Partial Algebras — An Introductory Survey

Abstract

Partial algebras are among the basic mathematical structures implemented on computers. Many-sorted algebras are basically partial algebras, too. These notes are meant to introduce into a theory of and a language for partial algebras in such a way that also a specification of (many-sorted) partial algebras as abstract data types can easily be performed. Besides the terminology and constructions from universal algebra (homomorphisms, generalized recursion theorem, epimorphism theorem, free partial algebras) also such from logic (existence equations and elementary implications), model theory (preservation and reflection of formulas by mappings) and from (elementary) category theory (factorization systems) prove to be quite useful for a good description of the arising concepts, as is shown at the end by the formulation of a “Meta Birkhoff Theorem”.