Weakly sp-Homogeneous Linear Orderings

• Weakly sp-Homogeneous Linear Orderings are countable linear orders with successor and predecessor functions that permit finitely many exceptional cuts, yielding a precise finite-decomposition structure.
• The work presents a complete structural classification showing that such orderings can be expressed as finite concatenations of sp-homogeneous segments separated by single blocks, clarifying their symmetry.
• Implications include a detailed computability and categoricity analysis, establishing that these orderings possess relative Δ⁰₄ categoricity and recognition complexity characterized as Σ⁰₆-complete.

Weakly $sp$-homogeneous linear orderings are countable linear orderings $L$ whose expansion by successor and predecessor,

$(L,<,s,p),$

is weakly homogeneous in the Adams–Cenzer sense: there is a finite exceptional set $a_1,\dots,a_n$ such that any isomorphism between finitely generated substructures fixing each $a_i$ extends to an automorphism. Recent work gives a complete structural classification of these orders, places them in the relative categoricity hierarchy, and relates the underlying strong notion $sp$-homogeneity to a family of finite relational approximations $C_{n,m}$ (Calvert et al., 29 Sep 2025).

1. Expanded order language and the meaning of weak $sp$-homogeneity

The ambient structures are linear orderings equipped with two total unary functions. The successor function $s(x)$ is the successor of $x$, if one exists, and otherwise $L$0; dually, $L$1 is either the predecessor of $L$2 or equals $L$3. The resulting structures are called $L$4-linear orderings. In this setting, a linear ordering is $L$5-homogeneous if its expansion $L$6 is homogeneous, and weakly $L$7-homogeneous if the same expansion is weakly homogeneous (Calvert et al., 29 Sep 2025).

The weak form differs from full homogeneity only by allowing finitely many named exceptional points. In the present context, that means that global symmetry may fail at finitely many locations while still holding uniformly on the remaining finitely generated $L$8-substructures. This finite-exception formulation is the source of both the structural decomposition theorem and the jump in logical complexity of the class (Calvert et al., 29 Sep 2025).

A central invariant is the block decomposition. The quotient $L$9 is obtained from $(L,<,s,p),$0 by the convex equivalence relation identifying points that are finitely far from each other; equivalently, two points are in the same block iff they are connected by finitely many successor/predecessor steps. The notation $(L,<,s,p),$1 means that $(L,<,s,p),$2 and $(L,<,s,p),$3 lie in the same block, and $(L,<,s,p),$4 denotes the block of $(L,<,s,p),$5. The block types relevant here are finite $(L,<,s,p),$6, $(L,<,s,p),$7, $(L,<,s,p),$8, and $(L,<,s,p),$9 (Calvert et al., 29 Sep 2025).

The terminology should be distinguished from the weak homogeneity used in the LOTS literature, where a linearly ordered topological space $a_1,\dots,a_n$0 is called weakly homogeneous if it is order isomorphic with every nonempty, bounded, open subinterval of itself. That interval-self-similarity notion is different from weak homogeneity in the expanded language $a_1,\dots,a_n$1 (Akin et al., 2021).

2. Structural classification

The defining theorem states that a linear ordering $a_1,\dots,a_n$2 is weakly $a_1,\dots,a_n$3-homogeneous if and only if it can be written as

$a_1,\dots,a_n$4

where each $a_1,\dots,a_n$5 is a possibly empty $a_1,\dots,a_n$6-homogeneous linear ordering and each $a_1,\dots,a_n$7 is a single non-empty block (Calvert et al., 29 Sep 2025).

This yields the basic geometric picture of the class: a weakly $a_1,\dots,a_n$8-homogeneous order is exactly a finite concatenation of $a_1,\dots,a_n$9-homogeneous pieces separated by finitely many individual blocks. Equivalently, after naming one element from each separator block, the ordering breaks into intervals on which full $a_i$0-homogeneity holds. The weak notion is therefore strictly broader than $a_i$1-homogeneity: every $a_i$2-homogeneous ordering is weakly $a_i$3-homogeneous by taking $a_i$4, but the weak class allows finitely many defects in the form of whole separator blocks (Calvert et al., 29 Sep 2025).

