Invariant sets and normal subgroupoids of universal \'etale groupoids induced by congruences of inverse semigroups

For a given inverse semigroup, one can associate an \'etale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated \'etale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept to consider quotients of inverse semigroup. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization.


Introduction
The relation among inverse semigroups, étale groupoids and C*-algebras has been revealed by many researchers. Paterson associated the universal groupoid to an inverse semigroup in [4]. He proved that the C*-algebras associated to universal groupoids are isomorphic to inverse semigroup C*-algebras. Furthermore, Paterson showed that the universal groupoid has a universal property about ample actions on totally disconnected spaces (also see [9]). In this paper we investigate a relation between inverse semigroups and universal groupoids. In particular, we focus on congruences of inverse semigroups, which is a fundamental concept for considering quotient inverse semigroups. Indeed, congruences of inverse semigroups induce closed invariant subsets and normal subgroupoids of universal groupoids. Our main theorem is that This work was supported by JSPS KAKENHI 20J10088. © Australian Mathematical Publishing Association Inc. 2021. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Since the element γ in (5) is uniquely determined by γ, γ is called the inverse of γ and denoted by γ −1 . We call G (0) the unit space of G. A subgroupoid of G is a subset of G that is closed under inversion and multiplication. For U ⊂ G (0) , we define G U · · = d −1 (U) and G U · · = r −1 (U). We define also G x · · = G {x} and G x · · = G {x} for x ∈ G (0) . The isotropy bundle of G is denoted by , G is called a group bundle over G (0) . A group bundle G is said to be abelian if G x is an abelian group for all x ∈ G (0) . A topological groupoid is a groupoid equipped with a topology where the multiplication and the inverse are continuous. A topological groupoid is said to be étale if the source map is a local homeomorphism. Note that the range map of an étale groupoid is also a local homeomorphism. The next proposition easily follows from the definition of étale groupoids. 2.2.1. Quotient étale groupoids. We recall the notion of quotient étale groupoids. See [1, Section 3] for more details. Let G be an étale groupoid. We say that a subgroupoid H ⊂ G is normal if: For a normal subgroupoid H ⊂ G, define an equivalence relation ∼ on G by declaring that α ∼ β if d(α) = d(β) and αβ −1 ∈ H hold. Then G/H · · = G/∼ becomes a groupoid such that the quotient map is a groupoid homomorphism. If H is open in G, then G/H is an étale groupoid with respect to the quotient topology. Moreover, the quotient map is a local homeomorphism (see [1, Section 3.1] for these facts).
We have the fundamental homomorphism theorem. The proof is left to the reader.
For an étale groupoid G, the author of [1] constructed the étale abelian group bundle G ab . We briefly recall the definition of G ab . We The reason why we consider G ab is that the next theorem holds. We denote the universal groupoid C*-algebra of G by C * (G). See [4] for the definition of the universal groupoid C*-algebras. THEOREM 2.3 [1,Theorem 4.12]. Let G be an étale groupoid such that G (0) is a locally compact Hausdorff space with respect to the relative topology of G. Then C * (G ab ) is the abelianization of C * (G) as a C*-algebra.
2.3. Étale groupoids associated to inverse semigroup actions. Let X be a topological space. We denote by I X the inverse semigroup of homeomorphisms between open sets in X. An action α : S X is a semigroup homomorphism S s → α s ∈ I X . For e ∈ E(S), we denote the domain of β e by D α e . Then α s is a homeomorphism from D α s * s to D α ss * . In this paper, we always assume that e∈E(S) D α e = X holds. For an action α : S X, we associate an étale groupoid S β X as follows. First, we put the set S * X · · = {(s, x) ∈ S × X | x ∈ D α s * s }. Then we define an equivalence relation ∼ on S * X by (s, x) ∼ (t, y) if x = y and there exists e ∈ E(S) such that x ∈ D α e and se = te. Set S α X · · = S * X/∼ and denote the equivalence class of (s, e . The source and range maps are defined by . Then S α X is a groupoid with these operations. For s ∈ S and an open set U ⊂ D α s * s , define These sets form an open basis of S α X. With this structure, S α X is an étale groupoid. Let S be an inverse semigroup. Now we define the spectral action β : S E(S).  for e ∈ E(S) and a finite subset P ⊂ E(S). Then these sets form a basis for the topology on E(S). For e ∈ E(S), we define D β e · · = {ξ ∈ E(S) | ξ(e) = 1}. For each s ∈ S and ξ ∈ D β s * s , define β s (ξ) ∈ D β ss * by β s (ξ)(e) = ξ(s * es), where e ∈ E(S). Then β is an action β : S E(S), which we call the spectral action of S. Now the universal groupoid of S is defined to be G u (S) · · = S β E(S).

