A survey of homogeneous structures

Dugald Macpherson
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
A relational first order structure is homogeneous if it is countable (possibly finite) and every isomorphism between finite substructures extends to an automorphism. This article is a survey of several aspects of homogeneity, with emphasis on countably infinite homogeneous structures. These arise as Fraissé limits of amalgamation classes of finite structures. The subject has connections to model theory, to permutation group theory, to combinatorics (for example through combinatorial
more &raquo; ... and through Ramsey theory), and to descriptive set theory. Recently there has been a focus on connections to topological dynamics, and to constraint satisfaction. The article discusses connections between these topics, with an emphasis on examples, and on special properties of an amalgamation class which yield important consequences for the automorphism group. language is a function h from A to B such that for any n > 0, any relation symbol R of arity n, and any a 1 , . . . , a n ∈ A, if R(a 1 , . . . , a n ) holds in A then R(h(a 1 ), . . . , h(a n )) holds in B. The article does not require much background from model theory. I assume familiarity with the notions of first order language and structure, formula, and interpretation of a formula in a language. Beyond this, I have tried to give definitions if they are needed. A good background source in model theory is [81] . Background to homogeneous structures Amalgamation classes and Fraissé limits In this article, I will adopt the following definition of homogeneity. By a relational structure, I mean a structure (M, (R i ) i∈I ). Such a structure has domain or universe M. Each R i has a prescribed arity, and a relation R i of arity a i is just a subset of M a i . The corresponding language L has relation symbols corresponding to the R i , of appropriate arity a i , and I do not distinguish notationally between a relation symbol (in the language) and the corresponding relation in the structure. I tend to use the same symbol for a structure and its domain, but where there is ambiguity (e.g. if several structures have the same domain) I may write M for (M, (R i ) i∈I ). Usually the language L is finite-this means that |I| is finite. Much of the theory of homogeneous structures can be developed for languages which also have function symbols and constant symbols, replacing 'finite' by 'finitely generated'. To keep with the combinatorial emphasis of the volume, I mostly avoid this, but it would be fairly harmless at least to allow finitely many constant symbols. Often, we will not be very specific about the language. For example, if we are talking about graphs, or digraphs, or partial orders, it is assumed that the language has a single binary relation symbol. The language for 3-hypergraphs would consist of a ternary relation symbol. Definition 2.1.1. A homogeneous structure is a countable, possibly finite, relational structure such that, for every isomorphism f : U → V between finite substructures U and V of M, there is an automorphism f ′ of M extending f . In some sources, the word 'ultrahomogeneous' is used for this notion (possibly with the requirement that |M| = ℵ 0 ), due to overload for the word 'homogeneous'. Also, we have built into the definition of homogeneity the requirement that |M| is countable. This is to save words, since in this paper we only consider countable structures. Some sources do not do this. Example 2.1.2. The structure (Q, <), where < is the usual order on the rationals, is homogeneous. For if f : U → V is a finite partial isomorphism and a ∈ Q \ U, then there is b ∈ Q \ V such that f extends to a partial isomorphism taking a to b; just choose any b in the appropriate interval. Using Cantor's back-and-forth procedure, one iterates this step, alternately putting new elements into the domain and range of f , until, at the limit, an automorphism, i.e. order-preserving permutation of Q, is constructed. This method for constructing automorphisms or isomorphisms is ubiquitous in this subject. In the particular case of (Q, <), the back-and-forth method is not needed: a piecewise linear automorphism extending f could be constructed directly. A few homogeneous structures, such as (Q, <), and disjoint unions of complete graphs all of the same size, require no special construction technique. However the standard method of construction of homogeneous structures, described next, is by Fraissé's Theorem.
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