Partial Group Actions

• Actions of partial groups are defined as families of bijections on restricted domains that satisfy modified group compatibility conditions, capturing local symmetries.
• They are modeled using inverse semigroups, groupoids, and globalization techniques that extend local actions to full global behaviors across set, topological, and algebraic frameworks.
• These methods enable classification via cohomology, Morita theory, and orbit analyses, with applications in understanding non-global symmetry and partial transformations.

Searching arXiv for relevant papers on partial actions, globalization, and related structures. Actions of partial groups are most commonly formalized as partial actions of groups: systems in which each group element acts not everywhere, but by a bijection or isomorphism between specified domains, while still satisfying the group law wherever compositions are defined. In the literature represented here, the term is not usually attached to an autonomous algebraic object called a “partial group”; rather, it describes a regime of partial symmetries encoded by families of partial bijections, partial algebra isomorphisms, inverse semigroups, groupoids, and globalization constructions. A standard formulation takes a group $G$, a set or structured object $X$, and maps $m_g:X_{g^{-1}}\to X_g$ satisfying $X_1=X$, $m_1=\mathrm{id}_X$, and compatibility conditions such as $m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$ together with $m_gm_h=m_{gh}$ on the common domain (Uzcategui et al., 2017). Across set-theoretic, topological, algebraic, cohomological, and categorical settings, the theory studies when such local symmetries can be extended to global actions, how their orbit structure behaves, and which algebraic and homological invariants classify them (Batista, 2016).

1. Formal definitions and basic models

A set-theoretic partial action of a group $G$ on a set $X$ may be given either as a partially defined map

$m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$

with domain

$X$0

or equivalently as a family

$X$1

of bijections between subsets of $X$2 satisfying $X$3, $X$4, and the usual compatibility axioms restricted to the domains where the compositions are defined (Uzcategui et al., 2017, Gómez et al., 2017, Batista, 2016). In the partially defined formulation, the axioms are: if $X$5 is defined then $X$6 is defined and equals $X$7; if $X$8 is defined then $X$9 is defined and the two values coincide; and $m_g:X_{g^{-1}}\to X_g$0 is always defined and equals $m_g:X_{g^{-1}}\to X_g$1 (Uzcategui et al., 2017).

For actions on sets, the domains $m_g:X_{g^{-1}}\to X_g$2 are arbitrary subsets. For actions on topological spaces, one requires each $m_g:X_{g^{-1}}\to X_g$3 to be open and each $m_g:X_{g^{-1}}\to X_g$4 to be a homeomorphism (Uzcategui et al., 2017, Gómez et al., 2017). For actions on algebras, one typically requires $m_g:X_{g^{-1}}\to X_g$5 to be ideals and $m_g:X_{g^{-1}}\to X_g$6 to be algebra isomorphisms, with the same formal compatibility conditions as in the set-theoretic case (Batista, 2016, Khrypchenko et al., 2016). In unital algebraic settings, the relevant ideals are often of the form $m_g:X_{g^{-1}}\to X_g$7 for central idempotents $m_g:X_{g^{-1}}\to X_g$8, a hypothesis that is central for globalization and crossed-product constructions (Batista, 2016).

A related but more algebraic presentation is via partial representations. A map $m_g:X_{g^{-1}}\to X_g$9 into a semigroup $X_1=X$0 is a partial homomorphism if

$X_1=X$1

$X_1=X$2

$X_1=X$3

for all $X_1=X$4 (Dokuchaev et al., 2017, Batista, 2016). Partial actions on sets are exactly partial representations into the symmetric inverse semigroup $X_1=X$5 of partial bijections of $X_1=X$6 (Batista, 2016). This identifies partial group actions with actions of an inverse semigroup encoding the partial symmetry data.

