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Partial groupoid

From Wikipedia, the free encyclopedia
Group-like structures
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[edit]

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:

  1. if and only if , and
  2. if (and, because of 1., also ).

References[edit]

  1. ^ 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.
  2. ^ 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.
  3. ^ 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[edit]

  • E.S. Ljapin; A.E. Evseev (1997). The Theory of Partial Algebraic Operations. Springer Netherlands. ISBN 978-0-7923-4609-8.