Strongly Topologically Orderable Gyrogroup

• Strongly topologically orderable gyrogroups are topological gyrogroups endowed with a total order whose induced topology matches the given topology and features gyration-invariant neighborhood bases.
• They exhibit a structural dichotomy where either metrizability holds or a totally ordered chain of clopen, gyration-stable L‐subgyrogroups governs the local neighborhood at the identity.
• This framework ensures hereditary paracompactness and the existence of suitable sets, linking discrete algebraic generation with the underlying topological order.

Searching arXiv for papers on strongly topologically orderable gyrogroups and related strongly topological gyrogroup structure. A strongly topologically orderable gyrogroup is a topological gyrogroup endowed with a total order whose order topology coincides with the given topology and whose neighborhood structure at the identity is invariant under all gyrations. In the terminology fixed in the recent literature, it is simultaneously a topologically orderable gyrogroup and a strongly topological gyrogroup; the “strong” modifier refers to gyration-invariance of a local base, not to any order-invariance axiom for the operation (He et al., 15 Jul 2025). The subject lies at the intersection of nonassociative topological algebra, generalized ordered spaces, and the structure theory of strongly topological gyrogroups developed earlier for paracompactness, metrizability, quotient constructions, and suitable sets (Bao et al., 2019).

1. Algebraic and topological definition

A gyrogroup is a groupoid $(G,\oplus)$ with identity element $1$, inverses $\ominus g$, and a family of automorphisms $\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$ satisfying gyroassociativity and the left gyroassociative loop property. Explicitly, for all $g,h,f\in G$,

$$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$

A topological gyrogroup $(G,\mathcal T,\oplus)$ is a gyrogroup whose operation $\oplus:G\times G\to G$ and inversion map are continuous. A subgyrogroup $H$ is an $L$-subgyrogroup when

$1$0

For such $1$1, the left cosets $1$2 partition $1$3 (He et al., 15 Jul 2025).

A linearly ordered set $1$4 carries the order topology generated by the rays $1$5 and $1$6. A topological gyrogroup $1$7 is topologically orderable if there exists a total order $1$8 on $1$9 such that the induced order topology is exactly $\ominus g$0. It is strongly topological if there exists a neighborhood base $\ominus g$1 at the identity $\ominus g$2 such that

$\ominus g$3

A strongly topologically orderable gyrogroup is therefore a quadruple $\ominus g$4 in which the topology is both an order topology and a strongly topological gyrogroup topology (He et al., 15 Jul 2025).

The associated notion of suitable set is also central. A subset $\ominus g$5 is a suitable set if $\ominus g$6 is discrete, $\ominus g$7 is closed, and the subgyrogroup $\ominus g$8 generated by $\ominus g$9 is dense in $\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$0. This adapts the classical concept from topological group theory to the gyrogroup setting (He et al., 15 Jul 2025).

2. Local ordered structure and the dichotomy theorem

The basic local fact is that in a topologically orderable gyrogroup the identity has a totally ordered neighborhood base under inclusion. In the strongly topological case, this can be refined to a totally ordered base of symmetric neighborhoods, each invariant under all gyrations. This leads to the principal structural dichotomy of the subject: if $\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$1 is a strongly topologically orderable gyrogroup, then either $\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$2 is metrizable, or $\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$3 has a totally ordered local base $\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$4 at the identity consisting of clopen $\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$5-subgyrogroups such that

$\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$6

Thus the nonmetrizable case is rigidly controlled by a chain of clopen, gyration-stable $\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$7-subgyrogroups (He et al., 15 Jul 2025).

This theorem has a converse flavor in the nonmetrizable regime. For a nonmetrizable strongly topological gyrogroup, topological orderability is equivalent to the existence of a totally ordered local base at the identity. The paper also states the corresponding specialization to the totally disconnected strongly topological case. In that sense, the ordered structure is not an auxiliary decoration on the topology; in the nonmetrizable case it is encoded by the very existence of a linearly nested local base (He et al., 15 Jul 2025).

A recurrent point of interpretation is that no additional algebraic invariance of the order is assumed. In particular, the definition does not require a left-invariant or bi-invariant order. The order is topological rather than translation-invariant in the classical ordered-group sense. This sharply distinguishes the notion from left-orderability in group theory and prevents a common misreading of the term “strongly topologically orderable” (He et al., 15 Jul 2025).