This perspective is sharpened by Exel’s universal inverse semigroup $X_1=X$7, generated by symbols $X_1=X$8 subject to the partial-representation identities. Any partial representation $X_1=X$9 factors uniquely through a semigroup homomorphism $m_1=\mathrm{id}_X$0 (Batista, 2016), and partial actions of $m_1=\mathrm{id}_X$1 on commutative monoids are equivalent to actions of $m_1=\mathrm{id}_X$2 in the sense of inverse semigroup modules (Dokuchaev et al., 2013). This suggests that “actions of partial groups” are often best understood as ordinary actions of a universal inverse semigroup associated to $m_1=\mathrm{id}_X$3, rather than as actions of a separate group-like object (Batista, 2016, Dokuchaev et al., 2013).

2. Globalization and enveloping actions

The central structural problem is whether a partial action is the restriction of a global action on a larger space or algebra. For sets, every partial action admits an admissible globalization. One forms an equivalence relation on $m_1=\mathrm{id}_X$4 by

$m_1=\mathrm{id}_X$5

defines the quotient

$m_1=\mathrm{id}_X$6

and lets $m_1=\mathrm{id}_X$7 act globally by

$m_1=\mathrm{id}_X$8

The embedding $m_1=\mathrm{id}_X$9 identifies the original partial action with the restriction of the global one (Batista, 2016, Uzcategui et al., 2017, Gómez et al., 2017). In topological language, this quotient is the enveloping space or globalization (Uzcategui et al., 2017).

For continuous partial actions of topological groups, the same quotient construction yields a continuous global action on $m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$0 and an embedding

$m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$1

with the property that the original partial action is recovered from the restriction of the global action to $m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$2 (Uzcategui et al., 2017). When $m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$3 is open in $m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$4, $m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$5 is open in $m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$6 (Uzcategui et al., 2017, Gómez et al., 2017). However, the quotient topology on $m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$7 can be pathological: it may fail to be Hausdorff, metrizable, or Polish (Uzcategui et al., 2017).

In algebraic categories, globalization is more delicate. For partial actions on algebras, an enveloping action $m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$8 consists of a global action on $m_g(X_{g^{-1}}\cap X_h)=X_g\cap X_{gh}$9, an algebra monomorphism $m_gm_h=m_{gh}$0, the condition $m_gm_h=m_{gh}$1, and admissibility $m_gm_h=m_{gh}$2 (Batista, 2016). A basic theorem states that a partial action on an algebra admits an enveloping action if and only if it is unital, meaning that every domain ideal is generated by a central idempotent (Batista, 2016).

The categorical refinement of globalization is developed in terms of reflectors. The category of global actions of $m_gm_h=m_{gh}$3 on sets is reflective inside the category of partial actions, and the universal globalization is the reflector (Khrypchenko et al., 2016). This extends to relational systems and to partial algebras. For algebras in a fixed variety $m_gm_h=m_{gh}$4, one always has a reflector into global $m_gm_h=m_{gh}$5-actions, but that reflector need not be a globalization; the obstruction is that the universal construction may identify distinct generators. The criterion is that the only identifications among letters $m_gm_h=m_{gh}$6 in the universal globalization be those already present in the set-level quotient $m_gm_h=m_{gh}$7 (Khrypchenko et al., 2016).

More recent work extends this strategy to nonassociative varieties $m_gm_h=m_{gh}$8. There, a partial action induced by an ideal partial representation admits a canonical globalization built from the module

$m_gm_h=m_{gh}$9

with multiplication

$G$0

and global action by left translation on the group coordinate (Dokuchaev et al., 23 Apr 2026). For generalized partial actions, a free-algebra quotient $G$1 provides a universal global action in $G$2 (Dokuchaev et al., 23 Apr 2026). This suggests that the globalization problem is robust far beyond associative algebra.

3. Topological and descriptive-set-theoretic structure

For topological partial actions of Polish groups, the orbit structure and quotient theory admit strong descriptive-set-theoretic control. If $G$3 is a continuous partial action of a Polish group $G$4 on a Polish space $G$5 and $G$6 is $G$7 in $G$8, then the enveloping space $G$9 is a standard Borel space: there exists a Polish topology $X$0 on the underlying set of $X$1 extending the quotient topology such that the quotient Borel structure coincides with $X$2 (Uzcategui et al., 2017). The global enveloping action is then $X$3-measurable, so $X$4 becomes a Borel $X$5-space in the sense of Becker–Kechris (Uzcategui et al., 2017).

