Download Decision Problems for Equational Theories of Relation by Hajnal Andreka, Steven R. Givant, Istvan Nemeti PDF

By Hajnal Andreka, Steven R. Givant, Istvan Nemeti

This paintings offers a scientific research of selection difficulties for equational theories of algebras of binary kin (relation algebras). for instance, an simply acceptable yet deep approach, in response to von Neumann's coordinatization theorem, is constructed for setting up undecidability effects. the strategy is used to resolve numerous awesome difficulties posed by way of Tarski. additionally, the complexity of durations of equational theories of relation algebras with appreciate to questions of decidability is investigated. utilizing principles that return to Jónsson and Lyndon, the authors exhibit that such durations may have an identical complexity because the lattice of subsets of the set of the common numbers. ultimately, a few new and really fascinating examples of decidable equational theories are given.

The equipment constructed within the monograph exhibit promise of huge applicability. they supply researchers in algebra and common sense with a brand new arsenal of thoughts for resolving selection questions in quite a few domain names of algebraic good judgment.

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Example text

Let M denote the 'universal domain' of algebraic geometry (an algebraically closed field of immense transcendence degree), and k a subfield of M which need not be a model of Th(Λl). Then polynomials over k are special kinds of formulas over fc, ideals in k[x] are (incomplete) types, and the spectrum of fc[x], the set of maximal ideals in fc[z], is in 1-1 correspondence with the Stone space of k. For ά, a sequence from Λt, this correspondence sends the ideal of polynomials over k which vanish at a to ί(ά fc).

Let A and AQ, . . , An-\ be finite sets. Show A C (Ji12 We will proceed by induction on quantifiers. p. formulas. p. p. For each i < n, let Hi denote the subgroup 0;(0;M). 13) to HQ + α0 <Ξ U 28 I. Groundwork for appropriate choice of the α^. Note that some of the Hi + α^ may be empty. 8, we see that ^(α; y) is equivalent to: HQ + CLQ C \JHi + ai. (*) i

P. formula φ, φ(N ) = φ(N) . 26 Corollary. For any module M and any /c, M κ -< M . 27 Historical Notes. The logical analysis of the theory of modules begins with Szmielew's quantifier elimination theorem for Abelian groups ([Szmielew 1955]). The work of Eklof, Fisher, and Sabbagh ([Eklof & Fisher 1972], [Eklof & Sabbagh 1970/71], [Baur 1976]) took a more model theoretic turn. -elimination theorem was proved at about the same time but independently by Baur, Garavaglia, and Monk; Garavaglia ([Garavaglia 1979], [Garavaglia 1980]) realized its significance for stability theory.

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