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 \( {\displaystyle (G,\circ )} \) is called a partial semigroup if the following associative law holds:[3]
Let \( {\displaystyle x,y,z\in G} \) such that \( {\displaystyle x\circ y\in G} \) and \( {\displaystyle y\circ z\in G} \), then
\( {\displaystyle x\circ (y\circ z)\in G} \) if and only if\( {\displaystyle (x\circ y)\circ z\in G} \)
and \( {\displaystyle x\circ (y\circ z)=(x\circ y)\circ z} \) if \( {\displaystyle x\circ (y\circ z)\in G} \) (and, because of 1., also \( {\displaystyle (x\circ y)\circ z\in G}). \)
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.
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.a
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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