Several immediate consequences are recorded. Every $a_i$5 categorical linear ordering is weakly homogeneous as an $a_i$6-linear ordering, because every $a_i$7-categorical ordering is a finite separated sum of $a_i$8, $a_i$9, $sp$0, and $sp$1, which fits the displayed decomposition. Conversely, if $sp$2 has infinitely many successors, then $sp$3 is not weakly $sp$4-homogeneous. A further non-example is provided by the $sp$5-representations

$sp$6

for which $sp$7; no $sp$8 representation is weakly $sp$9-homogeneous (Calvert et al., 29 Sep 2025).

The classification also clarifies a common misconception. Weak $C_{n,m}$0-homogeneity is not a vague local regularity condition: it is an exact finite-decomposition property. What is allowed is not arbitrary mild inhomogeneity, but only finitely many block-level separators between genuinely $C_{n,m}$1-homogeneous components (Calvert et al., 29 Sep 2025).

3. The $C_{n,m}$2-homogeneous components: blocks and shuffle sums

Understanding the weak class reduces to understanding the strong class. The same classification theorem gives a precise normal form for $C_{n,m}$3-homogeneous linear orderings: $C_{n,m}$4 is $C_{n,m}$5-homogeneous if and only if there is a pairwise disjoint family

$C_{n,m}$6

of subsets of $C_{n,m}$7 such that $C_{n,m}$8 is the union of suborderings of two kinds: for each $C_{n,m}$9, an open interval $sp$0 isomorphic to $sp$1, and for each $sp$2, a single block of size $sp$3 (Calvert et al., 29 Sep 2025).

Here $sp$4 denotes the unique colored linear ordering in which each color in the countable set $sp$5 is dense, and $sp$6 is the shuffle sum obtained by replacing each point of a color shuffle by a copy of the corresponding order. The paper also proves that for any $sp$7, the shuffle sum $sp$8 is $sp$9-homogeneous. Shuffle sums are therefore the basic source of homogeneous regions in the weak classification (Calvert et al., 29 Sep 2025).

The disjointness condition on the family $s(x)$0 is structurally decisive. A block type either occurs exactly once in the whole structure, or it occurs densely in a shuffle interval; it cannot appear in two different shuffle regions, nor both uniquely and densely. Weakly $s(x)$1-homogeneous orders inherit this regime componentwise, with the only additional freedom being the insertion of finitely many separator blocks between such components (Calvert et al., 29 Sep 2025).

From the block perspective, the classification is especially rigid. The building blocks are always convex classes of type $s(x)$2, $s(x)$3, $s(x)$4, or $s(x)$5, and the global organization is controlled by dense interleaving of block types together with finitely many exceptional concatenation points. A plausible implication is that many questions about weak $s(x)$6-homogeneity reduce to analyzing how separator blocks interact with the orbit structure already present inside the $s(x)$7-homogeneous pieces.

4. Categoricity and computability-theoretic properties

Weakly $s(x)$8-homogeneous linear orderings are always relatively $s(x)$9 categorical. The argument given is that $x$0 and $x$1 are computable from $x$2, while weak homogeneity in the $x$3-language gives relative $x$4-categoricity there; passing back to the pure order language yields relative $x$5-categoricity (Calvert et al., 29 Sep 2025).

The strong class receives a sharper $x$6 analysis. The paper determines exactly which $x$7-homogeneous orderings are uniformly relatively $x$8 categorical and exactly which are relatively $x$9 categorical. In the uniform case, the characterization excludes intervals of the form $L$00 where $L$01 includes an infinite block and finite blocks of arbitrary size, and imposes bounded-size finite-block neighborhoods around certain intervals such as $L$02, $L$03, $L$04, and $L$05. In the non-uniform case, the restrictions are slightly weaker but still formulated in terms of the local arrangement of infinite blocks and bounded finite neighborhoods (Calvert et al., 29 Sep 2025).

For weakly $L$06-homogeneous orderings, the decomposition

$L$07

shows that the $L$08 problem is controlled by the $L$09-homogeneous components. The paper gives a proposition for finitely cut decompositions of $L$10-homogeneous orderings and explicitly notes that its mechanism is what one uses for the weak case. This suggests that relative $L$11-categoricity for a weakly $L$12-homogeneous order is governed by whether each $L$13-homogeneous piece satisfies the uniform $L$14 conditions (Calvert et al., 29 Sep 2025).

