--- title: Bergman–Shelah Preorder Overview url: https://www.emergentmind.com/topics/bergman-shelah-preorder type: topic --- # Bergman–Shelah Preorder Overview Searching arXiv for the cited Bergman–Shelah preorder papers to ground the article in current records. The term *Bergman–Shelah preorder* denotes a family of related comparison relations that measure how one structure, class, or classification problem can be obtained from another with only limited auxiliary resources. In the original group-theoretic setting it compares subgroups of an infinite symmetric group by finite augmentation; in transformation semigroups it compares subsets of \(\mathbb N^{\mathbb N}\) under semigroup generation; in linear orders it is identified with the epimorphism preorder; in model theory it reappears as a preorder on types and, for \((\mathbb N,\mid)\), on ultrafilters under divisibility; and in generalized descriptive set theory it is used for Borel reducibility between isomorphism relations of theories [1109.2706] [1701.02020] [2509.09623] [1602.00605] [2509.04200]. ## 1. General schema and original setting A common pattern across the literature is that the preorder compares two objects \(A\) and \(B\) by asking whether \(A\) can be recovered from \(B\) after adjoining only finitely many, or countably many, auxiliary pieces. The induced equivalence relation identifies objects that simulate each other in this sense. | Context | Objects compared | Defining relation | |---|---|---| | Infinite symmetric group \(S_X\) | Subgroups \(H_1,H_2\le S_X\) | \(H_1\preccurlyeq H_2\) iff \(H_1\subseteq\langle H_2\cup U\rangle\) for some finite \(U\subseteq S_X\) | | Transformation semigroup \(\mathbb N^{\mathbb N}\) | Subsets or subsemigroups \(U,V\subseteq\mathbb N^{\mathbb N}\) | \(U\preccurlyeq V\) iff \(U\subseteq\langle V\cup C\rangle\) for some countable \(C\subseteq\mathbb N^{\mathbb N}\) | | Linear orders | Orders \(A,B\) | \(A\le_{\mathrm{epi}}B\) iff there is a surjective order-preserving map \(B\to A\) | | Types over ordered structures | Types \(p,q\in S_k(A)\) | \(p\lessapprox q\) iff some realizations satisfy \(\alpha\le\beta\) | | Generalized Baire space | Complete theories \(T,T'\) at cardinal \(\kappa\) | \(T\le_{\mathrm{BS},\kappa}T'\) iff \(\cong_T^\kappa\le_B\cong_{T'}^\kappa\) | In the original closed-subgroup setting for the infinite symmetric group on a countably infinite set, Bergman and Shelah worked with the function topology and classified closed subgroups up to the equivalence induced by \(\preccurlyeq\). Canonical representatives include \(S_X\) itself, pointwise stabilizers of finite nonempty subsets, setwise stabilizers of finite partitions into infinite blocks, and stabilizers of ultrafilters [2509.04200]. A recurring source of confusion is that the same expression does **not** designate a single universal preorder with fixed semantics. Rather, it labels a transferable comparison paradigm whose concrete meaning depends on the ambient category: groups, semigroups, linear orders, ultrafilters, types, or equivalence relations. ## 2. Transformation semigroups on \(\mathbb N^{\mathbb N}\) For the full transformation semigroup \(\mathbb N^{\mathbb N}\) with composition, the preorder is defined by \[ U \preccurlyeq V \iff \exists\, C \subseteq \mathbb N^{\mathbb{N}},\ |C|\le \aleph_0,\ \text{such that } U \subseteq \langle V \cup C \rangle. \] The associated equivalence is \[ U \approx V \iff \big(U\preccurlyeq V\ \text{and}\ V\preccurlyeq U\big). \] A key technical fact is that the witness \(C\) may be taken finite, or even two-element, by a classical result of Sierpiński presented via Banach’s argument; thus the