Certain least congruences
It is known that every inverse semigroup admits the least Clifford congruence and the least commutative congruence. For example, see [5, Ch. III, Proposition 6.7] for the least Clifford congruence and [6] for the least commutative congruence. In this section, we reprove that every inverse semigroup admits the least Clifford congruence and the least commutative congruence by a new method using the spectrum.

Invariant subset of E(S). Let S be an inverse semigroup. A subset
Note that F is invariant if and only if F is invariant as a subset of the groupoid G u (S). We omit the proof of the next proposition. Let S be an inverse semigroup and ρ be a normal congruence on E(S). Moreover, let q : E(S) → E(S)/ρ denote the quotient map. For ξ ∈ E(S)/ρ, we define q(ξ) ∈ E(S) by q(ξ)(e) = ξ(q(e)), where e ∈ E(S). Note that q(ξ) is not zero since q is surjective. Then q : E(S)/ρ → E(S) is a continuous map by the definition of the topology of pointwise convergence. One can see that We say that F ⊂ E(S) is multiplicative if the multiplication of two elements in F also belongs to F whenever it is not zero. PROOF. It is easy to show that F ρ ⊂ E(S) is a closed multiplicative set. We show that F ρ ⊂ E(S) is invariant. Take ξ ∈ F ρ and s ∈ S with ξ(s * s) = 1. By the definition of F ρ , there exists η ∈ E(S)/ρ such that ξ = η • q, where q : E(S) → E(S)/ρ denotes the quotient map. Then for all e ∈ E(S). Therefore, it follows that β s To show the reverse inclusion, assume that (e, f ) ∈ ρ F ρ . Define η q(e) ∈ E(S)/ρ by We say that F ⊂ E(S) is unital if F contains the constant function 1.

LEMMA 3.4. Let S be an inverse semigroup and F ⊂ E(S) be a unital multiplicative set. Assume that F separates E(S) (that is, for e, f ∈ E(S), e = f is equivalent to the condition that ξ(e) = ξ( f ) holds for all ξ ∈ F). Then F is dense in E(S).
PROOF. For e ∈ E(S) and a finite subset P ⊂ E(S), we define

Recall that these sets form an open basis of E(S).
Observe that N e P = N e eP holds, where eP · · = {ep ∈ E(S) | p ∈ P}. Now it suffices to show that F ∩ N e P ∅ holds for nonempty N e P such that p ≤ e holds for all p ∈ P. [7] Quotients of universal étale groupoids 105 In the case that P = ∅, the constant function 1 belongs to F ∩ N e P . We may assume that p ≤ e holds for all p ∈ P. Since N e P is nonempty, we have e p for all p ∈ P. Since F separates E(S), there exists ξ p ∈ F such that ξ p (e) = 1 and ξ p (p) = 0 for each p ∈ P. Define ξ · · = p∈P ξ p ; then ξ ∈ N e P ∩ F. PROPOSITION 3.5. Let S be an inverse semigroup. Then F = F ρ F holds for every unital multiplicative invariant closed set F ⊂ E(S).