The negative side of the $L$15 theory is expressed through $L$16-free tuples in the Ash–Knight sense. The paper states that a tuple is $L$17-free iff its automorphism orbit cannot be defined by a $L$18 formula, and that the presence of an $L$19-free tuple prevents uniform relative $L$20-categoricity. In the present setting, these tuples arise from badly behaved mixtures of infinite blocks, arbitrarily large finite blocks, and unfavorable adjacencies of infinite regions (Calvert et al., 29 Sep 2025).

5. Recognition complexity and descriptive-set-theoretic status

The class of weakly $L$21-homogeneous linear orderings has exact index-set complexity. The set

$L$22

is $L$23-complete, while the corresponding set for $L$24-homogeneous linear orderings is $L$25-complete (Calvert et al., 29 Sep 2025).

The upper bound for the weak class comes directly from the structural theorem. Weak $L$26-homogeneity can be expressed by saying that there exist finitely many cut points $L$27 such that the intervals $L$28, $L$29, and $L$30 are all either empty or $L$31-homogeneous. Since $L$32-homogeneity itself is a $L$33 property, the finite-existence quantifier raises the weak class to $L$34 (Calvert et al., 29 Sep 2025).

The paper also proves the boldface analogues: $L$35 is $L$36-complete, and

$L$37

is $L$38-complete. For the weak case, the reduction uses sums of building blocks of two types, one weakly $L$39-homogeneous and one not weakly $L$40-homogeneous, together with a boldface form of the Ash–Knight pair-of-structures theorem and a continuity lemma for sums of continuously produced orders (Calvert et al., 29 Sep 2025).

The one-level increase from $L$41 to $L$42 is not accidental. It exactly matches the passage from global homogeneity to “there exist finitely many exceptional cuts,” which is the model-theoretic content of weak homogeneity in the $L$43-language (Calvert et al., 29 Sep 2025).

6. $L$44-approximations, finite combinatorics, and present scope

A major auxiliary development is the hierarchy of $L$45-homogeneity. For $L$46, the language contains predicates $L$47 for having $L$48 successors, $L$49 for having $L$50 predecessors, and $L$51 for two points being exactly distance $L$52 apart. A linear ordering is $L$53-homogeneous if its definitional expansion by these predicates is homogeneous, and weakly $L$54-homogeneous if that expansion is weakly homogeneous (Calvert et al., 29 Sep 2025).

The hierarchy is a strong approximation to $L$55-homogeneity. For any $L$56 and $L$57, $L$58-homogeneous linear orderings are $L$59-homogeneous, and

$L$60

Moreover, the $L$61-homogeneous structures are classified exactly as follows: $L$62 For finite $L$63, there are only finitely many $L$64-homogeneous linear orderings, the number depends only on $L$65, and the counting sequence $L$66 has exact recurrence and closed form with asymptotic upper bound

$L$67

Among homogeneous colored linear orderings, these $L$68-homogeneous orders are asymptotically sparse: $L$69 (Calvert et al., 29 Sep 2025).

These results sharpen the surrounding strong theory, but they do not yet constitute a direct theory of weak $L$70-homogeneity. The enumerative paper explicitly states that it does not define weakly $L$71-homogeneous linear orderings, does not prove any theorem characterizing them, does not give counting results for weakly $L$72-homogeneous or weakly $L$73-homogeneous orders, and does not compare weakly $L$74-homogeneous with any weak $L$75-notion. Its contribution is instead to provide the block/shuffle description of $L$76-homogeneous orders, the relational approximation hierarchy $L$77, and finite combinatorial codings that organize the strong part of the theory (Gonzalez, 15 Apr 2026).

Within current knowledge, the topic therefore has a sharp center and a clear frontier. The sharp center is the theorem that weakly $L$78-homogeneous linear orderings are exactly finite concatenations of $L$79-homogeneous pieces separated by single blocks. The frontier is the absence of a parallel approximation or counting theory for the weak class itself.