Posets, homomorphisms, and homogeneity Peter J. Cameron p.j.cameron@qmul.ac.uk Dedicated to Jarik Neˇ setˇ ril on his sixtieth birthday Summary Fra¨ ıss´ e’s Theorem In about 1950, Fra¨ ıss´ e gave a necessary and suf- Jarik Neˇ setˇ ril has made deep contributions to ficient condition on a class C of finite structures for all three topics in the title, and we began think- it to be the age of a countable homogeneous struc- ing about connections between them when I spent ture M . six weeks in Prague in 2004. In this talk I want to survey the three topics and their connections. I The key part of this condition is the amalgama- will be reporting a theorem by my student Debbie tion property : two structures in C with isomorphic Lockett. substructures can be “glued together” so that the substructures are identified, inside a larger struc- • Homogeneous and generic structures ture in C . • Construction of the generic poset Moreover, if C satisfies Fra¨ ıss´ e’s conditions, then M is unique up to isomorphism; we call it the • Homomorphisms and homomorphism- Fra¨ ıss´ e limit of C . homogeneity Ramsey theory • Homomorphism-homogeneous posets There is a close connection between homogene- ity and Ramsey theory. Hubiˇ cka and Neˇ setˇ ril have shown that, if a Universality and homogeneity countably infinite structure carries a total order A countable relational structure M belonging to and the class of its finite substructures is a Ramsey a class P is class, then the infinite structure is homogeneous. • universal if every finite or countable structure This gives a programme for determining the in P is embeddable in M (as induced sub- Ramsey classes: first find classes satisfying the structure); amalgamation property, and then decide whether they have the Ramsey property. • homogeneous if every isomorphism between fi- The converse is false in general, but Jarik nite substructures of M can be extended to an Neˇ setˇ ril recently showed that the class of finite automorphism of M (an isomorphism M → metric spaces is a Ramsey class. M ). The random graph The age of a relational structure M is the class C The class of all finite graphs is obviously a of all finite structures embeddable in M . Fra¨ ıss´ e class. Let R be its Fra¨ ıss´ e limit. Then 1
• R is the unique countable universal homoge- Set theory with an atom neous graph; Take a countable model of set theory with a sin- gle atom ♦ . Now let M be any set not containing • R is the countable random graph ; that is, if ♦ . Put edges of a countable graph are chosen inde- M L = { A ∈ M : ♦ / ∈ A } , pendently with probability 1 2 , then the result- M R = { B \ {♦} : ♦ ∈ B ∈ M } . ing graph is isomorphic to R with probabil- Then neither M L nor M R contains ♦ . ity 1 (Erd˝ os and R´ enyi); In the other direction, given two sets P , Q whose elements don’t contain ♦ , let ( P | Q ) = P ∪ { B ∪ • R is the generic countable graph (this is an ana- {♦} : B ∈ Q } . Then ( P | Q ) doesn’t contain ♦ . logue of the Erd˝ os–R´ enyi theorem, with Baire Moreover, for any set M not containing ♦ , we category replacing measure). have M = ( M L | M R ) . Note that any set not containing ♦ can be repre- Constructions of R sented in terms of sets not involving ♦ by means There are a number of simple explicit construc- of the operation ( . | . ) tions for R , the first of which was given by Rado. For example, { ∅ , {♦}} is ( { ∅ } | { ∅ } ) . My favourite is the following: the vertices are the primes congruent to 1 mod 4; join p to q if p is The generic poset a quadratic residue mod q . Let P be the collection of the sets M not contain- Another one (relevant to what will follow) ing ♦ defined by the following recursive proper- is: Take any countable model of the Zermelo– ties: Fraenkel axioms for set theory; join x to y if either Correctness: M L ∪ M R ⊆ P and M L ∩ M R = ∅ ; x ∈ y or y ∈ x . Ordering: For all A ∈ M L and B ∈ M R , we have We do not need all of ZF for this; in particular, ( { A } ∪ A R ) ∩ ( { B } ∪ B L ) � = ∅ . Choice is not required. The crucial axiom turns out to be Foundation. Completeness: A L ⊆ M L for all A ∈ M L , and B R ⊆ M R for all B ∈ M R . The generic poset Now we put M ≤ N if In similar fashion, the class of all finite posets is a Fra¨ ıss´ e class; let P be its Fra¨ ıss´ e limit. We call P ( { M } ∪ M