Partial groupoid

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Group-like structures
Total Associative Identity Divisible Commutative
Partial magma UnneededUnneededUnneededUnneededUnneeded
Semigroupoid UnneededRequiredUnneededUnneededUnneeded
Small category UnneededRequiredRequiredUnneededUnneeded
Groupoid UnneededRequiredRequiredRequiredUnneeded
Magma RequiredUnneededUnneededUnneededUnneeded
Quasigroup RequiredUnneededUnneededRequiredUnneeded
Unital magma RequiredUnneededRequiredUnneededUnneeded
Loop RequiredUnneededRequiredRequiredUnneeded
Semigroup RequiredRequiredUnneededUnneededUnneeded
Monoid RequiredRequiredRequiredUnneededUnneeded
Group RequiredRequiredRequiredRequiredUnneeded
Abelian group RequiredRequiredRequiredRequiredRequired

In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation. [1] [2]

Contents

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:

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

References

  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