The proof passes through a second partial action $X$6 on $X$7, defined by

$X$8

whose orbit equivalence relation coincides with the relation defining the enveloping quotient (Uzcategui et al., 2017). Because the $X$9-orbits are $m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$0, the equivalence relation is smooth, and because it is also idealistic, it admits a Borel selector. The quotient can therefore be represented by a Borel transversal $m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$1, and the Polish topology on $m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$2 can be transported to $m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$3 (Uzcategui et al., 2017).

This result is complemented by partial-action versions of Burgess’s theorem and Vaught transforms. For a partial action of a Polish group on a Polish space, the orbit equivalence relation is idealistic: each orbit $m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$4 carries a $m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$5-ideal

$m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$6

and the associated Borel regularity condition is verified using Vaught transforms for partial actions (Uzcategui et al., 2017). If the orbit relation is smooth, then it has a Borel selector (Uzcategui et al., 2017). This is the partial-action analogue of the total-action Burgess theorem.

The corresponding Vaught transforms are defined, for a nonempty open $m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$7, by

$m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$8

$m:G\times X\to X,\qquad (g,x)\mapsto g\cdot x,$9

and retain the expected closure and Borelness properties from the global theory (Uzcategui et al., 2017). These transforms are used to transfer meagerness and comeagerness conditions along partial orbits, which is essential in the analysis of orbit equivalence.

The orbit theory of a partial action and that of its globalization are not merely related; they are Borel bireducible. If $X$00 is the orbit relation on $X$01 and $X$02 the orbit relation of the global enveloping action on $X$03, then

$X$04

in the sense of Borel reducibility (Uzcategui et al., 2017). The embedding $X$05 gives one reduction, and a Borel selector on $X$06 yields the other. This shows that, from the viewpoint of invariant descriptive set theory, partial actions of Polish groups are no more complicated than global Borel actions once their standard Borel globalization is taken into account (Uzcategui et al., 2017).

4. Orbits, open mapping, and homogeneous-space phenomena

Partial actions admit orbit and stabilizer notions parallel to those for global actions. For $X$07, one defines

$X$08

$X$09

$X$10

(Gómez et al., 2017). The orbit equivalence relation is

$X$11

(Gómez et al., 2017).

For transitive continuous partial actions of Polish groups on non-meager Hausdorff spaces, an open mapping principle holds. If $X$12 is open in $X$13 for some $X$14, then for every $X$15 the orbit map

$X$16

is open (Gómez et al., 2017). The proof adapts Baire-category arguments from the global case to the partial domains $X$17, using local orbit sets $X$18 built from a neighborhood basis at the identity (Gómez et al., 2017). This is a partial-action extension of the classical open mapping principle for transitive Polish group actions.

Effros-type theorems also extend. For a continuous partial action of a Polish group on a Polish space, the following are equivalent: the orbit equivalence relation is $X$19 in $X$20; every orbit is $X$21 in $X$22; and the quotient $X$23 is $X$24 (Gómez et al., 2017). For a fixed point $X$25, if $X$26 is $X$27 in $X$28 and $X$29 is closed, then the partial orbit space $X$30 is Polish and the map

$X$31

is continuous (Gómez et al., 2017). Under the extra assumption that $X$32 is open, the orbit $X$33 is $X$34 if and only if it is not meager in itself, if and only if $X$35 is a homeomorphism (Gómez et al., 2017). This is the partial-action analogue of the classical homogeneous-space representation of Polish orbits.

For transitive partial actions, the globalization is especially rigid. The enveloping action on $X$36 is transitive, and $X$37 is equivalent to the left coset action on $X$38 (Gómez et al., 2017). If $X$39 is Hausdorff and $X$40 is open, then $X$41 is Polish and homeomorphic to $X$42, while $X$43 itself is homeomorphic to the partial homogeneous space $X$44 (Gómez et al., 2017). This shows that many partial actions of Polish groups retain the local homogeneous structure of global actions, with the only modification being that the acting set of group elements is the domain $X$45 rather than all of $X$46.

