An introduction to mathematical logic covering all the usual topics: compactness and axiomatizability of semantical consequence; Lowenheim-Skolem-Tarski theorems; prenex and other normal forms; and characterizations of elementary classes with help of ultraproducts. Logic is based exclusively on semantics. Truth and satisfiability of formulas in structures are the basic notions, and there is no need to mention logical calculi with axioms and rules (they are the subjects of Volume Two). The methods are algebraic in the sense that notions such as homomorphisms and congruence relations are applied throughout, and this not just as abbreviations, but in order to gain new insights. These concepts are developed in an introductory chapter which, together with chapters five to nine on equations, can be viewed as a first course on universal algebra. The approach to algorithms generating semantical consequences is algebraic as well: for equations in algebras, for propositional formulas, for open formulas of predicate logic, and for the formulas of quantifier logic. The structural description of logical consequence is a straightforward extension of that of equational consequence as long as Bool

Collecting, curating, and illuminating over 75 years of progress since Tarski's seminal work in 1941, this textbook in two volumes offers a landmark, unified treatment of the increasingly relevant field of relation algebras. Clear and insightful prose guides the reader through material previously only available in scattered, highly-technical journal articles. Students and experts alike will appreciate the work as both a textbook and invaluable reference for the community. This set charts relation algebras from novice to expert level. The first volume, Introduction to Relation Algebras, offers a comprehensive grounding for readers new to the topic. The second, Advanced Topics in Relation Algebras, build on this foundation and advances the reader into the deeper mathematical results of the past few decades. Such material offers an ideal preparation for research in relation algebras and Boolean algebras with operators. Note that the second volume contains numerous, essential references to the first. Readers of the advanced material are encouraged to purchase the pair as a set, as access to the first book is necessary to make use of the second.