Ordered groupoid

A groupoid $H$ whose elements are partially ordered by a relation $\le$ satisfying the axioms $$ a \le b \Rightarrow ac \le bc\,,\ ca \le cb\ \ \text{for all}\ a,b,c \in H \ . $$

If an ordered groupoid $H$ satisfies the stronger axiom $$ a < b \Rightarrow ac < bc\,,\ ca< cb\ \ \text{for all}\ a,b,c \in H $$ then the order on $H$ is called strict, and $H$ is a strictly partially ordered groupoid. A partially ordered groupoid $H$ is said to be strong if $$ ac \le bc \ \text{and}\ ca \le cb \Rightarrow a \le b \ . $$

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

An element $a$ of an ordered groupoid $H$ is called positive (strictly positive) if the inequalities $ax \ge x$ and $xa \ge x$ (respectively, $ax > x$ and $xa > x$) hold for all $x \in H$. 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", see Magma. 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: $f \le g$ if $f$ is the restriction of $g$ 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) $S$ can be regarded as a groupoid, whose objects are the idempotent elements of $S$, and where the domain and codomain of an element $s$ are $s^{-1}s$ and $ss^{-1}$, 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)