definition is unchanged if “countable” is replaced by “finite” or “two” [1109.2706]. This semigroup preorder is the natural analogue of the Bergman–Shelah preorder on subgroups of \(\operatorname{Sym}(\mathbb N)\), but the resulting structure is much more complicated. In the subgroup case, closed subgroups fall into only four \(\approx\)-classes, whereas in the semigroup case there are infinitely many distinct \(\approx\)-classes containing closed subsemigroups. A basic witness is the strict chain of closed ideals \[ \mathfrak{I}_n=\{f\in\mathbb N^{\mathbb N}:\ |f(\mathbb N)|\le n\}\qquad(n\ge 2), \] with \[ \mathfrak{I}_2 < \mathfrak{I}_3 < \cdots. \] Moreover, \(U\preccurlyeq \mathfrak{I}_n\) iff \(U\setminus \mathfrak{I}_n\) is countable [1109.2706]. Several structural theorems calibrate the richness of the quotient poset. Theorem 2.1 proves that the Continuum Hypothesis is equivalent to the existence of a subsemigroup \(S\subseteq\mathbb N^{\mathbb N}\) such that \(S\approx \mathbb N^{\mathbb N}\) and every subsemigroup \(T\le S\) is either \(\approx\mathbb N^{\mathbb N}\) or \(\approx\{1_{\mathbb N}\}\). Theorem 3.1 shows that every closed subsemigroup \(S\subseteq\mathbb N^{\mathbb N}\) of size \(2^{\aleph_0}\) dominates, in the \(\preccurlyeq\)-sense, a closed subsemigroup \(T\subseteq \mathfrak I_2\) of the same size. This reduction to binary-valued behavior does not collapse the order: there are closed semigroups incomparable with every \(\mathfrak I_n\), obtained from almost disjoint families via \[ S_{\mathcal A}=\{s_A:A\in\mathcal A\}, \] where \(s_A\) is the characteristic function of \(A\) [1109.2706]. The quotient also contains substantial antichain and chain phenomena. For every \(i\in\mathbb N\), there are \(i\) distinct closed subsemigroups contained in \(\mathfrak I_2\) that are mutually incomparable; concretely, \[ \{L_{1,i+1},\,L_{2,i},\,L_{4,i-1},\,\dots,\,L_{2i-2,2}\} \] is an antichain of length \(i\), where \[ L_{k,m}=\Big\{f\in\mathbb N^{\mathbb N}: f(i)=i\text{ for }i0), \] that is, a finite multiple of an indecomposable countable ordinal. For countable non-scattered orders \(L\), the following are equivalent: \(L\) is strongly surjective; \(\mathbb Q\le_{\mathrm{epi}}L\); and \(L\) has no scattered initial or final segment. In this sense, \(\mathbb Q\) is, up to epi-equivalence, the unique countable non-scattered strongly surjective order [1701.02020]. The preorder preserves strong order-theoretic invariants. If \(K\) and \(L\) have no maximum and \(K\le_{\mathrm{epi}}L\), then \(\operatorname{cof}(K)=\operatorname{cof}(L)\); dually, if they have no minimum and \(K\le_{\mathrm{epi}}L\), then \(\operatorname{coi}(K)=\operatorname{coi}(L)\). Strongly surjective orders are short, hence have size at most \(2^{\aleph_0}\), and satisfy stringent admissibility conditions involving minima, maxima, cofinality, and coinitiality [1701.02020]. Methodologically, the paper develops “epimorphisms by pieces” and “family mash” constructions. These yield closure of strong surjectivity under finite lexicographic products: if \(L\) and \(M\) are strongly surjective, then \(L\cdot M\) is strongly surjective, and hence \(L^n\) is strongly surjective for every \(n\in\omega\). They also support many examples, including \(\mathbb Z^\alpha\cdot m\) for countable ordinals \(\alpha\), and more generally \(\mathbb Z^K\equiv_{\mathrm{epi}}\mathbb Q\) whenever \(K\) is countable and not a well-order [1701.02020]. The descriptive-set-theoretic complexity of the strongly surjective orders is also determined. The set \(\mathrm{StS}\) of countable strongly surjective linear orders is \(\check{D}_2(\Pi^1_1)\)-complete. Its