SWIR: Independence in Fraïssé Structures

• SWIR is a ternary relation defined on finitely generated substructures of countable homogeneous and Fraïssé structures, characterized by invariance, existence, stationarity, and monotonicity without symmetry.
• It establishes a canonical amalgamation framework where independence is linked directly to the geometric and type-theoretic properties of the structure, serving as a controlled amalgamation mechanism.
• SWIR underpins results on automorphism group simplicity and connects with Katětov functors and extensive embeddings, highlighting its role in bridging asymmetry and classical amalgamation techniques.

A stationary weak independence relation (SWIR) is a ternary relation $B \downarrow_A C$ on finite, or finitely generated, substructures of a countable homogeneous or Fraïssé structure that satisfies automorphism invariance, existence, stationarity, and monotonicity/transitivity, but does not require symmetry. In the homogeneous relational setting, SWIR was isolated to extend the Tent–Ziegler stationary-independence framework to asymmetric structures (Li, 2019). In later work on Fraïssé structures, the same notion is formulated on finitely generated substructures, split into local and global versions, and identified with a canonical amalgamation mechanism on the age of the structure (Kwiatkowska et al., 8 Aug 2025).

1. Ambient setting and formal definition

In the SWIR literature, the ambient objects are countable homogeneous structures. In the relational presentation, a countable $\mathcal{L}$-structure $\mathcal{M}$ is homogeneous if every partial isomorphism between finite substructures extends to an automorphism; by Fraïssé’s theorem, such structures arise as Fraïssé limits of amalgamation classes of finite structures (Li, 2019). In the later Fraïssé-oriented formulation, a countably infinite ultrahomogeneous structure is called a $i$ structure, and $\Age(M)$ denotes the class of finitely generated structures embeddable in $M$, while $\Ao(M)$ denotes the $\omega$-age, consisting of all embeddable structures (Kwiatkowska et al., 8 Aug 2025).

The relation is written $B \downarrow_A C$. When $A,B,C \subseteq M$, the notation $\mathcal{L}$0 denotes the substructure generated by $\mathcal{L}$1, and $\mathcal{L}$2 means that some automorphism fixing $\mathcal{L}$3 pointwise sends $\mathcal{L}$4 to $\mathcal{L}$5; equivalently, $\mathcal{L}$6 and $\mathcal{L}$7 have the same quantifier-free type over $\mathcal{L}$8 (Kwiatkowska et al., 8 Aug 2025).

Li’s definition, adopted in the Fraïssé setting, takes $\mathcal{L}$9 to be a ternary relation on finitely generated substructures and requires Invariance, Existence, Stationarity, and Monotonicity. A local SWIR is defined only on non-empty finitely generated substructures, whereas a global SWIR is also defined over the empty set; a global SWIR restricts to a local one (Kwiatkowska et al., 8 Aug 2025). In the earlier automorphism-group formulation, a SWIR on a homogeneous structure is given on finite substructures and is presented with Invariance, Monotonicity, Transitivity, Existence, and Stationarity, each in explicitly two-sided form (Li, 2019).

The omission of symmetry is the defining distinction from a stationary independence relation. If one adds symmetry,

$\mathcal{M}$0

then the relation is a stationary independence relation (SIR) in the sense of Tent–Ziegler (Kwiatkowska et al., 8 Aug 2025).

2. Axioms and equivalent presentations

The later Fraïssé presentation packages the definition into four axiom families, while the automorphism-group presentation isolates transitivity as a separate axiom. The core content is the same.

Axiom	Content
Invariance (Inv)	$\mathcal{M}$1 for $\mathcal{M}$2
Existence (Ex)	Realize either side over $\mathcal{M}$3 independently from the other
Stationarity (Sta)	Independent realizations of the same type over $\mathcal{M}$4 have the same type over $\mathcal{M}$5 or $\mathcal{M}$6
Monotonicity (Mon)	Independence persists under shrinking one side and enlarging the base
Transitivity (Tr)	Independence composes across intermediate bases

Several formal consequences are standard in this framework. First, left-Existence together with Invariance implies right-Existence, so the two-sided Existence axiom can be compressed (Kwiatkowska et al., 8 Aug 2025). Second, every SWIR satisfies base triviality: $\mathcal{M}$7 Third, the later Fraïssé treatment proves Transitivity from the SWIR axioms, and conversely shows that if a ternary relation satisfies Invariance, Existence, Stationarity, and Transitivity, then it also satisfies Monotonicity; this makes the Monotonicity-based and Transitivity-based presentations interchangeable (Kwiatkowska et al., 8 Aug 2025).