3. Covering properties, connected components, and metrizability

Every strongly topologically orderable gyrogroup is hereditarily paracompact. In the metrizable case this is immediate. In the nonmetrizable case, the clopen base from the dichotomy theorem can be refined to a base $\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$8 of clopen sets such that whenever $\mathrm{gyr}[g,h]\in\operatorname{Aut}(G,\oplus)$9 and $g,h,f\in G$0, one has $g,h,f\in G$1 or $g,h,f\in G$2. A theorem of Nyikos–Reichel then yields hereditary paracompactness. Consequently, every subspace is paracompact (He et al., 15 Jul 2025).

The connected component $g,h,f\in G$3 of the identity in a topological gyrogroup is always an $g,h,f\in G$4-subgyrogroup. In the orderable context, if $g,h,f\in G$5 is connected and has at least two elements, then $g,h,f\in G$6 is open. This has a decisive consequence: if a strongly topologically orderable gyrogroup is not totally disconnected, then it contains an open, first-countable $g,h,f\in G$7-subgyrogroup $g,h,f\in G$8, and hence the whole gyrogroup is metrizable. The proof uses local compactness of the connected component in the orderable setting together with a contradiction arising from compact connected homogeneous orderable spaces (He et al., 15 Jul 2025).

Local compactness leads to an even stronger conclusion. Every locally compact strongly topologically orderable gyrogroup is metrizable. In the totally disconnected locally compact case, every neighborhood of the identity contains a compact open $g,h,f\in G$9-subgyrogroup. Moreover, an infinite locally compact totally disconnected strongly topologically orderable gyrogroup is either discrete or contains a clopen subgyrogroup homeomorphic to the Cantor set. These statements show that the local ordered topology imposes a severe restriction on the nonassociative algebraic structure once local compactness is present (He et al., 15 Jul 2025).

A further metrization criterion comes from pseudocharacter. If $$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$0 is a $$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$1-set, then a strongly topologically orderable gyrogroup is metrizable and therefore has a suitable set. This criterion is parallel to the role of countable pseudocharacter in the earlier theory of strongly topological gyrogroups, where countable pseudocharacter already implied submetrizability (Bao et al., 2020).

4. Suitable sets

Suitable sets organize dense generation in a topologically sparse way. The existence theory in the orderable setting follows the metrizability results closely. Every strongly topologically orderable gyrogroup that is not totally disconnected has a suitable set, because such a gyrogroup is metrizable. Every locally compact strongly topologically orderable gyrogroup is likewise metrizable, and hence has a suitable set. The same conclusion holds when the identity is a $$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$2-point (He et al., 15 Jul 2025).

The paper places these results within the older suitable-set program for strongly topological gyrogroups. Earlier work had shown that every separable metrizable strongly topological gyrogroup has a suitable set, and had asked whether every locally compact strongly topological gyrogroup has one (Lin et al., 2020). That locally compact question was later answered affirmatively for strongly topological gyrogroups in general (Yang et al., 15 Jul 2025). The orderable theory recovers the locally compact case by proving metrizability inside the orderable subclass and then invoking the known existence of suitable sets in metrizable strongly topological gyrogroups (He et al., 15 Jul 2025).

The role of suitable sets is structural rather than merely combinatorial. They provide discrete algebraic skeleta whose generated subgyrogroups are dense, while the only possible accumulation occurs at the identity. In the orderable context, this suggests a direct interface between topological generation and local order structure, although the paper does not formulate an order-theoretic reconstruction theorem from a suitable set (He et al., 15 Jul 2025).

5. Dense subgyrogroups and permanence of suitable sets

One of the main new permanence results is a transfer theorem for dense subgyrogroups. If $$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$3 is a dense subgyrogroup of a strongly topologically orderable gyrogroup $$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$4, and $$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$5 has a suitable set, then $$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$6 also has a suitable set. If the suitable set in $$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$7 is closed, then the induced suitable set in $$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$8 may also be taken closed. This applies to dense substructures without any additional metrizability or local compactness hypothesis (He et al., 15 Jul 2025).

