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.[1][2]
As a motivation for the study of partial algebras, George Grätzer considered the case of "subsets of an algebra and properties of operations on these subsets, even if the subsets are not closed under all operations."[2]: 79
Example(s)
[edit]- partial groupoid
- field — the multiplicative inversion is the only proper partial operation[1]
- effect algebras[3]
Structure
[edit]There is a "Meta Birkhoff Theorem" by Andreka, Nemeti and Sain (1982).[1]
Relational systems
[edit]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
"Since an n-ary operation is a special case of an (n+1)-ary relation, we see that algebras may be regarded as a special case of relational structures."[4]
Though relational systems have greater generality than algebras and partial algebras, they do not have the rich theory of the algebras.[5] For example, defining a subalgebra of a relational system is not straight forward.[6]
References
[edit]- 1 2 3 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.
- 1 2 3 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.
- ↑ P. M. Cohn (1968) Universal Algebra, page 189
- ↑ Richard S. Pierce (1968) Introduction to the Theory of Abstract Algebras, page 17
- ↑ Pierce page 28
Further reading
[edit]- Peter Burmeister (2002) [1986]. A Model Theoretic Oriented Approach to Partial Algebras. CiteSeerX 10.1.1.92.6134.
- 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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