The Retract Construction

Abstract

The retract theme is a common one in the setting of ordered algebras. For instance, retracts play a considerable role in the algebraical theory of projective lattices (≡ retracts of free lattices). [9], [13] The study of retracts for general ordered sets is much less familiar.

A subset Q of an ordered set P is a retract of P if there is an order-preserving map g of P to Q which is the identity map on Q. The term is used up to isomorphism: an ordered set Q is a retract of an ordered set P if there are order-preserving maps f of Q to P and g of P to Q such that g ∘ f is the identity map on Q. In either case g is a retraction map.

For instance, if P contains a subset Q ≅ $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{2}$ (the two-element chain) then Q is a retract; in fact, if a < b in P then {a,b} ≅ $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{2}$ and

$$g(x) = \left\{ {_{b{\text{ if }}x\not \leqslant a}^{a{\text{ if }}x \leqslant a}} \right.$$

is a retraction. On the other hand, if P contains Q ≅ 2 (the two-element antichain) then Q is a retract of P if and only if P is disconnected. If $A \subseteq P$ is a (connected) component of P, $B = P - A \ne \not 0,a \in A,{\text{ and }}b \in B,$, then

$$g(x) = \left\{ {_{b{\text{ if }}x \in B}^{a{\text{ if }}x \in A}} \right.$$

is a retraction; if P is connected then each retract must be connected.

Retracts as Subobjects. It is fairly common to treat any subset of an ordered set with the inherited order as a ‘subobject’. The trouble with this convention is that very little of an ordered set is really inherited by an arbitrary subset. For instance, neither the completeness nor the connectivity of an ordered set is inherited by every subset. These and many other syntactical properties of an ordered set are, however, preserved by their retracts. One of the most interesting of these properties is the fixed point property. [5] The theme of ‘retract as subobject’ has recently been rather carefully exploited in a series of papers aimed at fashioning a structure theory for ordered sets. [7], [16], [20]

Retractions as Choice Functions. It is usually quite difficult to decide just which subsets of an ordered set are its retracts. As a rule we learn a great deal structurally about an ordered set when we can classify some or all of its retracts.

The retraction maps themselves are at times surprisingly simple. Among the earliest results are these two for lattices. The supremum map. Let P be a complete ordered set. Then $Q \subseteq P$ is a retract if and only if Q is complete. In this case,

$$g(x) = {\sup _Q}\{ y \in Q|y \leqslant x\} $$

is the retraction map. (Note that ${\sup _Q}\not 0 = {0_Q},$ the least element of Q.) [1]

The weaving map. Let ϕ be a homomorphism of a lattice P onto a countable lattice Q. Then Q is a retract of P. Let Q = {q ₁,q ₂,q ₃,…} and let p _i ∈ ϕ⁻¹(q _i ). Define the map f of Q to P by f(q ₁) = p ₁ and

$$f({q_i}) = \left( {{p_i} \vee \mathop \vee \limits_{j < i} (f({q_j})|{q_j} < {q_i})} \right) \wedge \mathop \wedge \limits_{j < i} (f({q_i})|{q_j} > {q_i}).$$

Then g = ϕ is a retraction (g ∘ f = id _Q ). [4]

Some retraction maps of more recent vintage are these. The folding map. Every crown of minimum order in an ordered set of length one is a retract. [14], [18] The well order map. Let C be a maximal chain in an ordered set P. Then C is a retract of P. Let $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\alpha }$ be a well order of C and for each x ∈ P let N(x) consist of all those y ∈ C such that either y = x or y is noncomparable with x. Then

$$g(x) = {\inf _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\alpha } }}N(x)$$

is a retraction map. [8]

Some retraction maps are quite cunning.

An example of a result from the structure theory is this. Every countable lattice is a retract of a direct product of chains each isomorphic to the chain of rational numbers. [16]