Order embedding

In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism. Both of these weakenings may be understood in terms of category theory.

Formal definition Formally, given two partially ordered sets (posets) ( S , ≤ ) {\displaystyle (S,\leq )} and ( T , ⪯ ) {\displaystyle (T,\preceq )} , a function f : S → T {\displaystyle f:S\to T} is an order embedding if f {\displaystyle f} is both order-preserving and order-reflecting, i.e. for all x {\displaystyle x} and y {\displaystyle y} in S {\displaystyle S} , one has

x ≤ y if and only if f ( x ) ⪯ f ( y ) . {\displaystyle x\leq y{\text{ if and only if }}f(x)\preceq f(y).}

Such a function is necessarily injective, since f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} implies x ≤ y {\displaystyle x\leq y} and y ≤ x {\displaystyle y\leq x} . If an order embedding exists from a poset S {\displaystyle S} to a poset T {\displaystyle T} , one says that S {\displaystyle S} can be embedded into T {\displaystyle T} .

Properties

An order isomorphism can be characterized as a surjective order embedding. As a consequence, any order embedding f restricts to an isomorphism between its domain S and its image f(S), which justifies the term "embedding". On the other hand, it might well be that two (necessarily infinite) posets are mutually order-embeddable into each other without being order-isomorphic. An example is provided by the open interval ( 0 , 1 ) {\displaystyle (0,1)} of real numbers and the corresponding closed interval [ 0 , 1 ] {\displaystyle [0,1]} . The function f ( x ) = ( 94 x + 3 ) / 100 {\displaystyle f(x)=(94x+3)/100} maps the former to the subset ( 0.03 , 0.97 ) {\displaystyle (0.03,0.97)} of the latter and the latter to the subset [ 0.03 , 0.97 ] {\displaystyle [0.03,0.97]} of the former, see picture. Ordering both sets in the natural way, f {\displaystyle f} is both order-preserving and order-reflecting (because it is an affine function). Yet, no isomorphism between the two posets can exist, since e.g. [ 0 , 1 ] {\displaystyle [0,1]} has a least element while ( 0 , 1 ) {\displaystyle (0,1)} does not. For a similar example using arctan to order-embed the real numbers into an interval, and the identity map for the reverse direction, see e.g. Just and Weese (1996). A retract is a pair ( f , g ) {\displaystyle (f,g)} of order-preserving maps whose composition g ∘ f {\displaystyle g\circ f} is the identity. In this case, f {\displaystyle f} is called a coretraction, and must be an order embedding. However, not every order embedding is a coretraction. As a trivial example, the unique order embedding f : ∅ → { 1 } {\displaystyle f:\emptyset \to \{1\}} from the empty poset to a nonempty poset has no retract, because there is no order-preserving map g : { 1 } → ∅ {\displaystyle g:\{1\}\to \emptyset } . More illustratively, consider the set S {\displaystyle S} of divisors of 6, partially ordered by x divides y, see picture. Consider the embedded sub-poset { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} . A retract of the embedding i d : { 1 , 2 , 3 } → S {\displaystyle id:\{1,2,3\}\to S} would need to send 6 {\displaystyle 6} to somewhere in { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} above both 2 {\displaystyle 2} and 3 {\displaystyle 3} , but there is no such place.

Additional perspectives

Posets can straightforwardly be viewed from many perspectives, and order embeddings are basic enough that they tend to be visible from everywhere. For example:

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