Symmetric subvarieties of infinite affine space

Let $\mathfrak{X}=\operatorname{Spec}(R)=\mathbf{A}^{\infty}$ be infinite dimensional affine space over the complex numbers (we allow arbitrary noetherian coefficient rings in the body of the paper). The goal of this paper is to classify the $\mathfrak{S}$-stable Zariski closed subsets of $\mathfrak{X}$ and understand their structure.

We say that a $\mathbf{C}$-point $x=(x_{i})_{i\geq 1}$ of $\mathfrak{X}$ is finitary if the coordinates of $x$ assume only finitely many values, that is, $\{x_{i}\mid i\geq 1\}$ is a finite subset of $\mathbf{C}$. This notion is closely related to discriminants. Indeed, let $\Delta_{r}=\prod_{1\leq i<j\leq r}(\xi_{i}-\xi_{j})$ be the $r$th discriminant. Then the point $x$ is finitary if and only if there is some $r$ such that $\Delta_{r}(\sigma x)=0$ for all $\sigma\in\mathfrak{S}$. It is not difficult to show that any non-zero $\mathfrak{S}$-stable ideal of $R$ contains some discriminant (Proposition 2.6). Thus any proper $\mathfrak{S}$-stable closed subset of $\mathfrak{X}$ is contained in the finitary locus $\mathfrak{X}_{\mathrm{fin}}$. For this reason, finitary points play a prominent role in this paper.

Suppose that $x$ is a finitary point. Let $a_{1},\ldots,a_{r}$ be the distinct values its coordinates assume, and let $\lambda_{i}$ be the number of times $a_{i}$ appears. Relabeling if necessary, we assume that $\lambda_{1}\geq\cdots\geq\lambda_{r}$; note that $\lambda_{1}$ is necessarily $\infty$. We call $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ the type of $x$, and generally refer to a tuple of this sort as an $\infty$-parition. For example, the point

has type $(\infty,\infty,3,2)$, since 6 and 7 occur infinitely, 3 occurs thrice, and 5 occurs twice. Let $\mathfrak{X}_{[\lambda]}$ be the set of all finitary points of type $\lambda$. We have $\mathfrak{X}_{\mathrm{fin}}=\bigsqcup\mathfrak{X}_{[\lambda]}$, where the union is over all $\infty$-partitions $\lambda$, and so we can regard the $\mathfrak{X}_{[\lambda]}$ as defining a stratification of $\mathfrak{X}_{\mathrm{fin}}$.

Before stating our theorem, we need to introduce one more concept. We say that a topological space $X$ with an action of a group $G$ is $G$-irreducible if it cannot be expressed as a union of two $G$-stable proper closed subsets. It is an easy consequence of Cohen’s theorem that any $\mathfrak{S}$-stable closed subvariety of $\mathfrak{X}$ is a finite union of $\mathfrak{S}$-irreducible closed subvarieties. It therefore suffices to study the $\mathfrak{S}$-irreducible subvarieties of $\mathfrak{X}$. We note that if $G$ is a finite group acting on a variety $X$ then any $G$-irreducible subvariety of $X$ is simply the union of the $G$-translates of an irreducible subvariety.

We can now state our classification theorem:

It is not difficult to describe the bijection in Theorem 1.1. To do this, it will be useful to introduce a refinement of the type stratification of $\mathfrak{X}_{\mathrm{fin}}$. Let $\mathcal{U}=\{\mathcal{U}_{1},\ldots,\mathcal{U}_{r}\}$ be a partition of the index set $[\infty]=\{1,2,\ldots\}$ into finitely many disjoint and non-empty pieces. Define $\mathfrak{X}_{\mathcal{U}}$ (resp. $\mathfrak{X}_{[\mathcal{U}]}$) be the subset of $\mathfrak{X}$ consisting of points $x$ such that $x_{i}=x_{j}$ if $i$ and $j$ belong to the same part of $\mathcal{U}$ (resp. $x_{i}=x_{j}$ if and only if $i$ and $j$ belong to the same part). The space $\mathfrak{X}_{\mathcal{U}}$ is isomorphic to $\mathbf{A}^{r}$, and under this isomorphism $\mathfrak{X}_{[\mathcal{U}]}$ corresponds to $\mathbf{A}^{[r]}$. Let $\lambda_{i}=\#\mathcal{U}_{i}$, relabeling if necessary so that $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ is an $\infty$-partition; we call this the $\infty$-partition associated to $\mathcal{U}$, and say $\mathcal{U}$ is of type $\lambda$. Then any point of $\mathfrak{X}_{[\mathcal{U}]}$ has type $\lambda$, and $\mathfrak{X}_{[\lambda]}$ is the union of the $\mathfrak{S}$-translates of $\mathfrak{X}_{[\mathcal{U}]}$.

