sp-Homogeneous Linear Orderings

• sp-Homogeneous linear orderings are defined as linear orderings enriched with successor and predecessor functions, where every isomorphism between finitely generated sp-substructures extends to an automorphism.
• The classification decomposes orderings into blocks (finite, ω, ω*, ζ) and shuffle sums, providing sharp criteria for relative Δ3^0 and Δ4^0 categoricity with computability-theoretic precision.
• Finite relational approximations via Cₙ₋ₘ-homogeneity translate infinite structural complexities into finite combinatorial data, enabling explicit enumeration and asymptotic analysis.

sp-homogeneous linear orderings are linear orderings studied in the expanded signature $(<,s,p)$, where $s$ and $p$ name successor and predecessor. A linear ordering $L$ is sp-homogeneous when the expanded structure $(L,<,s,p)$ is homogeneous, equivalently when every isomorphism between finitely generated sp-substructures extends to an automorphism. Recent work gives a complete structural classification of sp-homogeneous and weakly sp-homogeneous orderings, sharp relative categoricity bounds, optimal arithmetical complexity results, and a hierarchy of finite relational approximations $C_{n,m}$ whose isomorphism types can be counted explicitly (Calvert et al., 29 Sep 2025, Gonzalez, 15 Apr 2026).

1. Definition in the successor–predecessor language

An sp-linear ordering is a linear ordering $(L,<)$ expanded by successor $s(x)$ and predecessor $p(x)$ unary functions. Here $s(x)$ is the immediate successor of $s$0, or $s$1 itself if there is none, and $s$2 is the immediate predecessor of $s$3, or $s$4 itself if there is none. The structure $s$5 is sp-homogeneous if every isomorphism between finitely generated sp-substructures extends to an automorphism of the whole structure. It is weakly sp-homogeneous if there is a finite exceptional set $s$6 such that every isomorphism between finitely generated sp-substructures fixing $s$7 extends to an automorphism fixing $s$8 (Calvert et al., 29 Sep 2025).

This formulation is strictly finer than homogeneity for pure linear orderings, because the added successor and predecessor structure separates local configurations that pure order alone does not distinguish. In particular, endpoints, immediate neighbors, and points lying inside dense or block-like regions become first-order visible in the sp-language. The literature therefore treats sp-homogeneity as a strong homogeneity notion with direct computability-theoretic content.

2. Structural classification by blocks and shuffle sums

The structural theory is organized around blocks: points at finite distance form blocks. Quotienting by the convex equivalence relation $s$9 of being “finite apart” decomposes an ordering into blocks of types in $p$0, where the finite types record finite block size and the infinite types record one-sided or bi-infinite block structure (Calvert et al., 29 Sep 2025).

A complete characterization states that $p$1 is sp-homogeneous iff there exists a pairwise disjoint family $p$2 of subsets of $p$3 such that $p$4 is the union of suborderings with the following form: for each $p$5, an open interval $p$6 isomorphic to a shuffle sum $p$7; and for each $p$8, a single block of size $p$9 (Calvert et al., 29 Sep 2025). The same body of work reformulates this as a dense shuffling of allowed block types inside certain intervals, together with isolated blocks that occur only once.

Weakly sp-homogeneous orderings admit an equally explicit decomposition. They are precisely the orderings that can be written as a finite concatenation

$L$0

where each $L$1 is sp-homogeneous and each $L$2 is a single, possibly infinite, block (Calvert et al., 29 Sep 2025).

This classification identifies the locus of symmetry: homogeneous behavior is carried by shuffle-sum intervals, while departures from full homogeneity are confined to finitely many exceptional blocks in the weak setting. It also underlies the later enumerative correspondence with finite combinatorial data.

3. Relative categoricity and effective structure

sp-homogeneous linear orderings are always relatively $L$3 categorical as pure orderings, and weakly sp-homogeneous linear orderings are also relatively $L$4 categorical as orderings (Calvert et al., 29 Sep 2025). The $L$5 bound is not merely existential; the same work determines exactly when the complexity drops to relative $L$6.

For an sp-homogeneous ordering $L$7, relative $L$8 categoricity holds iff all of the following conditions are satisfied:

• No forbidden intervals: $L$9 has no intervals isomorphic to a shuffle sum $(L,<,s,p)$0 where $(L,<,s,p)$1 contains an infinite block and arbitrary large finite blocks, or contains $(L,<,s,p)$2 and an additional finite block.
• Buffer conditions: If an interval $(L,<,s,p)$3 is isomorphic to $(L,<,s,p)$4, $(L,<,s,p)$5, $(L,<,s,p)$6, or $(L,<,s,p)$7, or to a sum of at least two such types, then $(L,<,s,p)$8 must be buffered on both sides by intervals consisting only of finite blocks of bounded size.
• Adjacency restrictions: For $(L,<,s,p)$9, neither to the left nor to the right may there be a shuffle sum containing an infinite block.

The same analysis yields a criterion for computable categoricity in the sp-signature: an sp-homogeneous ordering is computably categorical as an sp-structure iff it is relatively computably categorical as an ordering and does not fall into certain forbidden cases where an endpoint of a unique infinite block is too exposed and cannot be $C_{n,m}$0-defined using $C_{n,m}$1 (Calvert et al., 29 Sep 2025).

