H-Hall matrix 175hered itary subset 12hom ogeneous linear equation 125homomorphism 115

-, complete 116-, countably-complete 116-, nonsingular 151- , singular 151-, strongly regular 133Hungarian method 176

-1-ideal, relative to a covering ... ... ... . 46-, of a semiring 5- , order 38ideal system, weak 2idempotent . 104-, additively 6- analysis 19-, multiplicatively 8- , semimodule 104incline 32increasing derivation . 38inductive semiring .. 158infinite element 14infinitesimal 95information algebra 7instantiation 31integral 113interval 30- analysis 31inverse spectral problem 152- , Thierrin-Vagner 68involution 68irreducible matrix 65isomorphism, of semimodules 115-, of semirings 23iteration function 158- theory 103iterative semiring 164

-J-jet 28

-K-kernei 22Kleene algebra 165-, standard 112- , with least fixed points 165Kuich semiring ........ .. 43

233

234

-L-lattice, action 50lattice-ordered group 52- semiring 33left multiplicatively cancellable 21- quasi-inverse 157- residual 50left-residuat ed bisemimodule 146- semiring 50Lehmarm semiring 164lexicographic order 138linear combination 107linear equation, homogeneous ........ .. 125- , nonhomogeneous 130linearly attached 106- dep endent 107- independent 107- unattached 106locally closed 161logic, basic 52- monoidal 51logical semiring 23lower element 86lewer-triangular matrix 61Lukasiewicz residual .... .... .. 51

-M-magnitude semimodule 117map, affine 129- , special affine 130matrix, adjacency 64- , aperiodic 65- , Hall 175-, irreducible 65-, lower-triangular 61-, orthogonal 76-, periodic 65- , permutation 61-, primitive 65-, reducible 65- , scalar 60- , transitive 79-, upper-triangular 61mean, cyclic 154measure .... 113metric, discrete 34

INDEX OF TERMINOLOGY

minor 189module 101monoid, free 11-, semilattice-ordered 6monoidallogic 51Moore-Penrose pseudo-inverse 121morphism, evaluation .......... 24, 60- of semirings 22multiplicatively cancellable .... .. .... 21- cancellative ..... 21- extremal 11- idempotent 8- regular element 68- regular semiring 69- subidempotent 8MV-algebra 55

-N -necessary summation 42, 147negative .... ...... 83, 95, 109nonhomogeneous linear equa tion .. .... .... 130- , strongly solvab le 135-, wea kly solvab le 135nonsingular 151nontrivial semimodule 102normal derivation 25normalization .. 133

- 0-optimal permutation 176optimization algebra 16order 88- ideal 38- , lexicographic 138- , Rees 34- , standard 34orthogonal matrix 76- projection 146

-p -partially-ordered sem imodule .. .. .... .. 137- semiring 27pattern 60periodic matrix 65permanent 175-, strong 176

235

236

permutation automorphism 119- matrix 61- , optimal .......... 176polar semiring 15polyhedral cone 108polynomial, characteristic 187- semiring 3positive 82, 95, 109- semimodule 137- semiring 27power stable 163precedent graph 64Presburger definable ...... 35- formula 35- semiring 36primitive matrix 65principal eigenelement 151product, Cauchy 13- , pseudodirect 8projection, orthogonal 146pseudodirect product 8pseudo-inverse, Moore-Penrose 121pseudonegative 86, 109pseudopositive 86, 109pseudosigned 86, 109

-Q-QLO-semimodule 148- -semiring 45quantale 46quantic lattice-ordered semimodule ....... ... 148- semiring 45quasi-inverse 159-,left 157quasiregular element of a ring 9- element of a semiring ..... .. ... 159-, right .. ........ 157

-R-rational closure 165- subsemimodule 143- subset .... ...... 35rationally additive 170- closed .......... 165reducible matrix ....... ... 65Rees order 34

INDEX OF TERMINOLOGY

regular 141- semimodule 142residual ... .... ... 50, 146- , Brouwer 51- , Gödel 51-, Lukasiewicz 51residuated semimodule 146- semiring 50- set 49right multiplicatively cancellable 21- quasi regular 157- residual 50right-residuated bisemimodule 146semiring 50rule 11