We now describe the bijection in Theorem 1.1. Suppose that $\mathfrak{Z}$ is an $\mathfrak{S}$-irreducible closed subvariety of $\mathfrak{X}$. Let $\Lambda_{\mathfrak{Z}}$ be the set of all $\infty$-partitions $\lambda$ such that $\mathfrak{Z}$ has a point of type $\lambda$. We show that $\Lambda_{\mathfrak{Z}}$ contains a unique maximal element with respect to a partial order $\preceq$ on $\infty$-partitions. Let $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ be this maximal element, and pick a partition $\mathcal{U}$ of $[\infty]$ of type $\lambda$. Let $Z=\mathfrak{Z}\cap\mathfrak{X}_{[\mathcal{U}]}$, where we identify $\mathfrak{X}_{[\mathcal{U}]}$ with $\mathbf{A}^{[r]}$. Then $\mathfrak{Z}$ corresponds to $(\lambda,Z)$. In the reverse direction, suppose that $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ and $Z$ are given. Again, pick $\mathcal{U}$ of type $\lambda$. Then the corresponding $\mathfrak{S}$-irreducible closed subvariety of $\mathfrak{X}$ is the Zariski closure of the $\mathfrak{S}$-orbit of $Z$, where we regard $Z$ as a subset of $\mathfrak{X}$ via the identification $\mathbf{A}^{[r]}=\mathfrak{X}_{[\mathcal{U}]}$.

While Theorem 1.1 does give a complete classification of $\mathfrak{S}$-subvarieties of $\mathfrak{X}$, it is not powerful enough to answer some finer questions, such as:

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  How does one describe the entire point set of $\mathfrak{Z}$ from $(\lambda,Z)$?

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  How does one determine the equations of $\mathfrak{Z}$ from those of $Z$?

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  How does one determine a containment $\mathfrak{Z}\subset\mathfrak{Z}^{\prime}$ in terms of the data $(\lambda,Z)$ and $(\lambda^{\prime},Z^{\prime})$?

In fact, Theorem 1.1 is a corollory of a stronger result, that we now describe, which allows us to address these questions (and more like them).

For an $\infty$-partition $\lambda=(\lambda_{1},\ldots,\lambda_{r})$, put $\mathcal{X}_{\lambda}=\mathbf{A}^{r}$. We regard the system $\mathcal{X}=\{\mathcal{X}_{\lambda}\}_{\lambda}$ as a single object. Given an $\mathfrak{S}$-stable closed subset $\mathfrak{Z}$ of $\mathfrak{X}$, we define a subsystem $\mathcal{Z}=\Phi(\mathfrak{Z})$ of $\mathcal{X}$ as follows. For an $\infty$-partition $\lambda$, let $\mathcal{U}$ be a partition of $[\infty]$ of type $\lambda$, and put $\mathcal{Z}_{\lambda}=\mathfrak{Z}\cap\mathfrak{X}_{\mathcal{U}}$, where, as usual, we idenify $\mathfrak{X}_{\mathcal{U}}$ with $\mathbf{A}^{r}=\mathcal{X}_{\lambda}$. This is well-defined (i.e., independent of the choice of $\mathcal{U}$) since $\mathfrak{Z}$ is $\mathfrak{S}$-stable. Note that $\mathcal{Z}_{\lambda}$ is Zariski closed in $\mathcal{X}_{\lambda}$, and so we can regard $\mathcal{Z}$ as a system of finite dimensional varieties.

Since $\Phi(\mathfrak{Z})$ records all the finitary points of $\mathfrak{Z}$, it follows that it completely determines $\mathfrak{Z}$. In other words, the construction $\Phi$ is injective. Much of the work in this paper goes into determining the image of $\Phi$, or, in other words, determining the precise conditions that the system $\mathcal{Z}$ of varieties must satisfy.