Concrete examples include Möbius transformations on $X$47, where each matrix acts by a fractional linear transformation on the domain where the denominator is nonzero, and flows of vector fields, where the time-$X$48 map is defined only on points whose integral curves exist at time $X$49 (Gómez et al., 2017). These illustrate that partial actions capture incomplete flows and partially defined geometric symmetries in a way compatible with the classical topological orbit theory.

5. Algebraic structure, Morita theory, and generalized matrix rings

Partial actions on rings and algebras support a rich extension of skew group constructions, Morita theory, and Galois theory. Given a partial action $X$50 of a group $X$51 on an algebra $X$52, the partial skew group algebra is

$X$53

with multiplication

$X$54

or, in the unital case,

$X$55

(Batista, 2016). This construction is associative in broad settings and becomes Morita equivalent to the corresponding global skew group algebra whenever an enveloping action exists (Batista, 2016).

A systematic matrix-theoretic framework is developed for generalized matrix rings $X$56 (Bagio et al., 2023). If each diagonal ring $X$57 carries a partial action $X$58 of a group $X$59, and the off-diagonal bimodules $X$60 satisfy a symmetry condition

$X$61

for each domain ideal, then the block matrix

$X$62

is an ideal of $X$63, and one can assemble compatible additive bijections on the blocks into a partial action $X$64 on the whole generalized matrix ring (Bagio et al., 2023). The resulting action restricts to the original $X$65 on each diagonal corner (Bagio et al., 2023).

This framework is particularly effective for Morita theory. The off-diagonal bimodules become $X$66-bimodules in the sense of Morita equivalence of partial actions, and under strict Morita-context hypotheses the diagonal partial actions are Morita equivalent (Bagio et al., 2023). Conversely, Morita equivalent regular partial actions can be encoded into a single partial action on a Morita ring or generalized matrix ring (Bagio et al., 2023). This suggests that generalized matrix rings provide a natural ambient object in which partial actions on Morita equivalent algebras can be studied simultaneously.

The groupoid case reduces to the group case in a similar spirit. For a connected groupoid $X$67, a transversal $X$68 and the isotropy group $X$69 at a chosen object $X$70, one can extract a datum consisting of ideals indexed by objects, transport isomorphisms along the transversal, and a partial action of the group $X$71 on the corner at $X$72 (Bagio et al., 2018). From such data one reconstructs a partial action of the whole groupoid; conversely, every partial groupoid action is an extension of a lifted one arising from a partial action of an isotropy group (Bagio et al., 2018). Under $X$73-globality and finiteness of the object set, the corresponding partial skew groupoid ring is isomorphic to a partial skew group ring over $X$74 (Bagio et al., 2018). This identifies connected groupoid actions as group actions in disguise, but with extra transport bookkeeping.

Galois and separability properties are preserved by these constructions. For a unital partial action $X$75 on a generalized matrix ring $X$76, the invariant subring has matrix form

$X$77

and the extension $X$78 is separable if and only if each diagonal extension $X$79 is separable (Bagio et al., 2023). Likewise, $X$80 is a partial Galois extension if and only if every diagonal extension is partial Galois (Bagio et al., 2023). This blockwise reflection principle is typical of the theory: generalized matrix constructions preserve the essential structure of the component partial actions rather than introducing new asymmetries.

6. Cohomology, homology, and universal algebraic structures

Partial actions admit intrinsic homology and cohomology theories. For a commutative monoid $X$81 with a unital partial action $X$82 of $X$83, the cochain group $X$84 consists of maps

$X$85

taking values in the unit group of the ideal

$X$86

and the coboundary operators are modified by the domain idempotents (Dokuchaev et al., 2013). The resulting cohomology groups $X$87 generalize ordinary group cohomology (Dokuchaev et al., 2013). In degree two, they classify twisted partial actions and embed into the partial Schur multiplier $X$88 (Dokuchaev et al., 2013, Dokuchaev et al., 2017).