-8-scalar matrix 60- multiplication 101schedule algebra .. 15Schur complement formula 189semifield 19- , tropical 21semilattice-ordered monoid 6semimodule .... ...... 101- , bounded-complete 112- , cancellative 106- , chain 137- , complete 112- , continuous 148- , countably-complete 112-, countably-continuous 148-, difference-chain 137-, difference-ordered 137-, extremal 104-, factor 123-, finitely-generated 108-, free 109-, idempotent 104-, magnitude 117-, partially-ordered .... ...... 137- , positive 137-, quantic lattice-ordered ........ .. 148-, regular 142-, residuated 146- , symmetrized 124

237

238 INDEX OF TERMINOLOGY

- , uniquely difference-ordered 137- , weakly cancellative 106- , weakly uniquely differ en ce-ordered 139- , zerosumfree 104semiring 1-, additively-idempotent 6- , additively-extremal 11- , basic 1- , boolean 12- , bounded complete 39- , Butkoviö 11- , cancellative 14

- , chain .. ........ 28- , commutative 1-, complete 39- , continuous 42- , Conway 164- , countably-complete 41- , countably-cont inuous 42-, countably-finitary 45- , difference-chain 30- , difference-ordered 29- , Dijkstra 14-, d ivision 18-, entire 7- , equatorial 16- , exot ic 16- , exp ectation 103- , factor 4- , finitary 45- , fuzz 2-, Gel'fand 20- , induc t ive 158- , iterative 164- , Kuich 43- , lattice-ordered 33-, left-residu ated 50- , Lehmann 164- , locally-closed 161- , logical 23- , multiplicatively-cancellative 21- , multiplicatively-extremal 11- , multiplicat ively-regular 69- of formal power series 13- of polynomials 3- , w-complete 41

-, partially-ordered 27-, polar 15- , positive 27-, Presburger 36-, rationally additive 170-, residuated 50-, right-residuated 50-, semitopological 96-, simple 14-, skew polynomial 25- , *- .......... 164-, syntactic 7

-, topological 96- , totally-ordered 28- , tropical 16-, uniquely difference-ordered 29-, quantic lattice-ordered 45-, Viterbi 3-, weak 4-, weak uniquely difference-ordered .. .... .. .. 31- , zerosumfree 7semi-Thue system 11semitopological semiring 96set of generators 108- of solutions to a homogeneous linear equation 126- of solutions to a nonhomogeneous linear equation . 130-, residuated .. .... .... 49signal 102signed 83, 109simple semiring .... .... .. 14singular 151skew polynomial semiring 25slope .. .. .... .. 32special affine map 130spectrum 151, 156stable with respect to an involution 69standard Kleene algebra 112- order 34*-semiring 164stationary Bellman problem 157-, generalized ...... .... 157stern .... .. .... 17string rewriting system 11strong permanent 176strongly regular homomorphism 133- solvable 135

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240 INDEX OF TERMINOLOGY

subidempotent , multiplieatively 8subsemimodule 101- , nontrivial 102- , rational .......... 143- , trivial .......... 102subsemiring 1-, rationally closed 165subset, hereditary . 12-, rational 35support, of a funetion 102- , of aPresburger formula 35symmetrie extension of a semiring 82symmetrized algebra 95- semimodule 124symmetry 83syntaetie semiring 7

-T-t-norrn 2tensor produet (in linear logic) 33Thierrin-Vagner inverse 68topological semiring 96- vector lattice 19topology, balance 95totally-ordered semiring 28transitive closure 171- matrix 79tri angular norm 2trivial semimodule 102tropical semifield 21- semiring 16Type I nonhomogeneous linear equation 131Type 11 nonhomogeneous linear equation 131Type 111 nonhomogeneous linear equation 131Type IV nonhomogeneous linear equation 131

-u-UDO semimodule 137- semiring 29un ipotent element 17uniquely differenee-ordered semimodule 137- semmng .. .. 29unit 17unsigned ...... 83upper element 86upper-triangular matrix 61

-v-vector lattice, topological 19Viterbi semiring 3

-w-weak ideal system 2- semiring 4- uniquely difference-ordered semimodule 139- uniquely difference-ordered semiring 31weakly cancellable 106- cancellative semimodule .. .... .. .. 106- solvable ...... 135weight 64- of an arborescence 186weighted edge 64width, of a tree 17word .. .. .... .. 13WUDO semimodule 139- semiring 31

-z-zeroid 7zerosumfree semimodule 104- semiring .......... 7

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