We now give an example of one of the simplest conditions that $\mathcal{Z}$ must satisfy. Suppose that $x=(b,a,a,\ldots)$ is a $\mathbf{C}$-point of $\mathfrak{Z}$. Let $x_{n}=(a,\ldots,a,b,a,a,\ldots)$, where all coordinates except the $n$th are equal to $a$. Then $x_{n}$ belongs to the $\mathfrak{S}$-orbit of $x$, and therefore belongs to $\mathfrak{Z}$. The sequence $\{x_{n}\}_{n\geq 1}$ converges coordinate-wise to $y=(a,a,\ldots)$, and so $y$ also belongs to $\mathfrak{Z}$. (This is a simple compactness argument, see §2.3.) We can rephrase the above observation in terms of $\mathcal{Z}$ as follows. The point $x$ shows that $\mathcal{Z}_{(\infty,1)}\subset\mathbf{A}^{2}$ contains the point $(a,b)$, while the point $y$ shows that $\mathcal{Z}_{(\infty)}\subset\mathbf{A}^{1}$ contains the point $a$. We thus see that if $p_{1}\colon\mathbf{A}^{2}\to\mathbf{A}^{1}$ is the projection map onto the first coordinate then $p_{1}(\mathcal{Z}_{(\infty,1)})\subset\mathcal{Z}_{(\infty)}$. This is a non-trivial condition on $\mathcal{Z}$.

By generalizing the above argument, we are led to a number of natural conditions that $\mathcal{Z}$ must satisfy. We define a “C3 subvariety” of $\mathcal{X}$ to be a subsystem satisfying these conditions. The following is the main theorem of this paper:

The inverse bijection $\Psi$ is described excplitily. We use this theorem to answer the questions posed above. Additionally, we apply the theorem in §9, where we prove a result on the support of $\mathfrak{S}$-equivariant $R$-modules. This result will play an important role in the follow-up paper [NS1].

We give some simple examples to illustrate Theorem 1.1.

In this paper, we classify the $\mathfrak{S}$-stable subvarieties of $\mathbf{A}^{\infty}$, or, equivalently, the $\mathfrak{S}$-stable radical ideals of $R$. We are currently pursuing two natural generalizations of this result. First, in forthcoming papers [NS1, NS2], we determine (in a certain sense) the structure of arbitrary $\mathfrak{S}$-stable ideals of $R$, and establish results about equivariant modules as well. And second, in ongoing joint work with Vignesh Jagathese, we are classifying the $\mathfrak{S}$-stable subvarieties of $X^{\infty}$ for an arbitrary variety $X$ (the present paper treating the case $X=\mathbf{A}^{1}$).

We list the most important notation used throughout the paper:

${[n]}$ :

the set $\{1,\ldots,n\}$ (we allow $n=\infty$)

$\mathfrak{S}$ :

the (small) symmetric group on $[\infty]$

$A$ :

the coefficient ring (often noetherian)

$W$ :

the spectrum of $A$

$R$ :

the polynomial ring over $A$ in variables $\{\xi_{i}\}_{i\geq 1}$

$\mathfrak{X}$ :

the spectrum of $R$

$\mathfrak{Z}$ :

a subset of $\mathfrak{X}$

$\mathcal{X}$ :

the system $\{\mathcal{X}_{\lambda}\}_{\lambda}$ indexed by $\infty$-compositions $\lambda$

$\mathcal{Z}$ :

a subset (really subsystem) of $\mathcal{X}$

Let $A$ be a commutative ring and let $W=\operatorname{Spec}(A)$ be its spectrum. Let $R=R_{A}=A[\xi_{i}]_{i\geq 1}$ be the infinite variable polynomial ring over $A$ in the variables $\xi_{i}$. Let $\mathfrak{X}=\mathfrak{X}_{W}=\operatorname{Spec}(R)$. We regard $\mathfrak{X}$ as a topological space (under the Zariski topology), and include its scheme-theoretic (non-closed) points.

We typically work with points in $\mathfrak{X}$ by using $K$-points, for variable fields $K$. Every point of $\mathfrak{X}$ comes from a $K$-point of $\mathfrak{X}$ for some field $K$, and a $K$-point $x$ and a $K^{\prime}$-point $x^{\prime}$ define the same point of $\mathfrak{X}$ if and only if there is a field $K^{\prime\prime}$ and embeddings $K\to K^{\prime\prime}$ and $K^{\prime}\to K^{\prime\prime}$ such that $x$ and $x^{\prime}$ define the same $K^{\prime\prime}$-points of $\mathfrak{X}$. We can therefore define a condition on points of $\mathfrak{X}$ by defining a condition on $K$-points that is invariant under field extension. For a subset $\mathfrak{Z}$ of $\mathfrak{X}$, we let $\mathfrak{Z}(K)$ be the subset of $\mathfrak{X}(K)$ consisting of $K$-points whose image lies in $\mathfrak{Z}$. We note that $X(K)=W(K)\times K^{\infty}$. For readers not familiar with scheme theory, one can assume that $A$ is an algebraically closed field and work only with closed points without losing much.