The partial Schur multiplier is not a single group but a semilattice of groups. It is described cohomologically in terms of the universal inverse semigroup $X$89 and ideals $X$90 containing a canonical ideal $X$91 (Dokuchaev et al., 2017). For each such ideal one defines a second partial cohomology group

$X$92

and the disjoint union over all $X$93 forms a semilattice of groups

$X$94

(Dokuchaev et al., 2017). For a field $X$95,

$X$96

so the components of the partial Schur multiplier are exactly the partial $X$97-groups indexed by ideals in $X$98 (Dokuchaev et al., 2017). These groups classify $X$99-cancellative central extensions of $m_g:X_{g^{-1}}\to X_g$00, making partial Schur theory a cohomology theory of partial symmetry semigroups rather than only of groups (Dokuchaev et al., 2017).

A homological counterpart for partial representations has been developed using simplicial methods. If $m_g:X_{g^{-1}}\to X_g$01 is a partial representation, there is a canonical partial action on $m_g:X_{g^{-1}}\to X_g$02, a universal globalization

$m_g:X_{g^{-1}}\to X_g$03

and an isomorphism

$m_g:X_{g^{-1}}\to X_g$04

where the left-hand side is partial group homology and the right-hand side is ordinary group homology with coefficients in the universal globalization (Jerez, 2024). Dually, there is a cohomological spectral sequence converging to partial cohomology

$m_g:X_{g^{-1}}\to X_g$05

(Jerez, 2024). This identifies partial group homology as ordinary homology after passing to a universal globalization, reinforcing the idea that partiality can often be resolved by a canonical enlargement.

For free groups acting partially on totally disconnected spaces, the associated action groupoids admit especially simple homological behavior. If $m_g:X_{g^{-1}}\to X_g$06 acts semi-saturatedly on a compact Hausdorff totally disconnected space $m_g:X_{g^{-1}}\to X_g$07, then the groupoid $m_g:X_{g^{-1}}\to X_g$08 has cohomological dimension at most $m_g:X_{g^{-1}}\to X_g$09, and its homology is computed by the explicit two-term complex

$m_g:X_{g^{-1}}\to X_g$10

(Steinberg, 16 Feb 2026). Consequently,

$m_g:X_{g^{-1}}\to X_g$11

$m_g:X_{g^{-1}}\to X_g$12

(Steinberg, 16 Feb 2026). This applies in particular to Deaconu–Renault groupoids and shows that partial free-group actions can have “free-like” homological dimension despite their local domains (Steinberg, 16 Feb 2026).

7. Finite actions, orbit–stabilizer formulas, and semigroup criteria

For finite groups acting partially on sets, the theory admits concrete orbit-counting formulas. If $m_g:X_{g^{-1}}\to X_g$13 is a partial action of a finite group $m_g:X_{g^{-1}}\to X_g$14 on a set $m_g:X_{g^{-1}}\to X_g$15, the partial orbit of $m_g:X_{g^{-1}}\to X_g$16 is

$m_g:X_{g^{-1}}\to X_g$17

the partial stabilizer is

$m_g:X_{g^{-1}}\to X_g$18

and the set of elements defined at $m_g:X_{g^{-1}}\to X_g$19 is

$m_g:X_{g^{-1}}\to X_g$20

(Sharma et al., 2016). The generalized orbit–stabilizer theorem states that $m_g:X_{g^{-1}}\to X_g$21 is partially $m_g:X_{g^{-1}}\to X_g$22-isomorphic to $m_g:X_{g^{-1}}\to X_g$23, hence

$m_g:X_{g^{-1}}\to X_g$24

(Sharma et al., 2016). If $m_g:X_{g^{-1}}\to X_g$25 is a globalization, then the full global orbit satisfies

$m_g:X_{g^{-1}}\to X_g$26

so the difference between the global and partial orbit sizes is exactly the contribution of the undefined group elements (Sharma et al., 2016). This gives a finite combinatorial picture of how partiality enlarges to globality.

