Partial algebra
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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 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
Further reading
- 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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