Let $\mathfrak{S}=\bigcup_{n\geq 1}\mathfrak{S}_{n}$ be the “small” infinite symmetric group and let $\mathfrak{S}^{\rm big}=\operatorname{Aut}([\infty])$ be the “big” infinite symmetric group; of course, $\mathfrak{S}$ is a subgroup of $\mathfrak{S}^{\rm big}$. The group $\mathfrak{S}^{\rm big}$ (and therefore $\mathfrak{S}$ as well) acts on $R$ by permuting the variables, and thus on $\mathfrak{X}$ as well. We prefer to work with $\mathfrak{S}$, since it is more algebraic in nature; however, the difference between these two groups is largely irrelevant for algebraic questions, as we see below (e.g., Proposition 2.4).

By an $\mathfrak{S}$-ideal of $R$, we mean an ideal of $R$ that is stable under the group $\mathfrak{S}$. Given a subset $S$ of $R$, we let $\langle\!\langle S\rangle\!\rangle$ be the ideal generated by the $\mathfrak{S}$-orbit of $S$; this is the minimal $\mathfrak{S}$-ideal containing $S$. By an $\mathfrak{S}$-subset of $\mathfrak{X}$, we mean a subset of $\mathfrak{X}$ that is stable under $\mathfrak{S}$. The following result is due to Cohen [Co, Co2]:

The space $\mathfrak{X}$ carries the usual Zariski topology. However, it also has a second topology, what we call the $\Pi$-topology, that will be useful in this paper. The definition is as follows: a subset $\mathfrak{Z}$ of $\mathfrak{X}$ is $\Pi$-closed if $\mathfrak{Z}(K)$ is closed in $\mathfrak{X}(K)=W(K)\times K^{\infty}$ for all fields $K$; here $W(K)$ and $K$ are given the discrete topology and $\mathfrak{X}(K)$ is given the product topology. Concretely, a sequence (or net) $\{x_{i}\}$ in $\mathfrak{X}(K)$ converges to a point $x$ if each coordinate of $x_{i}$ (including the $W$ coordinate) is equal to the corresponding coordinate of $x$ for $i\gg 0$.

Given a commutative ring $B$ and elements $b_{1},\ldots,b_{n}\in B$, we let $\Delta(b_{1},\ldots,b_{n})=\prod_{1\leq i<j\leq n}(b_{j}-b_{i})$ be their discriminant. We put $\Delta_{n}=\Delta(\xi_{1},\ldots,\xi_{n})$, which we regard as an element of $R$.

Let $x=(w,(x_{i})_{i\in[\infty]})$ be a $K$-point of $\mathfrak{X}$. We define the width of $x$ to be the cardinality of the subset $\{x_{i}\mid i\in[\infty]\}$ of $K$, and we say that $x$ is finitary if it has finite width. It is clear that width is invariant under field extension, and can thus be defined for points of $\mathfrak{X}$. We let $\mathfrak{X}_{\leq n}$ be the set of points of width $\leq n$, and $\mathfrak{X}_{\mathrm{fin}}$ the set of points of finitary points. We say that a subset of $\mathfrak{X}$ is bounded if it is contained in $\mathfrak{X}_{\leq n}$ for some $n$. Let $\mathfrak{d}_{n}=\langle\!\langle\Delta_{n}\rangle\!\rangle$ be the ideal of $R$ generated by the $\mathfrak{S}$-orbit of $\Delta_{n}$.

We now introduce an invariant that is finer than width. Recall that an $\infty$-partition is a sequence $\lambda=(\lambda_{1},\lambda_{2},\ldots)$ such that $\lambda_{1}=\infty$, $\lambda_{i}\geq\lambda_{i+1}$, and $\lambda_{i}=0$ for $i\gg 0$. Let $x=(w,(x_{i})_{i\in[\infty]})$ be a finitary $K$-point of $\mathfrak{X}$ of width $n$. Let $a_{1},\ldots,a_{n}$ be the $n$ values that the $x_{i}$ assume, and let $\mathcal{U}_{k}\subset[\infty]$ be the set of indices $i$ such that $x_{i}=a_{k}$. Permuting the $a$’s if necessary, we assume that $\#\mathcal{U}_{k}\geq\#\mathcal{U}_{k+1}$. Then putting $\lambda_{k}=\#\mathcal{U}_{k}$ for $1\leq k\leq n$, and $\lambda_{k}=0$ for $k>n$, we see that $\lambda$ is an $\infty$-partition. We define the type of $x$ to be $\lambda$. This quantity is clear invariant under field extension, and can thus be associated to fintary points of $\mathfrak{X}$. It is also clearly invariant under the action of the big symmetric group.