These results make the block calculus operational. Relative categoricity is governed by the arrangement of infinite blocks, the presence of large finite blocks inside shuffle sums, and the extent to which special configurations can be isolated by finite information in the sp-language.

4. Index sets and descriptive complexity

The classification is provably optimal from the standpoint of effective complexity. The set of computable indices for sp-homogeneous linear orderings is $C_{n,m}$2-complete, and the set of computable indices for weakly sp-homogeneous linear orderings is $C_{n,m}$3-complete (Calvert et al., 29 Sep 2025). Parallel boldface results are established for countable orderings as models: the class of orderings isomorphic to an sp-homogeneous model is $C_{n,m}$4-complete, and the class of orderings isomorphic to a weakly sp-homogeneous model is $C_{n,m}$5-complete.

Two distinct proof paradigms are used. One is a direct computability-theoretic and index-set analysis. The other uses descriptive set theory, including back-and-forth games, the boldface hierarchy, and connections with Scott analysis such as the Ash–Knight pair of structures theorem (Calvert et al., 29 Sep 2025). The coexistence of these methods is significant because it shows that the complexity bounds are not artifacts of one coding discipline; they reflect the intrinsic definability-theoretic complexity of recognizing the structural configurations appearing in the classification.

A common misreading is to regard the explicit structural decomposition as yielding a low-complexity decision procedure. The completeness results show the opposite: even with a complete classification in hand, the global recognition problem for computable presentations remains high in the arithmetical hierarchy.

5. $C_{n,m}$6-homogeneity as a finite relational approximation

To approximate sp-homogeneity inside finite relational languages, the theory introduces $C_{n,m}$7-homogeneity. The language is expanded by unary predicates $C_{n,m}$8, unary predicates $C_{n,m}$9, and binary predicates $(L,<)$0, where $(L,<)$1 means “has exactly $(L,<)$2 successors,” $(L,<)$3 means “has exactly $(L,<)$4 predecessors,” and $(L,<)$5 means that two elements are exactly $(L,<)$6 apart. A linear ordering is $(L,<)$7-homogeneous if this relational expansion is homogeneous (Calvert et al., 29 Sep 2025, Gonzalez, 15 Apr 2026).

The approximation is exact in the limit: $(L,<)$8 For finite parameters, the classification is explicit. For finite $(L,<)$9, the $s(x)$0-homogeneous orderings are exactly the sp-homogeneous orderings with all block sizes $s(x)$1. For $s(x)$2 and $s(x)$3 finite, they are the sp-homogeneous orderings without $s(x)$4-blocks. For $s(x)$5 and $s(x)$6 finite, they are the sp-homogeneous orderings without $s(x)$7-blocks. For $s(x)$8, one recovers all sp-homogeneous orderings (Calvert et al., 29 Sep 2025).

The later enumerative work emphasizes a useful warning: finite $s(x)$9-languages only “see” blocks of size up to $p(x)$0, and cannot distinguish blocks isomorphic to $p(x)$1 or $p(x)$2 beyond finite height; consequently, finite $p(x)$3-homogeneity is not identical with sp-homogeneity, but only a strong finite-stage approximation (Gonzalez, 15 Apr 2026). This finite relational reformulation is important in computable structure theory, where homogeneous structures over finite relational languages fit more naturally with standard categoricity machinery; in particular, the $p(x)$4-homogeneous orderings are always relatively $p(x)$5-categorical (Gonzalez, 15 Apr 2026).

6. Enumeration and correspondence with finite combinatorial data

The enumerative theory counts both $p(x)$6-homogeneous linear orderings and homogeneous colored linear orderings, and it does so by translating infinite order-theoretic objects into finite combinatorial data (Gonzalez, 15 Apr 2026). Although the structures being counted are generally infinite, there are only finitely many isomorphism types at each finite $p(x)$7 stage, and only countably many sp-homogeneous linear orderings in the limit.

The key correspondence proceeds through the block quotient by $p(x)$8. The ordering decomposes into blocks of type finite, $p(x)$9, $s(x)$0, or $s(x)$1, and the finite combination of how blocks can be ordered and shuffled, together with coloring or multi-coloring information, determines the isomorphism type. The paper states the central insight as follows: each sp-homogeneous linear ordering is determined by a finite arrangement data up to isomorphism (Gonzalez, 15 Apr 2026).

For $s(x)$2, let $s(x)$3 be the number of $s(x)$4-homogeneous linear orderings, and let $s(x)$5 be the number of homogeneous linear orderings in $s(x)$6 colors. The principal formulas are:

Quantity	Definition	Formula or asymptotic
$s(x)$7	Number of $s(x)$8-homogeneous orderings with $s(x)$9	$s$00
$s$01	Recursive form	$s$02
$s$03	Number of homogeneous colored orderings with $s$04 colors	$s$05

The asymptotic estimates are equally explicit: $s$06 and, writing $s$07 for the product logarithm, $s$08, $s$09,

$s$10

These formulas place sp-homogeneous orderings at an intersection of model-theoretic homogeneity, block-structured order theory, computable categoricity, and analytic combinatorics. The finite-stage approximation hierarchy $s$11 is the bridge: it converts a homogeneity notion defined using successor and predecessor functions into finite relational data that can be classified, counted, and asymptotically analyzed (Calvert et al., 29 Sep 2025, Gonzalez, 15 Apr 2026).