# Decision Problems for Equational Theories of Relation Algebras

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*Hajnal Andréka; Steven Givant; István Németi*

This work presents a systematic study of decision problems for equational theories of algebras of binary relations (relation algebras). For example, an easily applicable but deep method, based on von Neumann's coordinatization theorem, is developed for establishing undecidability results. The method is used to solve several outstanding problems posed by Tarski. In addition, the complexity of intervals of equational theories of relation algebras with respect to questions of decidability is investigated. Using ideas that go back to Jónsson and Lyndon, the authors show that such intervals can have the same complexity as the lattice of subsets of the set of the natural numbers. Finally, some new and quite interesting examples of decidable equational theories are given.

The methods developed in the monograph show promise of broad applicability. They provide researchers in algebra and logic with a new arsenal of techniques for resolving decision questions in various domains of algebraic logic.

#### Table of Contents

# Table of Contents

## Decision Problems for Equational Theories of Relation Algebras

- Contents vii8 free
- Introduction xi12 free
- Chapter I. Preliminaries 116 free
- Chapter II. Undecidability 1328
- Chapter III. A lattice embedding that preserves decidability and undecidability 5570
- Chapter IV. A finitely generated, infinite, simple relation algebra with a decidable equational theory 6782
- Bibliography 112127
- Index of symbols 115130 free
- Index of names and subjects 119134 free