Partial algebra

Short description: Algebraic structure

In abstract algebra, a partial algebra is a generalization of universal algebra to partial operations.^[1]^[2]

Example(s)

partial groupoid
field — the multiplicative inversion is the only proper partial operation^[1]
effect algebras^[3]

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 $𝔄$ 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

↑ ^1.0 ^1.1 ^1.2 Peter Burmeister (1993). "Partial algebras—an introductory survey". Algebras and Orders. Springer Science & Business Media. pp. 1–70. ISBN 978-0-7923-2143-9.
↑ ^2.0 ^2.1 George A. Grätzer (2008). Universal Algebra (2nd ed.). Springer Science & Business Media. Chapter 2. Partial algebras. ISBN 978-0-387-77487-9. https://archive.org/details/isbn_9780387774862.
↑ Foulis, D. J.; Bennett, M. K. (1994). "Effect algebras and unsharp quantum logics". Foundations of Physics 24 (10): 1331. doi:10.1007/BF02283036. Bibcode: 1994FoPh...24.1331F.
↑ Richard S. Pierce (1968) Introduction to the Theory of Abstract Algebras, page 17
↑ Pierce page 28

Peter Burmeister (2002). A Model Theoretic Oriented Approach to Partial Algebras.
Horst Reichel (1984). Structural induction on partial algebras. Akademie-Verlag.
Horst Reichel (1987). Initial computability, algebraic specifications, and partial algebras. Clarendon Press. ISBN 978-0-19-853806-6.

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[Burmeister1993-1] 1.0 ^1.1 ^1.2 Peter Burmeister (1993). "Partial algebras—an introductory survey". Algebras and Orders. Springer Science & Business Media. pp. 1–70. ISBN 978-0-7923-2143-9.

[GAG-2] 2.0 ^2.1 George A. Grätzer (2008). Universal Algebra (2nd ed.). Springer Science & Business Media. Chapter 2. Partial algebras. ISBN 978-0-387-77487-9. https://archive.org/details/isbn_9780387774862.

[3] Foulis, D. J.; Bennett, M. K. (1994). "Effect algebras and unsharp quantum logics". Foundations of Physics 24 (10): 1331. doi:10.1007/BF02283036. Bibcode: 1994FoPh...24.1331F.

[4] Richard S. Pierce (1968) Introduction to the Theory of Abstract Algebras, page 17

[5] Pierce page 28

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Partial algebra

Contents

Example(s)

Structure

Relational systems

References

Further reading