A further consequence is base enlargement: from $\mathcal{M}$8 one gets $\mathcal{M}$9 and $i$0 (Kwiatkowska et al., 8 Aug 2025). Under strong amalgamation, SWIR has a geometric consequence: if $i$1, then the sets $i$2 and $i$3 are disjoint (Kwiatkowska et al., 8 Aug 2025).

These consequences are important because later constructions use SWIR as a controlled amalgamation principle rather than merely as a type-theoretic relation.

3. Canonical amalgamation, SIR, and CIR

The modern structural theory of SWIR is built around its relation to canonical amalgamation. A standard amalgamation operator (SAO) on $i$4 assigns to each pair of embeddings $i$5, $i$6 an amalgam $i$7 satisfying Minimality, Invariance, and Transitivity (Kwiatkowska et al., 8 Aug 2025). Minimality says that the amalgam is generated by the images of $i$8 and $i$9; Invariance says that isomorphic amalgamation diagrams yield isomorphic amalgams; Transitivity says that the operator is compatible with further amalgamation on either side.

From an SAO, one defines independence by declaring $\Age(M)$0 precisely when the generated substructure $\Age(M)$1 is isomorphic to the canonical amalgam $\Age(M)$2. Conversely, from a SWIR one constructs a canonical amalgamation operator by taking an arbitrary amalgam inside $\Age(M)$3 and then using Existence and Stationarity to move one side into an independent position over the base (Kwiatkowska et al., 8 Aug 2025). This yields the equivalence

$\Age(M)$4

This equivalence is the main conceptual bridge in the recent Fraïssé treatment. It turns SWIR from a list of axioms into a statement that the age carries a functorial and canonical amalgamation rule.

Two neighboring notions are particularly relevant. The first is SIR, obtained by adding symmetry to SWIR. The second is the canonical independence relation (CIR) of Kaplan–Simon. For $\Age(M)$5 structures, the later paper states that

$\Age(M)$6

Thus SWIR can be viewed as the stationary strengthening of CIR in the Fraïssé context (Kwiatkowska et al., 8 Aug 2025).

4. Existence, local/global behavior, and examples

The SWIR landscape is highly non-uniform. Some classical Fraïssé limits admit symmetric relations, some only asymmetric ones, and some admit no local SWIR at all.

Structure	Status	Defining mechanism
Relational free-amalgamation limit	SIR	Free amalgamation over $\Age(M)$7
Random poset	SIR	Cross-side incomparability condition
Rational Urysohn space	Local SIR	Metric minimum formula over $\Age(M)$8
$\Age(M)$9	SWIR	Directed order condition across the base
Random tournament	SWIR	All cross edges point from $M$0 to $M$1
Oriented graphs omitting $M$2-anticliques, $M$3	SWIR	Same directional cross-edge rule
Generic two-graph	No local SWIR	Canonical amalgamation fails invariance
Betweenness $M$4	No local SWIR	Invariance obstruction
Separation $M$5	No local SWIR	Invariance obstruction

These examples are all worked out explicitly in the Fraïssé paper (Kwiatkowska et al., 8 Aug 2025). In particular, free-amalgamation classes give SIRs, while $M$6 and the random tournament give genuinely asymmetric SWIRs. The generic two-graph, the betweenness structure, and the separation structure fail even local SWIR because any putative canonical amalgam violates automorphism invariance of the base.

Local/global distinctions are also substantive. The generic $M$7-partite tournament exhibits three different regimes: if $M$8, it has no local SWIR but has extensible $M$9-age; if $\Ao(M)$0, it has a local but not global SWIR and has a Katětov functor; if $\Ao(M)$1, it has a global SWIR and a Katětov functor (Kwiatkowska et al., 8 Aug 2025). The same paper also notes closure under free superposition for relational strong-amalgamation classes: if the component limits have SWIRs, then the free superposition limit has a SWIR.

The automorphism-group paper gives another important source of examples. For a prioritised semi-free amalgamation class $\Ao(M)$2 of finite complete coloured digraphs omitting a prescribed set of forbidden triangles, the Fraïssé limit $\Ao(M)$3 carries a SWIR defined by

$\Ao(M)$4

Here $\Ao(M)$5 is the canonical prioritised semi-free amalgam (Li, 2019). This is the basic asymmetric construction used for Cherlin-style coloured tournaments.

5. SWIR and automorphism groups

The original motivation for isolating SWIR was group-theoretic. Tent–Ziegler had proved simplicity statements for automorphism groups of homogeneous structures equipped with a stationary independence relation; the SWIR framework shows that symmetry is not needed for the core argument (Li, 2019).

