Partial Algebras — An Introductory Survey

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”.

[1]  J. Schmidt.  Die Charakteristik einer allgemeinen Algebra. I , 1962 .

[2]  P. Burmeister.  A Model Theoretic Oriented Approach to Partial Algebras , 1986 .

[3]  Epireflection operators vs perfect morphisms and closed classes of epimorphisms , 1972, Bulletin of the Australian Mathematical Society.

[4]  Characterizations of congruence lattices of abstract algebras , 1963 .

[5]  William A. Lampe.  Notes on related structures of a universal algebra. , 1972 .

[6]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[7]  P. Kosiuczenko.  Mal'cev-type conditions for partial algebras , 1994 .

[8]  Faktorisierungssysteme in der Kategorie der partiellen Algebren, Kennzeichnung von (Homo)Morphismenklassen , 1979 .

[9]  Hajnal Andréka,et al.  Generalization of the concept of variety and quasivariety to partial algebras through category theory , 1983 .

[10]  R. John.  Gültigkeitsbegriffe für Gleichungen in partiellen Algebren , 1978 .

[11]  E. Specker,et al.  The Problem of Hidden Variables in Quantum Mechanics , 1967 .

[12]  Trevor Evans,et al.  Embeddability and the Word Problem , 1953 .

[13]  William Craig.  Near-Equational and Equational Systems of Logic for Partial Functions I , 1989, J. Symb. Log..

[14]  Hajnal Andréka,et al.  INJECTIVITY IN CATEGORIES TO REPRESENT ALL FIRST ORDER FORMULAS, I , 1979 .

[15]  J. Bell.  On the Problem of Hidden Variables in Quantum Mechanics , 1966 .

[16]  Schmidt Jürgen.  A homomorphism theorem for partial algebras , 1970 .

[17]  L. Rudak.  A completeness theorem for weak equational logic , 1983 .

[18]  Peter Burmeister,et al.  Partial algebras—survey of a unifying approach towards a two-valued model theory for partial algebras , 1982 .

[19]  J. Słomiński.  A theory of extensions of quasi-algebras to algebras , 1964 .

[20]  Trevor Evans,et al.  The Word Problem for Abstract Algebras , 1951 .

[21]  H. Höft.  Weak and strong equations in partial algebras , 1973 .

[22]  Horst Reichel.  Structural induction on partial algebras , 1984 .

[23]  Robert Kerkhoff.  Gleichungsdefinierbare Klassen partieller Algebren , 1970 .