R ) ∩ ( { N } ∪ N L ) � = ∅ . the generic poset . Theorem 1. The above-defined structure is isomorphic • P is the unique countable homogeneous uni- to the generic poset P . versal poset; Homomorphisms • P is the generic countable poset. (It is not A homomorphism f : M → N between relational clear how to define the notion of “countable structures of the same type is a map which pre- random poset”, but no sensible definition will serves the relations. For example, if M and N are give P .) posets with the strict order relation < , then a f is a homomorphism if and only if Schmerl classified all the countable homoge- neous posets. Apart from P , there are only an in- x < y ⇒ f ( x ) < f ( y ) . finite antichain and some trivial modifications of the totally ordered set Q . As usual, a monomorphism is a one-to-one homo- There is no known direct construction of P sim- morphism, and an isomorphism is a bijective ho- ilar to the constructions of R . I now outline a nice momorphism whose inverse is also a homomor- recursive construction by Hubiˇ cka and Neˇ setˇ ril. phism. 2
for any finite set Q , the set { z : z < Q } Thus, homomorphisms of the non-strict order has no maximal element and { z : z > Q } relation in posets are not the same as homomor- phisms of the strict order; but monomorphisms for has no minimal element. the two relations are the same. This is easy to see in the case Q = ∅ (so that P For most of this talk I will consider the strict or- has no least or greatest element). In general, sup- der. pose that Q < z , and z < z ′ . Extend the isomor- phism fixing Q and mapping z ′ to z ; if z ′′ is the Notions of homogeneity image of z , then Q < z ′′ < z . We say that a relational structure X has property HH if every homomorphism between finite sub- Taking Q to be a singleton, we see that P is structures of X can be extended to a homomor- dense. phism of X . Similarly, X has property MH if ev- ery monomorphism between finite substructures X-free posets extends to a homomorphism. There are six prop- We say that a countable poset is X-free if it satis- erties of this kind that can be considered: HH, MH, fies the following: IH, MM, IM, and II. (It is not reasonable to extend a map to one satisfying a stronger condition!) Note If A and B are 2-element antichains with that II is equivalent to the standard notion of homo- A < B , then there does not exist a point z geneity defined earlier . with A < z < B . These properties are related as follows Such a point z together with A and B would form (strongest at the top): the poset X. II MM HH Take a discrete tree T ; for each pair ( x , y ) in T ց ւ ց ւ such that y covers x , add a copy of the open ratio- IM MH nal interval ( 0, 1 ) between x and y ; and delete the ց ւ points of T . This poset is vacuously X-free, and IH also has the property that for any finite Q , { z : z < Q } has no maximal element and { z : z > Q } has no minimal element. Extensions of P Any poset with these two properties can be We can recognise P by the property that, if A , B shown to be HH and MM. This gives 2 ℵ 0 non- and C are pairwise disjoint finite subsets with the isomorphic HH and MM posets. properties that A < B , no element of A is above an element of C , and no element of B is below an Lockett’s Theorem element of C , then there exists a point z which is above A , below B , and incomparable with C . Theorem 2. • For a countable poset which is not Extensions of P (posets X with the same point an antichain, the properties IM, IH, MM, MH, set, in which x < y in P implies x < y in X ) can HH are all equivalent. be recognised by a similar property: if A and B are • A countable poset P has one of these properties if finite disjoint sets with A < B , then there exists a and only if one of the following holds: point z satisfying A < z < B . Using this, it can be shown that any extension – P is an antichain; of P has the properties MM and HH (and hence – P is the union of incomparable copies of Q ; all the earlier properties except II). – P is an extension of the generic poset P ; – P is X-free and, for any finite set Q, { z : z < Properties Q } has no maximal element and { z : z > If an IH poset P is not an antichain, then it has Q } has no minimal element. the following property: 3
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