# Ordered groupoid

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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).