Partial groupoid

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]

Let such that and , then

  1. if and only if
  2. and if (and, because of 1., also ).
gollark: So say "greater than X"?
gollark: I mean that it gives you a better reason to come up with more accurate information and not just wildly say whatever, because you have some (small) financial reason.
gollark: I don't see an issue with betting. It gives you incentives to make better predictions.
gollark: Having some specific mental thing preventing you from wearing a mask is probably very rare compared to, say, just having... severe asthma?
gollark: I think you missed the context.

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. Shelp, R. H. (1972). "A Partial Semigroup Approach to Partially Ordered Sets". Proc. London Math. Soc. (1972) s3-24 (1). London Mathematical Soc. pp. 46–58.

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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