The globalization problem has a precise algebraic solution in many varieties. For partial actions on algebras in a variety $m_g:X_{g^{-1}}\to X_g$27, one constructs a generalized amalgam $m_g:X_{g^{-1}}\to X_g$28 from copies of the algebra indexed by $m_g:X_{g^{-1}}\to X_g$29, glued along the domain subalgebras using the partial isomorphisms (Khrypchenko et al., 2016). The partial action globalizes in $m_g:X_{g^{-1}}\to X_g$30 if and only if this amalgam is embeddable in an algebra of $m_g:X_{g^{-1}}\to X_g$31 (Khrypchenko et al., 2016). In semigroups with ideal domains, there is an internal criterion: $m_g:X_{g^{-1}}\to X_g$32 for all $m_g:X_{g^{-1}}\to X_g$33, $m_g:X_{g^{-1}}\to X_g$34, and $m_g:X_{g^{-1}}\to X_g$35; this is equivalent to globalizability (Khrypchenko et al., 2016). If the domains are unital ideals $m_g:X_{g^{-1}}\to X_g$36, the globalization can be written explicitly on the quotient set $m_g:X_{g^{-1}}\to X_g$37 with multiplication

$m_g:X_{g^{-1}}\to X_g$38

(Khrypchenko et al., 2016). Inverse semigroups satisfy the relevant idempotency conditions automatically, so partial actions with ideal domains globalize in that setting (Khrypchenko et al., 2016).

These results clarify a possible misconception: partial actions are not merely restrictions of global actions in an ad hoc sense. In many categories the restriction viewpoint is correct, but the existence of a globalization depends on strong internal conditions such as unitality of ideals, embeddability of an associated amalgam, or semigroup identities like the one above (Batista, 2016, Khrypchenko et al., 2016). A plausible implication is that the success of globalization is controlled as much by the ambient category as by the local symmetry data.

8. Conceptual synthesis

The modern theory of actions of partial groups is therefore not centered on a single formalism, but on a network of equivalent or complementary models. At the set-theoretic level, a partial group action is a family of partial bijections satisfying group-compatibility on their common domains (Uzcategui et al., 2017, Batista, 2016). At the inverse-semigroup level, it is a global action of Exel’s semigroup $m_g:X_{g^{-1}}\to X_g$39 or of the symmetric inverse semigroup $m_g:X_{g^{-1}}\to X_g$40 (Batista, 2016, Dokuchaev et al., 2013). At the groupoid level, it is encoded by an action groupoid whose arrows are the defined instances of group elements acting on points (Batista, 2016, Gómez et al., 2017, Steinberg, 16 Feb 2026). At the algebraic level, it gives rise to partial crossed products, partial skew group rings, and partial central extensions classified by partial cohomology (Batista, 2016, Dokuchaev et al., 2017, Dokuchaev et al., 2013). At the descriptive-set-theoretic level, it admits Borel globalizations and orbit equivalence relations comparable to those of global Polish actions (Uzcategui et al., 2017, Gómez et al., 2017). At the categorical level, it is often the object reflected into the subcategory of global actions, with globalization measured by the injectivity of that reflection (Khrypchenko et al., 2016).

Although many sources do not use the phrase “partial groups” explicitly, the cumulative picture is precise. A total group $m_g:X_{g^{-1}}\to X_g$41 acting partially gives rise to a partial symmetry object—typically an inverse semigroup, groupoid, or algebroid—that behaves like a “partial group of transformations” (Batista, 2016). Globalization theorems then assert that these partial symmetries can often be realized as restrictions of full symmetries on larger objects (Uzcategui et al., 2017, Batista, 2016, Dokuchaev et al., 23 Apr 2026). Cohomology and homology theories show that these partial symmetries have intrinsic classification and deformation invariants, not reducible merely to those of the original group (Dokuchaev et al., 2017, Dokuchaev et al., 2013, Jerez, 2024).

This suggests a unifying interpretation: actions of partial groups are best understood as actions of local symmetries with controlled domain data, together with a suite of globalization, orbit, Morita, and homological mechanisms that allow one to compare them systematically to global group actions.