Conjugacy of Partial Semigroup Actions

• Conjugacy of partial semigroup actions is a framework for classifying equivalence classes of local symmetries under partially defined operations.
• Universal globalization techniques embed partial actions into global structures, enabling systematic comparison and invariant analysis.
• Classification employs combinatorial, algebraic, and cohomological methods to resolve conjugacy issues in inverse semigroups, twisted actions, and related contexts.

A conjugacy of partial semigroup actions refers to the study and classification of equivalence classes of partial symmetries under transformations that intertwine the structure of partially defined operations. The concept emerges as a broad generalization of group and semigroup conjugacy—where conjugacy classically involves elements being related by inner automorphisms—but is reshaped to accommodate localized or non-global symmetries, such as those found in inverse semigroups, categories, and their actions on sets or algebras. Current research consolidates disparate frameworks (partial semigroup, partial category, and partial group actions) via semigroupoid actions and universal globalizations, linking combinatorial, algebraic, and topological approaches.

1. Formal Definitions and Origination

Partial semigroup actions generalize ordinary semigroup actions by assigning to each $s \in S$ a partially defined map $\alpha_s$ with a specified domain ${}_sX$. The data $\{ {}_s X \}_{s\in S}$ and $\{ \alpha_s: {}_s X \to X \}_{s\in S}$ satisfy compatibility conditions reflecting compositional constraints (e.g., as in Kudryavtseva–Laan, $$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$ and $(\alpha_s \circ \alpha_t)(x) = \alpha_{st}(x)$ when defined) (Petasny et al., 2024).

Conjugacy, at its core, seeks to identify when two such partial actions are “equivalent” in a strong structural sense. Unlike group or full semigroup actions (where inner automorphisms and their induced conjugacy classes have a canonical form), for partial semigroup actions, the definitions must be localized and domain-aware to respect the partiality and possible lack of global symmetries.

For an inverse semigroup (the natural algebraic home for partial bijections), several explicit notions have been developed:

• Inverse semigroup conjugacy: $a\sim_i b$ if there exists $g \in S^1$ such that $g^{-1}ag = b$ and $\alpha_s$0 (Araujo et al., 2018), a direct generalization of group conjugacy but with adjusted invertibility.
• Partial conjugacy (in transformation semigroups): Characterizations often use cycle-chain-ray decompositions and compare the structure of digraph components, requiring “domain-preserving” intertwining maps (e.g., "rp-homomorphisms") (Araújo et al., 2013).
• Twisted partial conjugacy and module-theoretic equivalence: In the context of twisted partial actions (e.g., on semilattices of groups), conjugacy is encoded via a system of multipliers $\alpha_s$1 and an equivalence of associated “twisted module" structures, often mirroring equivalence of 2-cocycles in partial cohomology (Dokuchaev et al., 2016, Dokuchaev et al., 2013).
• Semigroupoid approach: The recent unification via partial semigroupoid actions recasts partial semigroup and partial category actions as instances of a more general framework. Here, conjugacy is best formulated in the category of partial semigroupoid actions, where the universal globalization is functorial and preserves conjugacy classes up to isomorphism (Petasny et al., 2024).

2. Universal Globalization and Conjugacy

Universal globalization is a construction whereby any partial semigroupoid action $\alpha_s$2 on $\alpha_s$3 can be canonically embedded into a global action $\alpha_s$4 on a larger set $\alpha_s$5 via a map $\alpha_s$6, satisfying a universal property: any morphism from $\alpha_s$7 to a global action factors uniquely through $\alpha_s$8 (Petasny et al., 2024).

This construction extends both the tensor product globalization for strong partial semigroup actions [Kudryavtseva–Laan] and the Nystedt–Lundström globalization for partial category actions, thus creating a unified setting for studying conjugacy. In this framework, conjugacy between partial semigroup actions reduces to isomorphism (intertwining via an invertible mapping) between their canonical globalizations.

Explicitly, if two partial actions are conjugate via a partial isomorphism, their universal globalizations $\alpha_s$9 are conjugate as global actions, i.e., there exists an automorphism ${}_sX$0 such that ${}_sX$1 for all ${}_sX$2 in the acting semigroupoid.

