Partially ordered set

In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. This relation formalizes the intuitive concept of an ordering, sequencing, or arrangement of that set's elements. Such an ordering does not necessarily need to be total, that is, it need not guarantee the mutual comparability of all objects in the set.

Category theory

When considered as a category where hom(x, y) = {(x, y) : x ≤ y} and (y, z)o(x, y) = (x, z), posets are equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if any, is an initial object, and the largest element, if any, a terminal object. Also, every pre-ordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.

Related Topics:
Isomorphic - Initial object - Terminal object - Isomorphism-closed

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