For a finite set $S$, we let $\mathbf{A}^{S}_{W}$ be the affine space over the base scheme $W$ with coordinates indexed by the set $S$. We let $\mathbf{A}^{[S]}_{W}$ be the open subscheme of $\mathbf{A}^{S}_{W}$ where the coordinates are all distinct. Given a function $f\colon S\to T$ of finite sets, we let $\alpha_{f}\colon\mathbf{A}_{W}^{T}\to\mathbf{A}_{W}^{S}$ be the map that takes $(x_{j})_{j\in T}$ to $(x_{f(i)})_{i\in S}$. We note that if $f$ is an injection then $\alpha_{f}$ is a projection, while if $f$ is a surjection then $\alpha_{f}$ is a multi-diagonal map (and thus a closed immersion).

A generalized partition is a tuple $\lambda=(\lambda_{1},\lambda_{2},\ldots)$ where $\lambda_{i}\in\mathbf{N}\cup\{\infty\}$ such that $\lambda_{i}\geq\lambda_{i+1}$ for all $i$ and $\lambda_{i}=0$ for $i\gg 0$. We define length of $\lambda$, denoted $\ell(\lambda)$, to be the largest $i$ such that $\lambda_{i}>0$. A generalized partition $\lambda$ is an $\infty$-partition if $\lambda_{1}=\infty$, otherwise $\lambda$ is called a finite partition. We denote the set of generalized partition by $\widetilde{\Lambda}$, and the set of partitions of infinity by $\Lambda$.

A generalized composition is a pair $(S,\lambda)$ consisting of a finite set $S$ and a function $\lambda\colon S\to\mathbf{Z}_{>0}\cup\{\infty\}$. We typically simply write $\lambda$ for the generalized composition, and denote the index set $S$ by $\langle\lambda\rangle$. Given a generalized composition $\lambda$, we let $|\lambda|=\sum_{i\in\langle\lambda\rangle}\lambda_{i}$ and $\ell(\lambda)=\#\langle\lambda\rangle$. We say that $\lambda$ is a composition of $\infty$ or an $\infty$-composition if $|\lambda|=\infty$, or equivalently, if $\lambda_{i}=\infty$ for some $i\in\langle\lambda\rangle$. We regard any generalized partition $\lambda$ as a generalized composition on the index set $[\ell(\lambda)]$.

Let $\lambda$ and $\mu$ be generalized compositions. A map $f\colon\lambda\to\mu$ is a function $f\colon\langle\lambda\rangle\to\langle\mu\rangle$ such that $f_{*}(\lambda)\leq\mu$, i.e., for all $j\in\langle\mu\rangle$ we have $\sum_{i\in f^{-1}(j)}\lambda_{i}\leq\mu_{j}$. We say that $f$ is a principal surjection if $f_{*}(\lambda)=\mu$, which implies that $f$ is a surjection. We write $f\colon\lambda\rightarrowtail\mu$ to indicate that $f$ is a principal surjection. Every map $f$ can be factored as $g\circ h$ where $h$ is a principal surjection and $g$ is an injection. The composition of two maps is again a map, and so we have a category of generalized compositions.

Let $\lambda$ and $\mu$ be generalized partitions. We define $\lambda\leq\mu$ if $\lambda$ can be obtained from $\mu$ by decreasing (or removing) parts, and we define $\lambda\preceq\mu$ if $\lambda$ can be obtained from $\mu$ by combining and decreasing (or removing) parts. We note here that if $f\colon\lambda\to\mu$ is an injection then $\lambda\leq\mu$ and if $f$ is a principal surjection then $\mu\preceq\lambda$. A poset is called a well-quasi-order (wqo) if it satisfies the descending chain condition and has no infinite antichains. We refer to [SS, §2] for general facts about wqo’s, though note there the term “noetherian” is used in place of “well-quasi-order.”