The key dynamical notions are “almost $\Ao(M)$6-maximal” and “almost $\Ao(M)$7-maximal” movement. An automorphism $\Ao(M)$8 moves almost $\Ao(M)$9-maximally if, for every finite set $\omega$0 and every type $\omega$1 over $\omega$2, some realization $\omega$3 satisfies

$\omega$4

It moves almost $\omega$5-maximally if one can realize $\omega$6 with

$\omega$7

Because symmetry is absent, both directions must be controlled separately (Li, 2019).

The main theorem states that if $\omega$8 is a countable structure with a SWIR and $\omega$9 is such that both $B \downarrow_A C$0 and $B \downarrow_A C$1 move almost $B \downarrow_A C$2-maximally and almost $B \downarrow_A C$3-maximally, then every element of $B \downarrow_A C$4 is a product of eight conjugates of $B \downarrow_A C$5 (Li, 2019). The same paper also observes that the Tent–Ziegler proof that $B \downarrow_A C$6 has a dense conjugacy class does not use symmetry, so it remains valid for SWIRs.

For prioritised semi-free amalgamation classes, SWIR becomes a direct route to simplicity. Under Condition 3.13 in the coloured-digraph setting, the paper proves that $B \downarrow_A C$7 is a prioritised semi-free amalgamation class, that its Fraïssé limit $B \downarrow_A C$8 has a canonical SWIR, and that for every non-trivial $B \downarrow_A C$9 there exist $A,B,C \subseteq M$0 such that $A,B,C \subseteq M$1 moves almost $A,B,C \subseteq M$2-maximally and almost $A,B,C \subseteq M$3-maximally; consequently $A,B,C \subseteq M$4 is simple (Li, 2019). This is the asymmetric analogue of the Tent–Ziegler simplicity theorem and is applied to some asymmetric examples due to Cherlin.

6. Extensive embeddings, Katětov functors, and open problems

The later Fraïssé paper connects SWIR to a different but related circle of ideas: extensive embeddings and Katětov functors. An embedding $A,B,C \subseteq M$5 is extensive if $A,B,C \subseteq M$6 is coinfinite in $A,B,C \subseteq M$7 and there is a group embedding

$A,B,C \subseteq M$8

such that each automorphism of $A,B,C \subseteq M$9 is extended by $\mathcal{L}$00 and the extension map respects composition. If every $\mathcal{L}$01 admits such an embedding, then $\mathcal{L}$02 is extensible (Kwiatkowska et al., 8 Aug 2025).

Katětov functors imply extensibility: if $\mathcal{L}$03 has a Katětov functor, then $\mathcal{L}$04 is extensible (Kwiatkowska et al., 8 Aug 2025). In the symmetric case, the link is strong. The paper rederives Müller’s theorem that if a $\mathcal{L}$05 structure has strong amalgamation and a local SIR, then it has a Katětov functor, hence extensible $\mathcal{L}$06-age and a universal automorphism group (Kwiatkowska et al., 8 Aug 2025).

For asymmetric independence, the relation is subtler. The same paper proves that a linearly ordered $\mathcal{L}$07 structure with strong amalgamation and a local SWIR has a Katětov functor, hence extensible $\mathcal{L}$08-age (Kwiatkowska et al., 8 Aug 2025). It also proves that if $\mathcal{L}$09 is relational, has strong amalgamation, and carries a local SIR, then its generic tournament expansion has a Katětov functor (Kwiatkowska et al., 8 Aug 2025).

However, SWIR and extensibility are independent in general. There is a relational $\mathcal{L}$10 structure with a Katětov functor but without a local SWIR, and there is a relational $\mathcal{L}$11 structure with a SWIR but without a universal automorphism group, hence with non-extensible $\mathcal{L}$12-age and no Katětov functor (Kwiatkowska et al., 8 Aug 2025). The second direction is witnessed by the limit of finite oriented graphs omitting $\mathcal{L}$13-anticliques for $\mathcal{L}$14: it has a SWIR, but its automorphism group is not universal (Kwiatkowska et al., 8 Aug 2025).

Several problems remain open. The Fraïssé paper asks whether the generic $\mathcal{L}$15-hypertournament or the generic $\mathcal{L}$16-enlarged tournament has a SWIR, whether the ultrahomogeneous oriented graph $\mathcal{L}$17 has a Katětov functor, and whether there are continuous-logic analogues of Katětov functors and unique extensibility, with the Gurarij space singled out as a test case (Kwiatkowska et al., 8 Aug 2025). These questions indicate that SWIR is now part of a broader program linking canonical amalgamation, automorphism groups, and extension phenomena across homogeneous structures.