Universal algebra for computer scientists
A new model-theoretic approach to universal algebra is offered in this book. Written for computer scientists, it presents a systematic development of the methods and results of universal algebra that are useful in a variety of applications in computer science. The notation is simple and the concepts are clearly presented. The book concerns the algebraic characterization of axiomatic classes of algebras (equational, implicational, and universal Horn classes) by closure operators generalizing the famous Birkhoff Variety Theorem, and the algebraic characterization of the related theories. The book also presents a thorough study of term rewriting systems. Besides basic notions, the Knuth-Bendix completion procedure and termination proof methods are considered. A third main topic is that of fixpoint techniques and complete ordered algebras. Algebraic specifications of abstract data types and algebraic semantics of recursive program schemes are treated as applications. The book is self-contained and suitable both as a textbook for graduate courses and as a reference for researchers.
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abstract class arbitrary assume called class of algebras closure operator conﬂuent congruence on Tg(V Consider countable critical pairs deﬁned deﬁnition denoted direct family direct limit direct products directed union Dom(T E-algebra E-tree equational theory equivalent Example family of algebras ﬁnitary ﬁnite ﬁnite set ﬁnite subset ﬁrst ﬁxpoint free algebras free w-completion fully invariant closure fully invariant congruence given ground terms Hence homomorphism iﬁ implications induction inequalities inference rules injective isomorphic least element Lemma logical mapping f monoid morphism multisets n-ary operation symbol natural numbers nonempty nontrivial partial order poset Proposition quasi-order recursive reduction rule reduction system reﬂexive relation satisﬁes Section semantics signature speciﬁcation strict ordered algebra subalgebras supremum sur-reﬂection surjective term algebra terminating Tg(X Tg(Z Theorem tree unary operation universal Horn clauses valid var(t variables VX(t w-chain w-complete ordered algebra w-complete poset w-continuous well-founded well-quasi-order