For an $\infty$-composition $\lambda$, we put $\mathcal{X}_{W,\lambda}=\mathbf{A}^{\langle\lambda\rangle}_{W}$ and $\mathcal{X}_{W,[\lambda]}=\mathbf{A}^{[\langle\lambda\rangle]}_{W}$. We regard $\mathcal{X}_{W}=\{\mathcal{X}_{W,\lambda}\}_{\lambda}$ as a single object. We omit the $W$ from the notation when possible.

By a “subset” of $\mathcal{X}$ we mean a rule $\mathcal{Z}$ assigning to every $\infty$-composition $\lambda$ a subset $\mathcal{Z}_{\lambda}$ of $\mathcal{X}_{\lambda}$; here we simply regard $\mathcal{X}_{\lambda}$ as set of points in its underlying topological space (including its non-closed points). We consider several conditions on such a subset $\mathcal{Z}$:

• •

  We say that $\mathcal{Z}$ is a C1 subset if for every principal surjection $f\colon\lambda\rightarrowtail\mu$ of $\infty$-compositions we have $\alpha_{f}^{-1}(\mathcal{Z}_{\lambda})=\mathcal{Z}_{\mu}$.

• •

  We say that a point $x\in\mathcal{X}_{\lambda}$ is $N$-approximable by $\mathcal{Z}$, for a positive integer $N$, if there exists an $\infty$-composition $\mu$ and an injective function $f\colon\langle\lambda\rangle\to\langle\mu\rangle$ such that $\mu_{f(i)}\geq\min(\lambda_{i},N)$ for all $i\in\langle\lambda\rangle$ and $x$ belongs to the image of $\mathcal{Z}_{\mu}$ under the projection map $\alpha_{f}\colon\mathcal{X}_{\mu}\to\mathcal{X}_{\lambda}$. (Note: $f$ need not define a map of $\infty$-compositions $\lambda\to\mu$.) We say that $x$ is approximable by $\mathcal{Z}$ if it is $N$-approximable for all $N$. We say that $\mathcal{Z}$ is a C2 subset if it is C1 and contains all of its approximable points (i.e., whenever $x\in\mathcal{X}_{\lambda}$ is approximable by $\mathcal{Z}$ we have $x\in\mathcal{Z}_{\lambda}$).

• •

  We say that $\mathcal{Z}$ is a C3 subvariety if it is C2 and $\mathcal{Z}_{\lambda}$ is Zariski closed in $\mathcal{X}_{\lambda}$ for all $\lambda$.

We note that if $\mathcal{Z}$ is a C1 subset then for any isomorphism $f\colon\lambda\to\mu$ of $\infty$-compositions we have $\alpha_{f}(\mathcal{Z}_{\mu})=\mathcal{Z}_{\lambda}$, since an isomorphism is a principal surjection. Thus $\mathcal{Z}$ is determined by $\mathcal{Z}_{\lambda}$ with $\lambda$ an $\infty$-partition. Ultimately, we are most interested in C3 subvarieties, but C1 and C2 subsets will be helpful intermediate objects.

We now establish some simple properties of C1, C2, and C3 subsets of $\mathcal{X}$. For a subset $\mathcal{Z}$ of $\mathcal{X}$, we let $\mathcal{Z}_{[\lambda]}=\mathcal{Z}_{\lambda}\cap\mathcal{X}_{[\lambda]}$.

Let $\lambda$ and $\mu$ be generalized compositions. A correspondence $f\colon\lambda\dashrightarrow\mu$ is a pair $(f_{1},f_{2})$ where $f_{1}\colon\rho\rightarrowtail\lambda$ is a principal surjection and $f_{2}\colon\rho\to\mu$ is an arbitrary map; here $\rho$ denotes another generalized composition. Suppose that $\lambda$ and $\mu$ are $\infty$-compositions; note that $\rho$ is then as well. For $S\subset\mathcal{X}_{\mu}$, we let $\alpha_{f}(S)=\alpha_{f_{1}}^{-1}(\alpha_{f_{2}}(S))$, a subset of $\mathcal{X}_{\lambda}$. Note that $x\in\alpha_{f}(S)$ if and only if $\alpha_{f_{1}}(x)\in\alpha_{f_{2}}(S)$. We say that a subset $\mathcal{Z}$ of $\mathcal{X}$ is closed under correspondences if $\alpha_{f}(\mathcal{Z}_{\mu})\subset\mathcal{Z}_{\lambda}$ for all correspondences $f\colon\lambda\dashrightarrow\mu$.