Fra ss es Theorem A class of finite structures is the age of a - PDF document

✄ ✄ ☎ ✄ ☎ ✂ ☎ ✄ ✂ Fra¨ ıss´ e’s Theorem A class of finite structures is the age of a countable homogeneous structure M if and only if is closed under isomorphism; is closed under taking induced substructures; An algebra related to enumeration contains only countably many members up to Peter J. Cameron isomorphism; School of Mathematical Sciences Queen Mary and Westfield College has the amalgamation property . London, U.K. If these conditions hold, then M is unique up to ıss´ isomorphism, and is called the Fra¨ e limit of the Fra¨ ıss´ e class ✂ . Thus, the enumeration problem for Fra¨ ıss´ e classes is equivalent to the orbit-counting problem for permutation groups. 1 3 Enumeration and orbit counting Oligomorphic groups Many combinatorial problems can be formulated as The permutation group G on X is called oligomorphic orbit-counting problems. if the number of G orbits on the set of n -subsets of X is finite for all n . (Equivalently, on X n .) The relational structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M . The Engeler–Ryll-Nardzewski–Svenonius Theorem states that a countable first-order structure M is � M ℵ 0 -categorical if and only if Aut The age of M is the class of all finite structures ✁ is oligomorphic. embeddable in M . Example: The ordered set ✆ . (Cantor’s Theorem So, if M is homogeneous, the number of n -element characterises it as the unique countable dense � M structures in Age ordered set without endpoints. Its automorphism ✁ (up to isomorphism) is equal to � M the number of orbits of Aut group is transitive on n -sets for all n .) ✁ on the set of n -subsets of M . 2 4

✟ ✓ ✞ ✎ ✁ ✒ ✆ ✡ ✞ ✓ ✌ ✞ ✞ ✁ ✁ ✔ ✁ ✁ ✞ ✁ An algebra, continued � M Examples If M is a relational structure on X , let A ✁ be the � M subalgebra of A of the form ✠ 0 V n ✁ , where n � M V n ✁ is the set of isomorphism-invariant functions Example 1 . The finite graphs form a Fra¨ ıss´ e class. on the n -subsets of X . ıss´ Its Fra¨ e limit is the random graph or Rado’s graph R . If G is a permutation group on X , let A G be the ✠ 0 V G n , where V G subalgebra of A of the form n is n Example 2 . Consider the set of finite sets which are the set of functions fixed by G . totally ordered and whose elements are coloured using the finite set A of colours. We can represent an � M Aut If M is homogeneous and G n -element structure in this class as a word of length n ✁ , then � M A G A in the alphabet A . The Fra¨ ✁ . In future we consider the group case, but ıss´ e limit is the set with its elements coloured using the set A in such a way most of this generalises. that each colour class is dense. � V n � G If G is oligomorphic, then dim ✁ is equal to the � G number F n ✁ of orbits of G on n -sets. 5 7 An algebra Let X be an infinite set. For any non-negative integer The element e n , let V n be the set of all functions from the set of n -subsets of X to ✝ . This is a vector space over ✝ . Let e be the constant function in V 1 with value 1 . Then it can be shown that e is not a zero-divisor. So Define multiplication by e is a monomorphism from V n to � V G � V G A V n ☞ 1 . In particular, dim dim V n ✁ , that is, the ☞ 1 n n � F n � G n ✠ 0 sequence ✁ is non-decreasing. with multiplication defined as follows: for f ☛ V m , g ☛ V n , let fg be the function in V m ☞ n whose value on Many results are known about the growth of this � m n ✁ -set A is given by the sequence. For example, Macpherson showed that, if � G � A � B � A 1 for all n , or G acts primitively, then either F n ∑ fg f ✁ g ✑ B the sequence grows at least exponentially. B ✍ A ✎ B ✏ m This is the reduced incidence algebra of the poset of finite subsets of X . 6 8

✖ ✄ ✁ ✄ ✁ ✁ ✒ ✁ ✌ ✄ ✄ ✄ ✂ ✂ ✔ Polynomial algebras Let M be the Fra¨ ıss´ e limit of ✂ . Under the following A conjecture � M hypotheses, it can be shown that A ✁ is a polynomial algebra: Conjecture . If G has no finite orbits on X , then there is a notion of disjoint union in ✂ ; (a) A G is an integral domain; there is a notion of involvement on the n -element ✕ eA G is an integral (b) e is prime in A G (that is, A G structures in ✂ , so that if a structure is domain). partitioned, it involves the disjoint union of the induced substructures on its parts; Conjecture (b) implies (a). These conjectures imply certain smoothness results about the growth of there is a notion of connected structure in ✂ , so � F n � G ✁ , for example, (a) implies that that every structure is uniquely expressible as � G � G � G the disjoint union of connected structures. 1 F m F m F n ☞ n � M The polynomial generators of A ✁ are the characteristic functions of the connected structures. 9 11 Examples revisited Some results Example 1 . If is the class of finite graphs, let We say that G is entire if A G is an integral domain, disjoint union and connectedness have their usual and strongly entire if e is prime in A G . meaning, and let involvement mean spanning � R subgraph. The conditions are satisfied. So A ✁ is a polynomial algebra, where R is the random graph. I can show the following. Example 2 . let consist of all finite words in the If G contains a (strongly) entire subgroup then G alphabet A . Let disjoint union mean concatenation in is (strongly) entire. decreasing lexicographic order, involvement be lexicographic order reversed, and connected structures be the Lyndon words (those which are If the point stabiliser, acting on the remaining lexicographically smaller than their cyclic shifts). The points, is (strongly) entire, then G is (strongly) conditions are satisfied. So the shuffle algebra on A entire. is a polynomial algebra generated by the Lyndon words. 10 12

Transitive extensions In each of the examples, the automorphism group of the infinite homogeneous structure has a transitive extension, which is strongly entire (by the earlier results). It is not known whether these algebras are polynomial algebras. These are interesting combinatorial problems. � R Example 1 . The transitive extension of Aut ✁ is the automorphism group G of the countable homogeneous random two-graph . If A G is a polynomial algebra, then the number of generators of dimension n is equal to the number of Eulerian graphs on n vertices. Example 2 . There are certain circular structures whose automorphism groups are transitive extensions of the groups of Example 2. They arise in model theory. 13