. Let S be an inverse semigroup. There is a one-to-one correspondence between normal congruences on E(S) and unital multiplicative invariant closed sets in E(S).
PROOF. Just combine Propositions 3.3 and 3.5.
3.2. The least Clifford congruences. Let S be an inverse semigroup. Recall that a congruence ρ on S is said to be Clifford if S/ρ is Clifford. For example, S × S is a Clifford congruence on S. In this subsection, we prove that every inverse semigroup admits the least Clifford congruence (Theorem 3.11). Our construction of the congruence is based on the fixed points of E(S). DEFINITION 3.7. Let S be an inverse semigroup. A character ξ ∈ E(S) is said to be fixed if ξ(s * es) = ξ(e) holds for all e ∈ E(S) and s ∈ S such that ξ(s * s) = 1. We denote the set of all fixed characters by E(S) fix .
One can see that E(S) fix is a closed subset of E(S). Moreover, E(S) fix is a multiplicative set. The fixed characters are characterized in the next proposition.
for all s ∈ S. Therefore, a semigroup homomorphism extension of ξ is unique if it exists.
It is obvious that ξ is fixed if ξ has a semigroup homomorphism extension. Assume Thus, ξ is a semigroup homomorphism. DEFINITION 3.9. Let S be an inverse semigroup. We define the normal congruence ρ Clif · · = ρ E(S) fix on E(S). Furthermore, we define the congruence ν Clif · · = ν ρ Clif ,min on S and S Clif · · = S/ν Clif . LEMMA 3.10. Let S be an inverse semigroup, ν be a Clifford congruence on S and q : S → S/ν be the quotient map. Then a set PROOF. Take ξ ∈ E(S/ν) and assume that ξ(q(s * s)) = 1 for some s ∈ S. For all e ∈ E(S), Applying what we have shown for the trivial congruence where e ∈ E(S). Since we assume that E(S) fix = E(S) and ξ s * s (s * s) = 1, Then we have s * s ≤ ss * . It follows that s * s ≥ ss * from the same argument. Now we have s * s = ss * and S is Clifford. Now we show that every inverse semigroup admits the Cliffordization.
THEOREM 3.11. Let S be an inverse semigroup. Then ν Clif is the least Clifford congruence on S.
PROOF. First, we show that the congruence ν Clif is Clifford. Take s ∈ S and ξ ∈ E(S) fix . Then one can see that ξ(s * s) = ξ(ss * ). Therefore, (s * s, ss * ) ∈ ν Clif and ν Clif is a Clifford congruence. Let ν be a Clifford congruence and q : S → S/ν be the quotient map. To show that The reverse inequality is obtained symmetrically and therefore q(t * t) = q(s * s) holds. Let η ∈ E(S/ν Clif ) be the above character. Since η • q is a fixed character and (s, t) ∈ ν Clif , there exists e ∈ E(S) such that η • q(e) = 1 and se = te hold. Since η • q(e) = 1, we have q(e) ≥ q(s * s) = q(t * t) by the definition of η. Now we have q(s) = q(s)q(e) = q(t)q(e) = q(t). Therefore, (s, t) ∈ ν. 3.3. The least commutative congruences. We say that a congruence on an inverse semigroup is commutative if the quotient semigroup is commutative. In this subsection, we show that every inverse semigroup admits the least commutative congruence. We denote the circle group by T · · = {z ∈ C | |z| = 1}. We view T ∪ {0} as an inverse semigroup with the usual products. By S, we denote the set of all semigroup homomorphisms from S to T ∪ {0}. DEFINITION 3.13. Let S be an inverse semigroup. We define the commutative congruence ν ab on S as the set of all pairs (s, t) ∈ S × S such that ϕ(s) = ϕ(t) holds for all ϕ ∈ S. We define S ab · · = S/ν ab .
Let S be a Clifford inverse semigroup and e ∈ E(S). We define H e · · = {s ∈ S | s * s = e}. One can see that H e is a group with the operation inherited from S. Note that the unit of H e is e.
In order to show that ν ab is the least commutative congruence, we need the next lemma.
Then one can check that ϕ is a semigroup homomorphism extension of ϕ. PROOF. Assume that ν is a commutative congruence. Let q : S → S/ν denote the quotient map. In order to show that ν ab ⊂ ν, take (s, t) ∈ ν ab . First, we show that q(s * s) = q(t * t). It suffices to show that ξ(q(s * s)) = ξ(q(t * t)) holds for all ξ ∈ E(S/ν). Note that ξ • q ∈ E(S) is a fixed point by Lemma 3.10. Since ξ • q is a restriction of an element in S by Proposition 3.8, ξ(q(s * s)) = ξ(q(t * t)) follows from (s * s, t * t) ∈ ν ab .
is a closed invariant subset of G u (S) as shown in Proposition 3.2.
We omit the proof of the next proposition.

