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A partially ordered set in which the compact elements below each element form a directed set with that element as their least upper bound.

Every subset having an upper bound has a least upper bound. If the empty subset is included, a nonempty such partially ordered set has a least element. This condition differs from a directed-complete partial order: a directed set need not have an upper bound.

A capital regular category, in the convention of Definitions 1.1–1.3, is a regular category whose well-supported objects are well-pointed objects in a category. It does not require arbitrary objects to be detected by global points. A partially ordered set with finite meets, regarded as a category, is capital: its only well-supported object is the top element. Such a category can have many proper subterminal objects, all with no points. This example distinguishes the definition from the stronger convention that the global-sections functor reflects every isomorphism.

A category specifies objects of a category, categorical morphisms between them, and composition in a category. An arrow $f:A\to B$ can be followed by $g:B\to C$, giving $g\circ f:A\to C$. Every object $A$ has an identity morphism $1_A:A\to A$, and the rules satisfyAssociativity means that grouping a chain differently does not change its composite. It does not assert commutativity, and the sources and targets must match before a composite is defined.

In the Category of sets, arrows are functions. Take two functions on the real numbers, $f(x)=x+1$ and $g(x)=2x$. Then $(g\circ f)(x)=2x+2$, whereas $(f\circ g)(x)=2x+1$. At $x=3$, the two orders give 8 and 7, respectively. Both orders are defined here because these maps share their source and target; for arbitrary arrows the reverse composite need not exist.

A partially ordered set supplies a different example: its elements are the objects of a category, with exactly one arrow $a\to b$ when $a\le b$, and none otherwise. Transitivity gives composition in a category, and reflexivity gives identity morphisms. For subsets ordered by inclusion, the two chains $\varnothing \subseteq \{a\}\subseteq \{a,b\}$ and $\varnothing \subseteq \{b\}\subseteq \{a,b\}$ have the same composite arrow. Thus arrows need not be maps of elements: they can record a relation.

Category theory studies mathematical structures through the morphisms between them. A category specifies which objects of a category and arrows are allowed; a functor translates these objects and arrows into another category; a natural transformation compares two such translations coherently. For example, the category of groups records both groups and group homomorphisms, so it remembers which maps respect multiplication.

The usefulness appears in constructions we already know. A Cartesian product combines two sets, while a direct product of groups combines two groups. Both satisfy the same universal property of a categorical product: any pair of maps into the factors comes from exactly one map into the product. This gives a reusable way to construct maps and establish isomorphisms, without depending on a particular choice of coordinates or element labels. In a partially ordered set regarded as a category, the corresponding categorical product is a greatest lower bound. The shared definition exposes a connection between pairing data and finding the best common bound.

Start with the computed examples under category, functor, natural transformation, and product in a category. Their arrow equations are practical tests: do both routes in a commutative diagram produce the same result? Later, the same language organizes tensor products, adjunctions, and the fundamental group of a topological space. Emily Riehl's textbook develops these basic notions before using them to compare constructions across different subjects. ^[ref]

In their 1945 paper, Samuel Eilenberg and Saunders Mac Lane set out the notions of category, functor, and natural transformation. The introduction uses the relation between a vector space and its dual space to ask which comparisons are natural, setting up a precise language for compatibility with maps. ^[ref]

A chain-complete partially ordered set has a supremum for every chain, with the empty-chain condition requiring a least element if empty chains are included. This implies the weaker upper-bound hypothesis of Zorn lemma, which requires only that each chain have some upper bound, not necessarily a least one. Some treatments use “chain-complete” for that weaker condition, so the convention should be specified. In the inclusion order on lattice ideals avoiding a specified element, the union of a nonempty chain is an ideal and is its supremum. This union observation is the step needed to apply Zorn in prime-ideal separation arguments.

A chain in a partially ordered set is a subset in which every two elements are comparable.

The order induced on this subset is a total order.

A subset of a partially ordered set on which the inherited order is total. The upper-bound hypothesis in Zorn lemma concerns all such subsets, not just countable increasing sequences.

