Real Semigroups, Real Spectra and Quadratic Forms over Rings
نویسندگان
چکیده
real spectrum (ARS)finitely closed subset of, 52morphism of, 31projective limits of, 31residue space of, 66–69saturated subset of, 56subspace of, 61algebraLukasiewicz, 84Kleene, 87Post, 84AOSclosed subset, 210dependent subset, 210AOS-fan, 166archimedean-equivalent, 188ARS-fan, 170connected components, 208finiteisomorphism theorem, 214standard generating system, 212involution, 201length, 195level in, 194associativitystrong, 19weak, 20axiom[RS3b], 20for spectral real semigroups, 154for Post algebra, 108reduction, 19strong associativity, 19weak associativity ([RS3a]), 20axiom-systemsFRS, 217–218 binary regular root-system (BRRS), 221Booleanelement, 86space, 6 categoryPA, 94, 99RS-PA, 94AOS, 166ARS, 166HNSS, 126, 136, 143RSG, 166RS, 166SRS, 126, 136, 143centerin Kleene algebra, 87in Post algebra, 84characterRS-, 24TS-, 3closed constructible set, 137closed set, 210combinatorial geometrynotions, 210complemented (Boolean) element, 86complete embedding, 100completely normaldistributive lattice, 154congruenceof real semigroup, 44of ternary semigroup, 10connected component, 196constructible topology, 6 dependent set, 210depth, 195distributive latticecompletely normal, 154Duality Theorem, 29 embeddingcomplete, 100pure, 109, 155 fan, 170AOS-, 166ARS-, 170connected components, 206–208involution, 201isomorphism theorem, 214length, 195level in, 194standard generating system, 212depth of element, 195depth of ideal, 195 SUBJECT INDEX258 finitecardinality of, 199of field, 187trivial, 187quotient, 230RS-, 170RSG-, 166fan1, 167, 170fan2, 167, 170fieldvaluationcompatible with preorder, 187fully compatible with preorder, 187finitely closed, 52formover, 26formulapositive-primitive, 108, 127, 135, 154FRSaxiom-systems, 217–218 generically closed, 55 hereditarily normal, 6, 127homomorphismRS-, 24TS-, 3 idealof a semigroup, 3prime, 3of ringT -compatible, 40T -convex, 40T -radical, 40implicationLukasiewicz, 85inequalityKleene, 84 Kleene algebra, 87Kleene inequality, 84 languagefor real semigroups, 19for ternary semigroups, 2lengthof ARS-fan, 195level, 193in ARS-fan, 194localization, 55Lukasiewicz algebrathree-valued, 84 matrixpositive-primitive, 108modal operators, 85, 86morphismof abstract real spectrum, 31of real semigroup (see homomorphism),24preserves 3-products, 82transversally 2-regular, 80 necessity operator, 85negationin Kleene algebra, 87in Lukasiewicz algebra, 84 operatormodal, 85, 86MV-algebra, 85necessity, 85possibility, 84 positive-primitive formula, 108, 127, 135, 154positive-primitive matrix, 108possibility operator, 84Post algebra, 84axiom, 108injective, 110of order 3, 84representation theorem, 86RS-, 92Post hullof real semigroup, 97preserves 3-products, 82prime cone (ordering) of ring, 2prime idealof a semigroup, 3prime subsemigroup, 3proconstructible, 6propermultiplicative subset of ring, 71subsemigroup of RS, 71pure embedding, 109, 155 q-fan, 167, 176–178quasi-compact, 6quotientof real semigroup, 44 real semigroup (RS), 19 SUBJECT INDEX259 axiom [RS3b], 20axioms for, 19character, 24congruence, 44homomorphism, 24language for, 19localization, 55Post hull, 97quotient, 44reduction axiom, 19spectral, 133spectral hull of, 140strong associativity axiom, 19weak associativity, 20real spectrumobjects of, 2topology of, 2real spectrum (of ring), 31representation, 19in Boolean algebra, 89in Post algebra, 89partial order, 34transversal, 19representation partial order, 34residue space, 66–69ringidealT -compatible, 40T -convex, 40T -radical, 40lattice-ordered, 162multiplicative subsetproper, 71prime cone of, 2real spectrum, 31semi-real, 2von Neumann-regular, 113, 116root-systembinary regular (BRRS), 221RS-fan, 170RS-Post algebra, 92RS-quotient, 44RSG-fan, 166 saturatedideal, 24set, 24, 56subsemigroup, 24semi-real ring, 2semigroupreal, 19ternary, 1separates points, 166spacespectral, 6specialization, 6spectral hull, 140spectral real semigroup, 133axioms, 154spectral space, 6closed constructible subset of, 137hereditarily normal, 6, 127standard generating system, 212subsemigroup, 3prime, 3proper, 71subsemigroup of RStransversally saturated (tss)non-trivial, 62subspace, 61supportof prime cone, 2symmetric difference, 88 T-compatiblering ideal, 40T-convexring ideal, 40T-convex hullof ring ideal, 41T-radicalof ring ideal, 41ring ideal, 40ternary semigroup (TS), 1character, 3congruence, 10homomorphism, 3language for, 2Theoremanti-equivalence, 143duality, 29idempotency, 140isomorphism (finite ARS-fans), 214representation (Post algebras), 86topological spaceBoolean, 6quasi-compact, 6topologyconstructible, 6spectral, 6 SUBJECT INDEX260 transversal representation, 19transversally 2-regularmorphism, 80set of characters, 80transversally saturatedset, 24subsemigroup (tss)non-trivial, 62tri-semigroup property, 32TS-congruence, 10 valuationin fieldfully compatible with preorder, 187compatible with preorder, 187von Neumann-regular ring, 113, 116 weak associativity, 20
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