Via the map in the above proposition, we identify G u (H) with an open subgroupoid of G u (S).
Let S be an inverse semigroup, ν be a congruence on S and q : S → S/ν be the quotient map. Define ker ν · · = q −1 (E(S/ν)) ⊂ S. Then ker ν is a normal subsemigroup of S. Although G u (ker ν) is not necessarily a normal subgroupoid of G u (S), the following holds. PROOF. We know that G u (ker ν) F ν is an open normal subgroupoid of G u (S) F ν . We show that G u (ker ν) F ν is normal in G u (S) F ν . Let q : S → S/ν denote the quotient map.

THEOREM 4.3. Let S be an inverse semigroup and ν be a congruence on S. Then
PROOF. Let q : S → S/ν denote the quotient map. Note that a map Using Proposition 2.1, one can see that Φ is a groupoid homomorphism which is a local homeomorphism and injective on G u (S) (0) F ν . Observe that Φ is surjective. We show that ker Φ = G u (ker ν) F ν holds. The inclusion ker Φ ⊃ G u (ker ν) F ν is obvious. In order to show that ker Φ ⊂ G u (ker ν) F ν , take [s, q(ξ)] ∈ ker Φ. Since we have [q(s), ξ] ∈ G u (S/ν) (0) and q(E(S)) = E(S/ν), there exists e ∈ E(S) such that [q(s), ξ] = [q(e), ξ]. There exists f ∈ E(S) such that ξ(q( f )) = 1 and q(s)q( f ) = q(e)q( f ). Now we have s f ∈ ker ν, so it follows that This shows that ker Φ = G u (ker ν) F ν . By Proposition 2.2, Φ induces an isomorphism Φ that makes the following diagram commutative: where Q denotes the quotient map.

4.2.
Universal groupoids associated to special quotient inverse semigroups.

Minimum congruences associated to normal congruences on semilattices of idempotents.
Let S be an inverse semigroup. Recall that a congruence ρ on E(S) is normal if (e, f ) ∈ ρ implies (ses * , s f s * ) ∈ ρ for all s ∈ S and e, f ∈ E(S). Note that one can construct the least congruence ν ρ,min whose restriction to E(S) coincides with ρ.
Recall that we can associate the closed invariant subset F ρ of G u (S) as shown in Proposition 3.2.
Next we show that G u (ker is easy to show. Let q : S → S ab and q : S → S Clif denote the quotient maps. Since a commutative inverse semigroup is Clifford, there exists a semigroup homomorphism σ : S Clif → S ab such that q = σ • q . To show the reverse inclusion take [n, q(ξ)] ∈ G u (ker ν ab ) fix , where n ∈ ker ν ab and ξ ∈ E(S ab ). Since n ∈ ker ν ab , there exists e ∈ E(S) such that q(n) = q(e). Then we have q(n * n) = q(e). Since ν ab coincides with ν Clif on E(S), it follows that q (n * n) = q (e). Define then H q (e) is a group in the operation inherited from S Clif . Observe that a unit of H q (e) is q (e) and we have q (n) ∈ H q (e) . Fix a group homomorphism χ : H q (e) → T arbitrarily. By Proposition 3.14, χ is extended to the semigroup homomorphism χ : S Clif → T ∪ {0}. Since T ∪ {0} is commutative, there exists a semigroup homomorphism χ : S ab → T ∪ {0} that makes the following diagram commutative: Now χ(q (n)) = χ(q(n)) = χ(q(e)) = χ(q (e)).

Applications and examples
5.1. Clifford inverse semigroups from the viewpoint of fixed points. A 0-group is an inverse semigroup isomorphic to Γ ∪ {0} for some group Γ. For a group Γ, we denote the 0-group associated to Γ by Γ 0 · · = Γ {0}. It is clear that every 0-group is a Clifford inverse semigroup. Conversely, we see that every Clifford inverse semigroup is embedded into a direct product of 0-groups. We remark that this fact is already known (see [5,Theorem 2.6]). Using fixed characters, we obtain a new proof.