The proof depends on a detailed analysis of discrete subsets against the clopen ordered base. First, every discrete subset of a strongly topologically orderable gyrogroup can be separated by pairwise disjoint neighborhoods; in the nonmetrizable case these neighborhoods are chosen as cosets $$g\oplus(h\oplus f)=(g\oplus h)\oplus \mathrm{gyr}[g,h](f), \qquad \mathrm{gyr}[g,h]=\mathrm{gyr}[g\oplus h,h].$$9 of members of the clopen chain from the dichotomy theorem. Second, if $(G,\mathcal T,\oplus)$0 is a decreasing base of clopen $(G,\mathcal T,\oplus)$1-subgyrogroups and $(G,\mathcal T,\oplus)$2 is a disjoint family of cosets, then a point is an accumulation point of the coset family if and only if it is an accumulation point of the underlying set $(G,\mathcal T,\oplus)$3. Third, any discrete closed subset $(G,\mathcal T,\oplus)$4 contained in the closure of a subgyrogroup $(G,\mathcal T,\oplus)$5 can be approximated by a discrete subset $(G,\mathcal T,\oplus)$6 with the same accumulation behavior. These three ingredients produce a suitable set inside the dense subgyrogroup (He et al., 15 Jul 2025).

The paper ends this line of inquiry with two open questions: whether the existence of a suitable set always passes from a strongly topologically orderable gyrogroup to an arbitrary subgyrogroup, and whether closed suitable sets pass to subgyrogroups that are not closed. Those questions indicate that density is essential in the current transfer theorem (He et al., 15 Jul 2025).

6. Position within the broader theory of strongly topological gyrogroups

The orderable theory is built on earlier structural results for strongly topological gyrogroups. Bao and Lin defined strongly topological gyrogroups by the existence of a gyration-invariant base at the identity and proved that a strongly topological gyrogroup is feathered if and only if it contains a compact $(G,\mathcal T,\oplus)$7-subgyrogroup $(G,\mathcal T,\oplus)$8 such that the quotient space $(G,\mathcal T,\oplus)$9 is metrizable; they also proved that every feathered strongly topological gyrogroup is paracompact (Bao et al., 2019). In the orderable subclass, the newer hereditary paracompactness theorem is therefore a much stronger conclusion, but it is obtained under the additional orderability hypothesis (He et al., 15 Jul 2025).

The later metrization theory sharpened this background. $\oplus:G\times G\to G$0-strongly topological gyrogroups were shown to be completely regular, and countable pseudocharacter was shown to imply submetrizability; for admissible $\oplus:G\times G\to G$1-subgyrogroups $\oplus:G\times G\to G$2, the quotient $\oplus:G\times G\to G$3 was also shown to be submetrizable (Bao et al., 2020). Further work on feathered strongly topological gyrogroups established that countable $\oplus:G\times G\to G$4-character or an $\oplus:G\times G\to G$5-base forces metrizability, and that compact resolutions are equivalent to the existence of compact $\oplus:G\times G\to G$6-subgyrogroups with Polish quotients (Bao et al., 2022). These results explain why the orderable theory repeatedly drives toward metrizability: orderability is being imposed on a class whose weak countability assumptions already have strong metric consequences.

The topic also interacts with embedding theory. Hartman–Mycielski type constructions show that Hausdorff topological gyrogroups can be embedded as closed subgyrogroups of Hausdorff path-connected and locally path-connected topological gyrogroups, and in the strong setting this can be done inside strongly topological gyrogroups (Jin et al., 2021). This suggests that strongly topologically orderable gyrogroups belong to a larger ecosystem of topological gyrogroup envelopes, even though no order-theoretic Hartman–Mycielski theorem is presently stated.

A final conceptual clarification is that the orderable theory does not collapse to topological groups. Strongly topological gyrogroups include non-group examples such as the Möbius gyrogroup, where gyrations are nontrivial but preserve the basic neighborhoods at the identity (Bao et al., 2019). The orderable results therefore concern genuinely nonassociative objects. What they show is that once a compatible linear order topology is imposed together with gyration-invariant local structure, the resulting space behaves with remarkable regularity: it is hereditarily paracompact, often metrizable, and frequently rich enough to carry suitable sets (He et al., 15 Jul 2025).