Games with finitely generated structures

Abstract

We study the abstract Banach-Mazur game played with finitely generated structures instead of open sets. We characterize the existence of winning strategies aiming at a single countably generated structure. We also introduce the concept of weak Fraïssé classes, extending the classical Fraïssé theory, revealing its relations to our Banach-Mazur game. Finally, we exhibit connections between the universality number and the weak amalgamation property.

Introduction

Infinite mathematical structures are often built inductively as unions of chains (sometimes called towers or nests) of finite structures of the same type, for example, at each stage adding just one element. Such a procedure could also be random, typical examples come from graph theory where one builds an infinite graph starting from a single vertex, at each stage adding one more, choosing randomly the new edges. Another possibility of building a structure from small “pieces” is using a natural infinite game in which two players alternately extend given finite structures, building an infinite chain. In the end, after infinitely many steps, one can test the union of the resulting chain. A general question is when one of the players has a strategy for obtaining an object with prescribed properties. Ultimately, the property could be “being isomorphic to a specific model”. A more relaxed version is “being embeddable into a specific model”.

The aim of this note is to characterize the existence of winning strategies in these games. It turns out that the crucial point is the weak amalgamation property, necessary and sufficient for the existence of winning strategies. We prove that if the goal of the game is “being isomorphic to M” then the second player has a winning strategy if and only if M is a special model characterized by some kind of extension property. Otherwise the first player has a winning strategy, that is, the game is determined. We also show that if the goal of the game is “being embeddable into N” then the second player has a winning strategy if and only if N contains an isomorphic copy of the special model M mentioned before. In order to make precise statements of our main results, we now introduce our main concepts.

Throughout this note, $\mathcal{F}$ will denote a fixed class of finitely generated models (structures) of a fixed countable signature. We assume that $\mathcal{F}$ is closed under isomorphisms. $\sigma \mathcal{F}$ will denote the class of all models representable as unions of countable chains in $\mathcal{F}$.

We consider the following infinite game for two players Eve and Odd. Namely, Eve starts by choosing $A_0\in \mathcal{F}$. Odd responds by choosing a bigger structure $A_1\supseteq A_0$ in $\mathcal{F}$. Eve responds by choosing $A_2\in \mathcal{F}$ containing $A_1$. And so on; the rules for both players remain the same. After infinitely many moves, we receive a model $A_{\infty }={\bigcup }_{n\in \omega }{A}_n$, called the result of the play. Now let $\mathcal{A}\subseteq \sigma \mathcal{F}$ be a class of models. We can say that Odd wins if the resulting structure is isomorphic to some model in $\mathcal{A}$. Otherwise, Eve wins. The game is particularly interesting when $\mathcal{A}=\{G\}$ for some concrete model G. We shall denote this game $\mathrm{BM}(\mathcal{F},\mathcal{A})$ and $\mathrm{BM}(\mathcal{F},G)$ in case where $\mathcal{A}=\{G\}$.

This is in fact an abstract version of the well known Banach-Mazur game [16], in which open sets are replaced by abstract objects. In [11] it was shown that Odd has a winning strategy in $\mathrm{BM}(\mathcal{F},G)$ whenever $\mathcal{F}$ is a Fraïssé class (equivalently: G is homogeneous with respect to its finitely generated substructures). The paper [11] contains also examples of non-homogeneous graphs G for which Odd still has a winning strategy. One needs to admit that our Banach-Mazur game is a particular case of infinite games often considered in model theory, where sometimes the players are denoted by ∃ and ∀. For more information, see the monograph of Hodges [5].

Our goal is to present a characterization of the existence of a winning strategy in $\mathrm{BM}(\mathcal{F},G)$, where G is a countable first-order structure. We also develop the theory of limits of weak Fraïssé classes, where the amalgamation property is replaced by a weaker condition exhibited first by Ivanov [7], then used by Kechris and Rosendal [8], and recently studied by Kruckman [10]. We show that Odd has a winning strategy in $\mathrm{BM}(\mathcal{F},G)$ if and only if $\mathcal{F}$ is a weak Fraïssé class and G is its limit. We note that similar results, using the topological Banach-Mazur game, were recently obtained by Kruckman in his Ph.D. thesis [10]. Our approach is direct, we do not use any topology. Finally, we discuss a variant of the game where the second player wins if the resulting structure is embeddable into a fixed in advance model V. We show that the situation is the same: the second player has a winning strategy if and only if the class in question is a weak Fraïssé class and its limit is embeddable into V. Finally, we show that small universality number implies the weak amalgamation property.

No prerequisites, except a very basic knowledge in model theory, are required for understanding our concepts and arguments.

Prehomogeneity and ubiquity in category.  As it happens, there have been some works addressing the question when a fixed countable structure occurs residually in a suitably defined space of structures (typically: all structures whose universe is the set of natural numbers). This idea was first studied by Cameron [3], later explored by Pouzet and Roux [15], although it actually goes back to Pabion [13] who invented and characterized the concept of prehomogeneity. A structure M is prehomogeneous if for every finitely generated substructure A of M there is a bigger finitely generated substructure B containing A such that every embedding of A into M extends to an automorphism of M as long as it is extendible to an embedding of B into M. It may easily happen that no automorphism of M extends a given embedding of B into M. Pouzet and Roux proved that a countable structure is ubiquitous in category (i.e. its isomorphic type forms a residual set in a suitable space) if and only if it is prehomogeneous (Cameron [3] showed earlier that homogeneous structures are ubiquitous). The theorem of Pouzet and Roux can be easily derived from our results, due to Oxtoby's characterization [12] of winning strategies in the original Banach-Mazur game played with open sets.

A structure M is pseudo-homogeneous (we would rather call it cofinally homogeneous) if every finitely generated substructure A of M extends to a finitely generated substructure B of M such that every embedding of B into M extends to an automorphism of M. Clearly, this is a natural strengthening of prehomogeneity. Pseudohomogeneity was, according to our knowledge, first studied by Calais [1], [2], around ten years after Fraïssé's work [4]. The first example of a countable prehomogeneous and not cofinally homogeneous structure is contained in Pabion [13], attributed to Pouzet. None of these works mentions the weak amalgamation property, an essential tools for constructing prehomogeneous structures from “small pieces”. Also, none of the above-mentioned works combines prehomogeneity or the weak amalgamation property with the existence of winning strategies in the abstract Banach-Mazur game played with finitely generated models.