scattered part is \(\Pi^1_1\)-complete and its non-scattered part is \(\Sigma^1_1\)-complete. This places the Bergman–Shelah phenomenon for linear orders in a precise definability hierarchy rather than only a structural one [1701.02020]. At the uncountable level, the situation depends on additional axioms. Under \(BA_\kappa\), there are strongly surjective orders of size \(\kappa\); under \(\Diamond^+\), the lexicographic order of a Baumgartner tree is strongly surjective. By contrast, under CH there is no uncountable strongly surjective linear order, so existence of such an order is not provable in ZFC [1701.02020]. ## 4. Orders on types and the divisibility preorder on ultrafilters A model-theoretic extension of the Bergman–Shelah idea starts with an expansion \((M,\le,\ldots)\) of an infinite partial order. For \(p,q\in S_k(A)\), the preorder on complete types is defined by \[ p\lessapprox q \quad\Longleftrightarrow\quad \exists\alpha\models p\,\exists\beta\models q\ \big(\alpha\le \beta\big). \] The induced equivalence is \(p\approx q\) iff \(p\lessapprox q\lessapprox p\). Proposition 2.4 shows that \(p\lessapprox q\) is equivalent to a Bergman–Shelah-style closure condition: every upward closed formula in \(p\) belongs to \(q\), or equivalently every downward closed formula in \(q\) belongs to \(p\). When \(k=1\), or when \(\operatorname{dcl}(A)\) contains two comparable points, this is also equivalent to monotonicity under pushforward along \(A\)-definable increasing partial functions [2509.09623]. In definably complete linear orders, the quotient of the space of \(1\)-types is classified exactly. For \(p\in S_1(A)\), define \[ L_p=\{a\in\dcl(A): p(x)\vdash x\ge a\},\qquad R_p=\{a\in\dcl(A): p(x)\vdash x\le a\}. \] The resulting cut of \(p\) determines a point of \(\operatorname{CC}(A)\), the linear order of consistent cuts in \(\dcl(A)\). The main theorem states that \[ \pi:S_1(A)/{\approx}\ \longrightarrow\ \operatorname{CC}(A),\qquad [p]\mapsto \operatorname{cut}(p), \] is an isomorphism of linear orders. Thus, in this setting, the quotient preorder on \(1\)-types is not merely partially ordered but canonically identified with the cut structure of the definable closure [2509.09623]. Specializing to \((\mathbb N,\mid)\) in the full language yields the divisibility preorder on ultrafilters. Since definable subsets are then all subsets of \(\mathbb N\), the preorder on \(S_1(\mathbb N)\cong\beta\mathbb N\) agrees with the Bergman–Shelah extension of divisibility: \[ p\ \widetilde{\mid}\ q \quad\Longleftrightarrow\quad \exists\alpha\models p\,\exists\beta\models q\ (\alpha\mid\beta). \] Equivalently, for every upward closed subset \(U\subseteq\mathbb N\) under divisibility, \(U\in p\Rightarrow U\in q\); or dually, for every downward closed subset \(D\subseteq\mathbb N\), \(D\in q\Rightarrow D\in p\). Principal ultrafilters behave exactly as expected: \[ \mathcal U_n\,\widetilde{\mid}\,\mathcal U_m \iff n\mid m. \] This identifies the ultrafilter divisibility relation as a direct instance of the type-space construction [2509.09623]. For a prime ultrafilter \(p\in\overline{\mathbb P}\), the paper studies the suborder \(\mathcal E_p\) consisting of classes represented by ultrafilters of the form \(tp(\gamma^\delta/\mathbb N)\), where \(\gamma\models p\). Comparison reduces to comparison of exponents in the prime model \(\mathbb N(\gamma)\), and Theorem 4.4 proves \[ \mathcal E_p\cong \operatorname{CC}(\gamma). \] If \(p\) is a standard prime, then \(\mathcal E_p\cong \omega+1\). The isomorphism type of \(\mathcal E_p\) is set-theoretically sensitive: under CH, all \(\mathcal E_p\) with \(p\) nonprincipal are