Partial algebra

In abstract algebra, a partial algebra is a pair <A, P> where A is a set and P is a collection of partial operations on A. In universal algebra, when P consists of operations that are defined on all arguments taken from A, then the algebra is a total algebra. Frequently the adjective total is omitted when there are no partial operations.¹²

Example(s)

partial groupoid
field — the multiplicative inversion is the only proper partial operation¹
effect algebras³

Structure

There is a "Meta Birkhoff Theorem" by Andreka, Nemeti and Sain (1982).¹

Relational systems

Operations and partial operations may be written as finitary relations, where there is no requirement of totality. "A relational system ${\mathfrak {A}}$ is a pair <A, R>, where A is a non-void set and R is a family of (finitary) relations on A."²^: 8

Though relational systems have greater generality than algebras and partial algebras, they do not have the rich theory of the algebras.⁴ For example, defining a subalgebra of a relational system is not straight forward.⁵

References

Peter Burmeister (1993). "Partial algebras—an introductory survey". In Ivo G. Rosenberg; Gert Sabidussi (eds.). Algebras and Orders. Springer Science & Business Media. pp. 1–70. ISBN 978-0-7923-2143-9.
George A. Grätzer (2008). Universal Algebra (2nd ed.). Springer Science & Business Media. Chapter 2. Partial algebras. ISBN 978-0-387-77487-9.
Foulis, D. J.; Bennett, M. K. (1994). "Effect algebras and unsharp quantum logics". Foundations of Physics. 24 (10): 1331. Bibcode:1994FoPh...24.1331F. doi:10.1007/BF02283036. hdl:10338.dmlcz/142815. S2CID 123349992.
Richard S. Pierce (1968) Introduction to the Theory of Abstract Algebras, page 17
Pierce page 28

Example(s)

Structure

Relational systems

References

Further reading