POSET LIMITS AND EXCHANGEABLE RANDOM POSETS 1 ...
POSET LIMITS AND EXCHANGEABLE RANDOM POSETS Abstract. We develop a theory of limits of finite posets in close analogy to the recent theory of graph limits. In particular, we study representations of the limits by functions of two variables on a probability space, and connections to exchangeable random infinite posets.
1. Introduction and main results A deep theory of limit objects of (finite) graphs has in recent years been created by Lov´ asz and Szegedy [19] and Borgs, Chayes, Lov´asz, S´os and Vesztergombi [6, 7], and further developed in a series of papers by these and other authors. It is shown by Diaconis and Janson [10] that the theory is closely connected with the Aldous–Hoover theory of representations of exchangeable arrays of random variables, further developed and described in detail by Kallenberg [18]; the connection is through exchangeable random infinite graphs. (See also Tao [24] and Austin [2].) The basic ideas of the graph limit theory extend to other structures too; note that the Aldous–Hoover theory as stated by Kallenberg [18] includes both multi-dimensional arrays (corresponding to hypergraphs) and some different symmetry conditions (or lack thereof). For bipartite graphs and digraphs (i.e., directed graphs), some details are given by Diaconis and Janson [10]. For hypergraphs, an extension is given by Elek and Szegedy [11]; see also [10] (where no details are given) and Tao [24] and Austin [2]. It seems possible that some future version of the theory will be formulated in a general way that includes all these cases as well as others. While waiting for such a theory, it is interesting to study further structures. Brightwell and Georgiou [8] have initiated the study of limits of finite posets (i.e., partially ordered sets). In the present paper, we develop this theory further. The theory for posets can be developed in analogy with the theory for graph limits, but it can also be obtained as a special case of the theory for digraphs. We will in this paper use both views. Our main results are parallel to results for graph limits. In this paper, all posets (and graphs) are assumed to be non-empty. They are usually finite, but we will sometimes use infinite posets as well. If (P,
1. Introduction and main results A deep theory of limit objects of (finite) graphs has in recent years been created by Lov´ asz and Szegedy [19] and Borgs, Chayes, Lov´asz, S´os and Vesztergombi [6, 7], and further developed in a series of papers by these and other authors. It is shown by Diaconis and Janson [10] that the theory is closely connected with the Aldous–Hoover theory of representations of exchangeable arrays of random variables, further developed and described in detail by Kallenberg [18]; the connection is through exchangeable random infinite graphs. (See also Tao [24] and Austin [2].) The basic ideas of the graph limit theory extend to other structures too; note that the Aldous–Hoover theory as stated by Kallenberg [18] includes both multi-dimensional arrays (corresponding to hypergraphs) and some different symmetry conditions (or lack thereof). For bipartite graphs and digraphs (i.e., directed graphs), some details are given by Diaconis and Janson [10]. For hypergraphs, an extension is given by Elek and Szegedy [11]; see also [10] (where no details are given) and Tao [24] and Austin [2]. It seems possible that some future version of the theory will be formulated in a general way that includes all these cases as well as others. While waiting for such a theory, it is interesting to study further structures. Brightwell and Georgiou [8] have initiated the study of limits of finite posets (i.e., partially ordered sets). In the present paper, we develop this theory further. The theory for posets can be developed in analogy with the theory for graph limits, but it can also be obtained as a special case of the theory for digraphs. We will in this paper use both views. Our main results are parallel to results for graph limits. In this paper, all posets (and graphs) are assumed to be non-empty. They are usually finite, but we will sometimes use infinite posets as well. If (P,