isomorphic, whereas in forcing extensions obtained by adding \(\kappa\ge\mathfrak c\)-many Cohen reals there are nonprincipal primes \(p,q\) with \(\mathcal E_p\not\cong\mathcal E_q\) [2509.09623]. For ultrafilters with finitely many prime divisors, the paper gives a five-way classification of \(=_{\sim}\)-classes by the behavior of exponent tuples: lower-dimensional, antichain cases (“nrac”), full-dimensional (“box”), ladder, and mixed. Singleton classes arise both in the lower-dimensional and antichain cases, while the box and ladder cases describe genuinely higher-dimensional local behavior of the divisibility preorder. The paper explicitly leaves open the general description of \(S_k(A)/{\approx}\) for \(k\ge 2\) in linearly ordered definably complete structures [2509.09623]. ## 5. Borel reducibility as a Bergman–Shelah preorder on theories In generalized descriptive set theory, the Bergman–Shelah idea is transferred from algebraic generation to comparison of classification problems. For complete first-order theories \(T\) and \(T'\) at an uncountable cardinal \(\kappa\), define \[ T\le_{\mathrm{BS},\kappa}T' \quad\Longleftrightarrow\quad \cong_T^\kappa\le_B\cong_{T'}^\kappa, \] where \(\cong_T^\kappa\) is the isomorphism relation on \(2^\kappa\)-codes of models of \(T\) of size \(\kappa\). The ambient space is the generalized Baire space \(2^\kappa\) or \(\kappa^\kappa\) with the bounded topology, and reducibility is by \(\kappa\)-Borel functions [1602.00605]. The paper develops a counterpart of Shelah’s Main Gap for this preorder. The central consistency statement is that, for all complete theories \(T,T'\), if \(T\) is classifiable and \(T'\) is non-classifiable, then \[ \cong_T^\kappa\le_c\cong_{T'}^\kappa \qquad\text{and}\qquad \cong_{T'}^\kappa\nleq_B\cong_T^\kappa. \] Equivalently, consistently, \[ T<_{\mathrm{BS},\kappa}T' \] whenever \(T\) is classifiable and \(T'\) is not. The hypotheses used include cardinals satisfying \(\kappa^{<\kappa}=\kappa\), \(\kappa=\lambda^+=2^\lambda\), \(\lambda^{<\lambda}=\lambda\), and suitable diamond principles; the paper also shows that these can be arranged in forcing extensions by a \(\kappa\)-closed, \(\kappa\)-cc forcing [1602.00605]. The benchmark equivalence relations between the two sides of the gap are the non-stationary ideal relations \[ E_X=\Big\{(\eta,\xi)\in (2^\kappa)^2:\ (\eta^{-1}[1]\ \Delta\ \xi^{-1}[1])\cap X\text{ is non-stationary}\Big\}, \] and especially \(E_{\mu\text{-club}}=E_{S_\mu}\) for regular \(\mu<\kappa\). For classifiable \(T\), EF-game arguments and diamond principles give continuous reductions \(\cong_T^\kappa\le_c E_X\), while earlier results imply \(E_{\mu\text{-club}}\le_B\cong_T^\kappa\). For non-classifiable theories \(T'\), several model-theoretic dividing lines yield reductions in the opposite direction: unstable and superstable with OTOP, and under additional arithmetic also superstable with DOP, satisfy \(E_{\lambda\text{-club}}\le_c \cong_{T'}^\kappa\); stable unsuperstable theories satisfy \(E_{\omega\text{-club}}\le_c\cong_{T'}^\kappa\) under suitable hypotheses [1602.00605]. The interval between the two sides of the gap is itself highly structured. Under the same forcing framework, the paper embeds \((\mathcal P(\kappa),\subseteq)\) into the Borel degrees between \(\cong_T^\kappa\) and \(\cong_{T'}^\kappa\) by stationary sets \(K(\mu,A)\), obtaining \[ E_{K(\mu,A)}\le_B E_{K(\mu,B)}\quad\text{iff}\quad A\subseteq B. \] Consequently, the interval of Borel degrees has cardinality \(2^\kappa\). This suggests that the Bergman–Shelah preorder on theories is not merely a binary classifiable/non-classifiable dichotomy, but a large