PROPOSITION 5.1. Let S be a Clifford inverse semigroup. Then the semigroup homomorphism
is injective. In particular, every Clifford inverse semigroup is embedded into a direct product of 0-groups.

PROPOSITION 5.2. Let S be a finitely generated Clifford inverse semigroup. Then E(S) is a finite set.
PROOF. Take a finite set F ⊂ S which generates S. Let X denote the set of all nonzero semigroup homomorphisms from S to {0, 1}. Then a map is injective since F generates S. By Proposition 3.8 and Lemma 3.10, the map

COROLLARY 5.3. Let S be a finitely generated Clifford inverse semigroup. Then S is embedded into a direct sum of finitely many 0-groups.
Let S be a Clifford inverse semigroup and ξ ∈ E(S). Recall that G u (S) ξ is a discrete group. In [2], the authors gave a way to calculate G u (S) ξ . Then ξ −1 ({1}) is a directed set with respect to the order inherited from E(S). S(e).
We give a way to realize lim − − →ξ(e)=1 S(e) as a quotient of S. PROPOSITION 5.5. Let S be a Clifford inverse semigroup and ξ ∈ E(S). Then we have the following isomorphism:
Combining Theorem 5.4 with Proposition 5.5, we obtain the next corollary.

COROLLARY 5.6. Let S be a Clifford inverse semigroup and ξ ∈ E(S). Then G u (S) ξ is isomorphic to S(ξ).
Let I be a discrete set and {Γ i } i∈I be a family of discrete groups. Then the disjoint union i∈I Γ i is a discrete group bundle over I in the natural way. Using Proposition 5.2 and Corollary 5.6, we obtain the next corollary. For an étale groupoid G with the locally compact Hausdorff unit space G (0) , we write C * (G) (respectively C * λ (G)) to represent the universal (respectively reduced) groupoid C*-algebra of G (see [4] for the definitions). Corollary 5.7 immediately implies the next corollary. COROLLARY 5.8. Let S be a finitely generated Clifford inverse semigroup. Then we have isomorphisms

