Partial groupoid
Closure | Associative | Identity | Cancellation | Commutative | |
---|---|---|---|---|---|
Partial magma | Unneeded | Unneeded | Unneeded | Unneeded | Unneeded |
Semigroupoid | Unneeded | Required | Unneeded | Unneeded | Unneeded |
Small category | Unneeded | Required | Required | Unneeded | Unneeded |
Groupoid | Unneeded | Required | Required | Required | Unneeded |
Commutative Groupoid | Unneeded | Required | Required | Required | Required |
Magma | Required | Unneeded | Unneeded | Unneeded | Unneeded |
Commutative magma | Required | Unneeded | Unneeded | Unneeded | Required |
Quasigroup | Required | Unneeded | Unneeded | Required | Unneeded |
Commutative quasigroup | Required | Unneeded | Unneeded | Required | Required |
Associative quasigroup | Required | Required | Unneeded | Required | Unneeded |
Commutative-and-associative quasigroup | Required | Required | Unneeded | Required | Required |
Unital magma | Required | Unneeded | Required | Unneeded | Unneeded |
Commutative unital magma | Required | Unneeded | Required | Unneeded | Required |
Loop | Required | Unneeded | Required | Required | Unneeded |
Commutative loop | Required | Unneeded | Required | Required | Required |
Semigroup | Required | Required | Unneeded | Unneeded | Unneeded |
Commutative semigroup | Required | Required | Unneeded | Unneeded | Required |
Monoid | Required | Required | Required | Unneeded | Unneeded |
Commutative monoid | Required | Required | Required | Unneeded | Required |
Group | Required | Required | Required | Required | Unneeded |
Abelian group | Required | Required | Required | Required | Required |
In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation.[1][2]
A partial groupoid is a partial algebra.
Partial semigroup
A partial groupoid is called a partial semigroup if the following associative law holds:[3]
For all such that and , the following two statements hold:
- if and only if , and
- if (and, because of 1., also ).
References
- ^ Evseev, A. E. (1988). "A survey of partial groupoids". In Ben Silver (ed.). Nineteen Papers on Algebraic Semigroups. American Mathematical Soc. ISBN 0-8218-3115-1.
- ^ Folkert Müller-Hoissen; Jean Marcel Pallo; Jim Stasheff, eds. (2012). Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift. Springer Science & Business Media. pp. 11 and 82. ISBN 978-3-0348-0405-9.
- ^ Schelp, R. H. (1972). "A partial semigroup approach to partially ordered sets". Proceedings of the London Mathematical Society. 3 (1): 46–58. doi:10.1112/plms/s3-24.1.46. Retrieved 1 April 2023.
Further reading
- E.S. Ljapin; A.E. Evseev (1997). The Theory of Partial Algebraic Operations. Springer Netherlands. ISBN 978-0-7923-4609-8.
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