degree structure with canonical intermediate benchmarks furnished by stationary-set combinatorics [1602.00605]. ## 6. Symmetric inverse monoids and current extensions For an infinite set \(X\), the symmetric inverse monoid \[ I_X=\{\,f:A\to B\mid A,B\subseteq X,\ \text{$f$ is a bijection}\,\} \] extends the symmetric group by allowing partial bijections. Its idempotents are the partial identities \[ E(I_X)=\{\,e_A\mid A\subseteq X\,\}. \] A basic identity is that any \(f\in I_X\) can be written in the form \[ f=e_B\,g\,e_A, \] where \(g\in S_X\) extends \(f\). Hence \[ I_X=\langle S_X\cup E(I_X)\rangle. \] The Bergman–Shelah preorder on subsemigroups of \(I_X\) is defined by \[ S_1\preccurlyeq S_2 \quad\text{iff}\quad \exists\,U\subseteq I_X\text{ finite such that }S_1\subseteq \langle S_2\cup U\rangle, \] with the corresponding equivalence \(S_1\approx S_2\) [2509.04200]. This extension changes the algebraic landscape because idempotents become decisive. Subsemigroups containing \(S_X\) and a nontrivial idempotent immediately generate large families of partial maps, and subsemigroups containing \(S_X\) together with all idempotents lie in the top \(\approx\)-class, namely \(\approx I_X\). The thesis studies maximal subsemigroups of \(I_X\) containing, respectively, \(S_X\), the pointwise stabilizer of a finite nonempty subset, the stabilizer of an ultrafilter, and the stabilizer of a finite partition [2509.04200]. A second theme is topological. In semigroup topologies on \(I_X\) introduced by Elliot et al. in 2023, the closed subsemigroups containing all idempotents are exactly the semigroups of partial endomorphisms or partial automorphisms of relational structures on \(X\). Concretely, for a relational structure \(R\), \[ \operatorname{End}_p(R)=\{f\in I_X: f\text{ is a partial endomorphism of }R\}, \qquad \operatorname{Aut}_p(R)=\{f\in I_X: f\text{ is a partial automorphism of }R\}. \] For a countable set \(X\) and a structure \(R\) with finitely many relations, there exists a finite subset \(U\subseteq I_X\) such that \[ \langle \operatorname{Aut}_p(R)\cup U\rangle = I_X. \] In preorder language, \(\operatorname{Aut}_p(R)\preccurlyeq I_X\) [2509.04200]. The thesis formulates a conjecture directly analogous to the classical Bergman–Shelah theorem for closed subgroups of \(S_X\): among closed inverse subsemigroups \(S\le I_X\) containing \(E(I_X)\), there should be only finitely many \(\approx\)-equivalence classes under the preorder. Proposed canonical representatives include \(I_X\) itself, inverse subsemigroups generated by idempotents together with pointwise stabilizers of finite nonempty sets, inverse subsemigroups stabilizing finite partitions, inverse subsemigroups stabilizing ultrafilters, and semigroups of partial automorphisms of finite-signature relational structures [2509.04200]. Taken together, these extensions indicate that the Bergman–Shelah preorder is best viewed as a robust comparative template rather than a single invariant. In groups it yields a finite coarse classification of closed subgroups; in transformation semigroups it produces strict chains, finite antichains, and CH-sensitive phenomena; in linear orders it becomes the epimorphism preorder and isolates strong surjectivity; in model theory it identifies quotients of \(1\)-type spaces with cut orders and recovers divisibility on ultrafilters; in generalized Baire space it organizes classification problems by Borel complexity; and in inverse monoids it interacts with idempotents, topology, and partial automorphism semigroups in ways that remain only partially classified. Source: https://www.emergentmind.com/topics/bergman-shelah-preorder