Free Clifford inverse semigroups.
We investigate universal groupoids and C*-algebras associated to free Clifford inverse semigroups on finite sets. First, we recall the definition of the free groups. DEFINITION 5.9. Let X be a set. A free group on X is a pair (F(X), κ) consisting of a group F(X) and a map κ : X → F(X) such that: (1) κ(X) generates F(X); and (2) for every group Γ and a map ϕ : X → Γ, there exists a group homomorphism ϕ : F(X) → Γ such that ϕ(x) = ϕ(κ(x)) holds for all x ∈ X.
We define free inverse semigroups in a similar way.
DEFINITION 5.10. Let X be a set. A free inverse semigroup on X is a pair (FIS(X), ι) consisting of an inverse semigroup FIS(X) and a map ι : X → FIS(X) such that: (1) ι(X) generates FIS(X); and (2) for every inverse semigroup T and a map ϕ : X → T, there exists a semigroup homomorphism ϕ : FIS(X) → T such that ϕ(x) = ϕ(ι(x)) holds for all x ∈ X.
It is known that free inverse semigroups exist and are unique up to isomorphism. See [3, Section 6.1] for the existence of free inverse semigroups. The uniqueness is obvious. DEFINITION 5.11. A free Clifford inverse semigroup on X is a pair (FCIS(X), ι) consisting of a Clifford inverse semigroup FCIS(X) and a map ι : X → FCIS(X) such that: (1) ι(X) generates FCIS(X); and (2) for every Clifford inverse semigroup T and a map ϕ : X → T, there exists a semigroup homomorphism ϕ : FCIS(X) → T such that ϕ(x) = ϕ(ι(x)) holds for all x ∈ X.
Free Clifford inverse semigroups exist and are unique up to isomorphism. Indeed, for a free Clifford inverse semigroup (FIS(X), ι) and the quotient map q : FIS(X) → FIS(X) Clif , one can see that (FIS(X) Clif , q • ι) is a free Clifford inverse semigroup on X. The uniqueness is obvious.
Let X be a set. For A ⊂ X, define a map χ A : X → {0, 1} by Since {0, 1} is Clifford, χ A can be extended to the semigroup homomorphism from FCIS(X) to {0, 1}, which we also denote by χ A . Every semigroup homomorphism from FCIS(X) to {0, 1} is of the form χ A for a unique A ⊂ X. By Proposition 3.8, χ A | E(FCIS(X)) is a fixed character if A is not empty. By Lemma 3.10, all characters on E(FCIS(X)) are fixed characters. Therefore, we obtain the next proposition. PROPOSITION 5.12. Let X be a finite set. Put S = FCIS(X). Then the map is bijective, where P(X) denotes the power set of X.
We identify χ A | E(FCIS(X)) with χ A since we can recover χ A from the restriction For a nonempty set A ⊂ X, define e A · · = x∈A ι(x) * ι(x) ∈ E(FCIS(X)). For e ∈ E(FCIS(X)), the condition that χ A (e) = 1 is equivalent to the condition that e ≥ e A . Using this fact, one can prove the next proposition.
PROPOSITION 5.13. The map is bijective.
In order to apply Corollary 5.6 for free Clifford inverse semigroups, we prepare with the next proposition.
PROPOSITION 5.14. Let X be a set and A ⊂ X be a nonempty set. Put S = FCIS(X). Then S(χ A ) is isomorphic to the free group F(A) generated by A.
PROOF. If X = A, S(χ A ) is the maximal group image of S. Therefore, S(χ A ) is isomorphic to F(A).
We assume that A X. Let ϕ A : S → S(χ A ) 0 denote the map defined by where Q : S → S/ν χ A denotes the quotient map. By the universality of F(A), define a group homomorphism τ : F(A) → S(χ A ) such that τ(κ(a)) = ϕ A (ι(a)) for all a ∈ A.
We construct the inverse map of τ. Using the universality of S = FCIS(X), define a semigroup homomorphism σ : S → F(A) 0 that satisfies for x ∈ X. We claim that (s, t) ∈ ν χ A implies σ(s) = σ(t) for s, t ∈ S. If χ A (s * s) = 0, we have σ(s) = σ(t) = 0. We may assume that χ A (s * s) = 1. By (s, t) ∈ ν χ A , we have se A = te A . Since σ(e A ) is the unit of F(A), we have σ(s) = σ(t). Therefore, we obtain a semigroup homomorphism σ : S(χ A ) 0 → F(A) 0 that makes the following diagram commutative: Now one can verify that σ| S(χ A ) is the inverse map of τ.
Now we have the following theorem. By Proposition 5.14, we obtain the isomorphism in the statement. It is known that G u (S) has the following universal property for Boolean actions.
THEOREM 5.16 [9,Proposition 5.5]. Let S be an inverse semigroup, X be a Boolean space and α : S X be a Boolean action. Then S α X is isomorphic to G u (S) F for some closed invariant subset F ⊂ E(S).
COROLLARY 5.17. Let S be a finitely generated inverse semigroup and α : S X be a Boolean action. Then α has finitely many fixed points. More precisely, the number of fixed points of α is less than or equal to the number of nonzero semigroup homomorphisms from S to {0, 1}.
PROOF. Since we assume that S is finitely generated, the set of all nonzero semigroup homomorphisms from S to {0, 1} is a finite set. By Proposition 3.8, there exists a bijection between the set of all nonzero semigroup homomorphisms from S to {0, 1} and E(S) fix . Now Theorem 5.16 completes the proof. for all i, j ∈ {1, 2, . . . , n}. Define ξ : S n → {0, 1} by ξ(x) = 1 for all x ∈ S n . Then ξ is the unique nonzero semigroup homomorphism from S n to {0, 1}. Since 0 ∈ S n , ξ is an isolated point of E(S). Therefore, every Boolean action of S n has at most one fixed point, which becomes an isolated point.