On a general construction of countable universal homogeneous - PowerPoint PPT Presentation

On a general construction of countable universal homogeneous algebraic systems Dragan Maˇ sulovi´ c Department of Mathematics and Informatics University of Novi Sad, Serbia (joint work with Wiesłav Kubi´ s) AAA 88, Warsaw, 19–22 June 2014

Homogeneous structures automorphism A isomorphism

Fra¨ ıss´ e theory age ( A ) — the class of all finitely generated struct’s which embed into A amalgamation class — a class K of fin. generated struct’s s.t. ◮ there are only countably many pairwise noniso struct’s in K ; ◮ K has (HP); ◮ K has (JEP); and ◮ K has (AP): for all A , B , C ∈ K and embeddings v f : A ֒ → B and g : A ֒ → C , there exist D ∈ K C → D ֒ and embeddings u : B ֒ → D and v : C ֒ → D → → g u ֒ ֒ such that u ◦ f = v ◦ g . A ֒ → B f

Fra¨ ıss´ e theory Theorem. [Fraisse, 1953] 1 If A is a countable homogeneous structure, then age ( A ) is an amalgamation class. 2 If K is an amalgamation class, then there is a unique (up to isomorphism) countable homogeneous structure A such that age ( A ) = K . 3 If B is a countable structure younger than A (that is, age ( B ) ⊆ age ( A ) ), then B ֒ → A . Definition. If K is an amalgamation class and A is the countable homogeneous structure such that age ( A ) = K , we say that A is the Fra¨ ıss´ e limit of K .

Some prominent Fra¨ ıss´ e limits ıss´ ( Q , < ) — the Fra¨ e limit of the class of all linear orders U Q — Fra¨ ıss´ e limit of the class of finite metric spaces with rational distances (the rational Urysohn space) R — Fra¨ ıss´ e limit of the class of all finite graphs (the Rado graph) P — Fra¨ ıss´ e limit of the class of all finite posets (the random poset)

The Urysohn space P. U RYSOHN : Sur un espace m´ etrique universel. Bull. Math. Sci. 51 (1927), 43–64, 74–90 U — complete separable metric space which is homogeneous and embeds all separable metric spaces. U = U Q

Kat´ etov’s construction of the Urysohn space M. K AT ´ ETOV : On universal metric spaces. General topology and its relations to modern analysis and algebra. VI (Prague, 1986), Res. Exp. Math. vol. 16, Heldermann, Berlin, 1988, 323–330 A Kat´ etov function over a finite rational metric space X is every function α : X → Q such that | α ( x ) − α ( y ) | � d ( x , y ) � α ( x ) + α ( y ) K ( X ) = all Kat´ etov functions over X , which is a rational metric space under sup metric → K 2 ( X ) ֒ → K 3 ( X ) ֒ colim ( X ֒ → K ( X ) ֒ → · · · ) = U Q

Kat´ etov’s construction of the Urysohn space M. K AT ´ ETOV : On universal metric spaces. General topology and its relations to modern analysis and algebra. VI (Prague, 1986), Res. Exp. Math. vol. 16, Heldermann, Berlin, 1988, 323–330 Observation 1. K ( X ) is the set of all 1-types over X (in an appropriate first-order language). Observation 2. K is functorial.

� � �� � Kat´ etov functors A — a category of fin generated L -struct’s with (HP) and (JEP) C — the category of all colimits of ω -chains in A Definition. A functor K : A → C is a Kat´ etov functor if K preserves η A A � � K ( A ) embeddings and there exists a natural � � transformation η : ID → K such that for · every embedding f : A ֒ → B in A where g B is a 1-point extension of A there is an B embedding g : B ֒ → K ( A ) satisfying − →

Kat´ etov functors A Kat´ etov functor exists for the following categories A : ◮ finite linear orders with order-preserving maps, ◮ finite graphs with graph homomorphisms, ◮ finite K n -free graphs with embeddings, ◮ finite digraphs with digraph homomorphisms, ◮ finite rational metric spaces with nonexpansive maps, ◮ finite posets with order-preserving maps, ◮ finite boolean algebras with homomorphisms, ◮ finite semilattices with embeddings, ◮ finite lattices with embeddings, ◮ finite distributive lattices with embeddings. A Kat´ etov functor does not exist for the category of finite K n -free graphs and graph homomorphisms.

Existence of Kat´ etov functors A — a category of fin generated L -struct’s with (HP) and (JEP) C — the category of all colimits of ω -chains in A Theorem. If there exists a Kat´ etov functor K : A → C , then A is ıss´ an amalgamation class, and its Fra¨ e limit F can be obtained by the “Kat´ etov construction” starting from an arbitrary A ∈ A : → K 2 ( A ) ֒ → K 3 ( A ) ֒ F = colim ( X ֒ → K ( A ) ֒ → · · · ) .

Kat´ etov functors for categories of algebras L — algebraic language V — a variety of L -algebras understood as a category of L -algebras with embeddings A — the full subcategory of V spanned by all finitely generated algebras in V C — the full subcategory of V spanned by all countably generated algebras in V Theorem. Suppose that there are only countably many isomorphism types in A . There exists a Kat´ etov functor K : A → C if and only if A is the amalgamation class.

The Importance of Being ✘✘✘✘✘ ✘ Earnest Functor Theorem. Let K : A → C be a Kat´ etov functor and let F be the Fra¨ ıss´ e limit of A . Then for every object C in C : ◮ Aut ( C ) ֒ → Aut ( F ) ; ◮ End C ( C ) ֒ → End C ( F ) . Corollary. For the following Fra¨ ıss´ e limits F we have that End ( F ) embeds all transformation monoids on a countable set: ◮ Q , ◮ the random graph [Bonato, Deli´ c, Dolinka 2010], ◮ the random digraph, ◮ the rational Urysohn space, ◮ the random poset [Dolinka 2007], ◮ the countable atomless boolean algebra.

The Importance of Being ✘✘✘✘✘ ✘ Earnest Functor ıss´ Corollary. For the following Fra¨ e limits F we have that Aut ( F ) embeds all permutation groups on a countable set: ◮ Q [Truss], ◮ the random graph [Henson 1971], ◮ Henson graphs [Henson 1971], ◮ the random digraph, ◮ the rational Urysohn space [Uspenskij 1990], ◮ the random poset, ◮ the countable atomless boolean algebra, ◮ the random semilattice, ◮ the random lattice, ◮ the random distributive lattice.