Ordered groupoid

A groupoid whose elements are partially ordered by a relation satisfying the axioms

If an ordered groupoid satisfies the stronger axiom

then the order on is called strict, and is a strictly partially ordered groupoid. A partially ordered groupoid is said to be strong if

A strongly partially ordered groupoid is always strict, and for totally ordered groupoids the two concepts coincide.

An element of an ordered groupoid is called positive (strictly positive) if the inequalities and (respectively, and ) hold for all . Negative and strictly negative elements are defined by the opposite inequalities. An ordered groupoid is called positively (negatively) ordered if all its elements are positive (negative). Some special types of ordered groupoids are of particular interest (cf. Naturally ordered groupoid; Ordered semi-group; Ordered group).

Comments

The above definition refers to the first of the two meanings of the word groupoid. Groupoids in the second sense also occur naturally with orderings in various contexts: for example, the groupoid of all partial automorphisms of an algebraic or topological structure (that is, isomorphisms between its substructures — e.g. the groupoid of diffeomorphisms between open subsets of a smooth manifold) is naturally ordered by the relation: if is the restriction of to a subset of its domain. Ordered groupoids of this type are of importance in differential geometry (see [a1]). More generally, any inverse semi-group (cf. Inversion semi-group) can be regarded as a groupoid, whose objects are the idempotent elements of , and where the domain and codomain of an element are and , respectively; here the objects have a natural meet semi-lattice ordering, and the order can also be defined on morphisms in a natural way (see [a2]).

References

[a1]	Ch. Ehresmann, "Structures locales et catégories ordonnés" , Oeuvres complètes et commentées , Supplément aux Cahiers de Topologie et Géométrie Différentielle Catégoriques , Partie II (1980)
[a2]	J.M. Howie, "An introduction to semigroup theory" , Acad. Press (1976)