The construction of ${}_sX$3 uses ${}_sX$4 modulo elementary identifications, ensuring the preservation of conjugacy phenomena present in the original partial dynamical system (Petasny et al., 2024).

3. Classification and Characterization Techniques

The study of conjugacy classes in partial semigroup actions employs a rich array of combinatorial, algebraic, and cohomological techniques:

• Decomposition in I(X): For the symmetric inverse semigroup ${}_sX$5 (of partial bijections), every element can be uniquely decomposed into cycles, chains, and rays. Two elements are conjugate in the inverse semigroup sense if and only if they have the same cycle–chain–ray type (Araujo et al., 2018).
• Cycle set and domination in P(X): In semigroups of partial transformations, conjugacy classes are classified by invariants such as the set of (prime) lengths of cycles, the presence of double rays, and the dominance of maximal right rays, with formal conditions ensuring equivalence (Araújo et al., 2013).
• Equivalence of twisted data: For twisted partial actions on semilattices of groups, conjugacy is witnessed by families of invertible multipliers satisfying certain cocycle relations (changes by coboundaries), leading to a bijective correspondence with Sieben twisted modules. This is made precise by identities such as ${}_sX$6 (Dokuchaev et al., 2016).
• Cohomological invariants: Partial cohomology groups ${}_sX$7, capturing 2-cocycle data, control the classification of twisted partial actions up to conjugacy: if their associated cocycles lie in the same cohomology class, the partial actions are (twisted) conjugate (Dokuchaev et al., 2013).
• Canonical forms and globalized morphisms: In the partial semigroupoid setting, conjugacy classes can be transferred to the global model via the universal globalization, with the embedding ${}_sX$8 providing a direct link between partial and global conjugacy (Petasny et al., 2024).

4. Variants and Comparative Notions of Conjugacy

Multiple notions of conjugacy have been studied, each exhibiting distinct properties and consequences for the structure of partial actions:

Conjugacy Notion	Defining Relation	Typical Domain	Equivalence Property
${}_sX$9 (Primary)	$\{ {}_s X \}_{s\in S}$0	Arbitrary semigroups	May fail transitivity; transitive in commutative/idempotent cases (Borralho, 2019, Araújo et al., 2015)
$\{ {}_s X \}_{s\in S}$1	Domain-preserving $\{ {}_s X \}_{s\in S}$2 and vice versa	Transformation semigroups, restriction semigroups	Always an equivalence relation (Araújo et al., 2013, Araújo et al., 2015)
$\{ {}_s X \}_{s\in S}$3 (Inverse)	$\{ {}_s X \}_{s\in S}$4, $\{ {}_s X \}_{s\in S}$5	Inverse semigroups	Equivalence; aligns with group conjugacy when restricted (Araujo et al., 2018)
$\{ {}_s X \}_{s\in S}$6 (Trace)	Pseudoinverse-based, involving group part $\{ {}_s X \}_{s\in S}$7	Epigroups	Coincides with $\{ {}_s X \}_{s\in S}$8 in completely regular semigroups (Araújo et al., 2015)
$\{ {}_s X \}_{s\in S}$9 (Natural)	Unifies $\{ \alpha_s: {}_s X \to X \}_{s\in S}$0, included in all other conjugacies	General semigroups	Aligns with $\{ \alpha_s: {}_s X \to X \}_{s\in S}$1 in inverse semigroups, used for defining partial inner automorphisms (Araújo et al., 2023)

Transitivity of $\{ \alpha_s: {}_s X \to X \}_{s\in S}$2 is a subtle property; under the condition $\{ \alpha_s: {}_s X \to X \}_{s\in S}$3 $\{ \alpha_s: {}_s X \to X \}_{s\in S}$4 for all $\{ \alpha_s: {}_s X \to X \}_{s\in S}$5, primary conjugacy becomes transitive (Borralho, 2019).

Partition-covering properties are established for many of these relations: for any set $\{ \alpha_s: {}_s X \to X \}_{s\in S}$6 and partition $\{ \alpha_s: {}_s X \to X \}_{s\in S}$7, there exists a semigroup with universe $\{ \alpha_s: {}_s X \to X \}_{s\in S}$8 whose conjugacy classes coincide with $\{ \alpha_s: {}_s X \to X \}_{s\in S}$9 for $$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$0 (Jack, 2024, Jack, 2021).

