Partially ordered set
Mappings between partially ordered sets

Given two partially ordered sets (S, ≤) and (T, ≼), a function f: S → T {\displaystyle f:S\to T} is called order-preserving, or monotone, or isotone, if for all x, y ∈ S, {\displaystyle x,y\in S,} x ≤ y {\displaystyle x\leq y} implies f (x) ≼ f (y).If (U, ≲) is also a partially ordered set, and both f: S → T {\displaystyle f:S\to T} and g: T → U {\displaystyle g:T\to U} are order-preserving, their composition g ∘ f: S → U {\displaystyle g\circ f:S\to U} is order-preserving, too.A function f: S → T {\displaystyle f:S\to T} is called order-reflecting if for all x, y ∈ S, {\displaystyle x,y\in S,} f (x) ≼ f (y) implies x ≤ y. {\displaystyle x\leq y.} If f is both order-preserving and order-reflecting, then it is called an order-embedding of (S, ≤) into (T, ≼).In the latter case, f is necessarily injective, since f (x) = f (y) {\displaystyle f(x)=f(y)} implies x ≤ y and y ≤ x {\displaystyle x\leq y{\text{ and }}y\leq x} and in turn x = y {\displaystyle x=y} according to the antisymmetry of ≤. {\displaystyle \leq.} If an order-embedding between two posets S and T exists, one says that S can be embedded into T. If an order-embedding f: S → T {\displaystyle f:S\to T} is bijective, it is called an order isomorphism, and the partial orders (S, ≤) and (T, ≼) are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagrams (see Fig. 7a). It can be shown that if order-preserving maps f: S → T {\displaystyle f:S\to T} and g: T → U {\displaystyle g:T\to U} exist such that g ∘ f {\displaystyle g\circ f} and f ∘ g {\displaystyle f\circ g} yields the identity function on S and T, respectively, then S and T are order-isomorphic.

For example, a mapping f: N → P (N) {\displaystyle f:\mathbb {N} \to \mathbb {P} (\mathbb {N})} from the set of natural numbers (ordered by divisibility) to the power set of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. It is order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. However, it is neither injective (since it maps both 12 and 6 to { 2, 3 } {\displaystyle \{2,3\}}) nor order-reflecting (since 12 does not divide 6). Taking instead each number to the set of its prime power divisors defines a map g: N → P (N) {\displaystyle g:\mathbb {N} \to \mathbb {P} (\mathbb {N})} that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it, for instance, does not map any number to the set { 4 } {\displaystyle \{4\}}), but it can be made one by restricting its codomain to g (N). {\displaystyle g(\mathbb {N}).} Fig. 7b shows a subset of N {\displaystyle \mathbb {N} } and its isomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to a wide class of partial orders, called distributive lattices; see Birkhoff's representation theorem.
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