Let $\mathbf{FinPos}$ contain finite partially ordered sets and all order-preserving functions, including the empty object. The covariant functor category $[\mathbf{FinPos},\mathbf{Set}]$ classifies weak partial orders. Finite positive presentations reduce to finite preorders with mutually comparable cycles identified. An internal model $M$ gives a flat functor $P\mapsto \mathrm{Hom}(P,M)$ on ${\mathbf{FinPos}}^{\mathrm{op}}$; conversely the corresponding filtered presentation by finite models recovers its carrier and order relation.

A complete lattice is a partially ordered set in which every subset has a meet and a join, including the empty subset.

A partially ordered set in which, for every $x$, the elements way-below $x$ form a directed set with least upper bound $x$. Directed least upper bounds other than these approximating ones need not exist.

In a partially ordered set, $y$ covers $x$ when $x<y$ and there is no $z$ with $x<z<y$. Thus a cover relation records an immediate increase, rather than every comparable pair. For a finite partially ordered set, every strict comparison can be obtained by chaining cover relations.

A differential poset is a locally finite graded partially ordered set with a unique minimum, finite ranks, and up and down cover operators satisfying $DU-UD=rI$ for a fixed positive integer $r$. The Young lattice has $r=1$: distinct equal-rank diagrams have equally many common upper and lower covers, and each diagram has one more upper cover than lower cover.

Every finite partially ordered set can be partitioned into as many chains in a partial order as its largest antichain. Reachability in an acyclic directed graph is a partial order, so the theorem supplies chains which can be expanded into directed paths, possibly overlapping at intervening vertices.

A partially ordered set is directed-complete if every directed set in it has a least upper bound. A complete lattice is directed-complete. Conversely, directed completeness together with all finite joins, including the empty join, gives arbitrary joins: take the directed set of joins of finite subsets.

A fixed point of a function on a partially ordered set that is greater than or equal to every other fixed point. For the order-preserving function $H(X)=A\cap \{s:\mathrm{Post}(s)\subseteq X\}$ on the power set of a finite transition system with no terminal states, its greatest fixed point is the set of states from which every infinite path in a transition system remains in $A$. Iteration from the whole state space decreases to this set; every strict decrease removes at least one state.

A Hasse diagram draws a finite partially ordered set using one point for each element and an upward line from $x$ to $y$ for each cover relation $x\prec y$. It omits reflexive loops and comparisons already implied by paths. Line crossings are not additional elements. In a finite partially ordered set, following upward paths recovers exactly the original comparisons. The diagram is named after Helmut Hasse.

A real-valued function on a partially ordered set is increasing when it preserves its order. Indicators of increasing events are examples. In spin or edge configuration spaces the order is coordinatewise, so increasing observables grow when a zero coordinate is replaced by one. Positive association of random variables concerns pairs of such observables, and stochastic domination compares their expectations.

For a subset $A$ of a partially ordered set, its infimum is its greatest lower bound and its supremum is its least upper bound. An upper bound only has to lie above every member; the supremum is the smallest such bound. A lower bound lies below every member; the infimum is the largest such bound. These definitions identify the exact edges of a set even when neither edge belongs to it.

For example, $(0,1)$ and $[0,1]$ both have infimum zero and supremum one. Only $[0,1]$ has a minimum and a maximum: those must be actual members of the set. The set $C=\{1/n:n\in {\mathbb{N}}_{>0}\}$ has infimum zero without a minimum, but its supremum one is also its maximum. Such examples distinguish a bound from an attained extremal value. ^[ref].

For a nonempty subset of the real numbers, completeness of the real numbers guarantees a real supremum when it is bounded above, and a real infimum when it is bounded below. The bound and the ambient partially ordered set both matter: $\{q\in \mathbb{Q}:q\ge 0,q^2<2\}$ has supremum $\sqrt{2}$ in the real numbers, but no supremum in the rational numbers. The detailed explanation belongs to completeness of the real numbers.

Why are unattained bounds useful? They specify a precise limiting target. In mathematical optimization, $f(x)=x$ on $(0,1)$ has infimum zero but no global minimizer. The inputs $x_n=1/n$ for $n\ge 2$ give a minimizing sequence: their objective values approach zero without ever reaching it. Bounds therefore describe what approximation can achieve even when no feasible input is best. The approximation property of an infimum and approximation property of a supremum express this idea quantitatively.

An inflationary map on a partially ordered set satisfies $x\le f(x)$ for every $x$. This condition is distinct from being order-preserving.