5. Algebraic and Topological Tools

Several algebraic and analytic frameworks facilitate the analysis of conjugacy of partial semigroup actions:

• Prefix expansions and universal semigroups: The prefix expansion $$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$1 (or Birget–Rhodes expansion) upgrades partial actions of an inverse semigroup or group to global actions, converting conjugacy problems for partial actions into equivalent questions for global actions (Buss et al., 2011, Shourijeh et al., 2015, Martínez et al., 2021). The crossed products $$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$2 constructed from partial actions have isomorphic structure to crossed products by the universal expansion, transferring conjugacy invariants.
• Partial semigroup algebras and representations: Partial semigroup algebras encapsulate the structure underlying all partial representations and their functorial equivalence classes, allowing conjugacy questions to be functorially transported to algebraic settings (Shourijeh et al., 2015, Meenakshi et al., 2023).
• Recurrence sets and groupoids: Inverse semigroup partial actions admit analysis via recurrence sets, which serve as robust invariants under conjugacy. When passing to associated groupoids of germs, these recurrence sets reflect orbit or conjugacy equivalence in the dynamical system (Mantoiu, 2021).
• Topological extensions: A minimal Hausdorff inverse semigroup topology on the semigroup of partial homeomorphisms ensures that continuous partial actions admit extensions to continuous global actions, preserving conjugacy up to homeomorphism (Martínez et al., 2021).

6. Applications, Algorithmic Aspects, and Further Research

• Algorithmic decision: Polynomial-time and logspace algorithms are established for checking conjugacy in finite inverse semigroups and transformation semigroups. Many related decision problems (e.g., nilpotency, R-triviality) are classified with precise complexity; checking whether $$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$3 is the identity relation is in $$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$4 (Jack, 2021).
• Cohomological, categorical, and dynamical implications: The correspondence between twisted partial actions, module-theoretic data, and partial cohomology $$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$5 classes places the study of conjugacy within a broader homological and categorical context, with applications to the classification of crossed products, dynamical systems, and operator algebras (Dokuchaev et al., 2013, Dokuchaev et al., 2016).
• Unification and generalization: The semigroupoid globalization framework subsumes earlier results for partial semigroup and category actions, giving a unified language for conjugacy and enabling results, techniques, and invariants to be transferred systematically between settings previously studied in isolation (Petasny et al., 2024).

Several open questions pertain to the correspondence and complexity of conjugacy classes for partial actions beyond inverse semigroups, deeper structural invariants arising from partial cohomology, and potential extensions of the partition-covering and transitivity results to broader classes and more general topological settings (Dokuchaev et al., 2016, Araújo et al., 2023, Jack, 2024). The development of a more refined “lattice” of conjugacy relations in the partial setting remains an active topic.

7. Summary Table: Notions of Conjugacy (Selection)

Symbol	Equivalence Type	Partial/Global	Structure dependence	Partition-Covering
$$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$6	Primary	Full/Partial	Factorization $$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$7	Yes (Jack, 2024)
$$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$8	Domain-preserving	Partial	Existence of rp-homomorphisms	Yes
$$\alpha_t^{-1}({}_s X \cap X_t) = {}_t X \cap {}_{st} X$$9	Inverse semigroup	Partial/Global	Two-sided conjugator	Yes in appropriate classes
$(\alpha_s \circ \alpha_t)(x) = \alpha_{st}(x)$0	Trace (Epigroup)	Partial/Global	Group part comparisons	Yes (finite)
$(\alpha_s \circ \alpha_t)(x) = \alpha_{st}(x)$1	Natural	Partial/Global	Includes all above in general	Yes (Jack, 2024)

This table encapsulates several of the main conjugacy relations, their setting, and structural features.

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In conclusion, conjugacy in partial semigroup actions is a field that synthesizes algebra, category theory, combinatorics, and topology, driven by a shift from global to local notions of symmetry. The universal globalization principle, classification by combinatorial and cohomological invariants, compatibility with a range of conjugacy definitions, and transfer to modules and crossed products provide a broad and systematic structural theory. This unification greatly facilitates the transport of invariants, algorithms, and classification results across the many areas where partial actions and their symmetries arise.