Automatic Continuity of Homomorphisms Between Topological Inverse Semigroups Received April 26, 2017; Accepted August 27, 2017

Topol. Algebra Appl. 2018; 6:60–66

Research Article

Iryna Pastukhova* Automatic continuity of homomorphisms between topological inverse semigroups https://doi.org/10.1515/taa-2018-0004 Received April 26, 2017; accepted August 27, 2017

Abstract: We nd conditions on topological inverse semigroups X, Y guaranteeing the continuity of any homomorphism h : X → Y having continuous restrictions to any subsemilattice and any subgroup of X.

Keywords: topological inverse semigroup

MSC: 22A15, 22A26, 54C08, 54E52

1 Introduction

This paper was motivated by the following old result of Yeager [10] who generalized an earlier result of Bowman [4].

Theorem 1.1 (Yeager). A homomorphism h : X → Y between compact topological inverse Cliord semigroups is continuous if and only if for any subgroup H ⊂ X and any subsemilattice E ⊂ X the restrictions h|H and h|E are continuous.

Let us dene a homomorphism h : X → Y between topological semigroups to be EH-continuous if – the restriction h|EX to the set of idempotents of X is continuous; – for every subgroup H ⊂ X the restriction h|H is continuous.

In terms of EH-continuity, Theorem 1.1 says that each EH-continuous homomorphism h : X → Y between compact topological inverse Cliord semigroups is continuous. For compact topological inverse Cliord semigroup X with Lawson maximal semilattice EX = {x ∈ X : xx = x} this result of Yeager was proved by Bowman [4] in 1971. In [3], the Bowman’s result was generalized to non-compact case. Namely, the authors proved the continuity of any EH-continuous homomorphism h : X → Y from a topological inverse Cliord U-semigroup X to a ditopological inverse semigroup Y. Another result of [3] states that each EH-continuous homomorphism h : X → Y from a topological inverse U0-semigroup X to a ditopological inverse semigroup Y is continuous. A topological inverse semigroup X is an U-semigroup (resp. U0-semigroup) if its maximal semilattice E(X) is a U-semilattice (resp. a U0-semilattice). Ditopological inverse semigroups were introduced and studied in [1]. In this paper we shall introduce weakly ditopological inverse semigroups and prove the * continuity of any EH-continuous homomorphism h : X → Y from a topological inverse U0-semigroup X to * a weakly ditopological inverse semigroup Y. Topological inverse U0-semigroups are introduced in Section 2 and weakly ditopological inverse semigroups are introduced in Section 3. In Example 3.4 we shall construct a weakly ditopological inverse semigroup, which is not ditopological. The main result (Theorem 4.1 on the continuity of EH-continuous homomorphisms) is proved in Section 4. In Section 5 we apply Theorem 4.1 to establish the continuity of Souslin-measurable homomorphisms between topological inverse semigroups.

*Corresponding Author: Iryna Pastukhova: Ivan Franko National University of Lviv, Universytetska 1, 79000, Lviv, Ukraine, E-mail: [email protected]

Open Access. © 2018 Iryna Pastukhova, published by De Gruyter. This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivs 4.0 License. Automatic continuity of homomorphisms between topological inverse semigroups Ë 61

2 Preliminaries

1.1. Semigroups. A semigroup is a non-empty set endowed with an associative binary operation. A semigroup S is called – inverse if for every x ∈ S there is a unique element x−1 ∈ S such that x = xx−1x and x−1 = x−1xx−1; – Cliord inverse if it is inverse and xx−1 = x−1x for every x ∈ S; – a semilattice if it is commutative and every element x ∈ S is an idempotent, that is xx = x.

Each semilattice E carries the natural partial order ≤ dened by x ≤ y i xy = yx = x. For a point x ∈ E let ↓x = {y ∈ E : y ≤ x} and ↑x = {y ∈ E : x ≤ y} be the lower and upper cones of x, respectively. For a semigroup S by E(S) = {e ∈ S : ee = e} we denote the set of idempotents of S and for each idempotent e ∈ E(S) let

He = {x ∈ S : ∃y ∈ S xy = e = yx, xe = x = ex, ye = y = ey} be the maximal subgroup of S containing e. If the semigroup S is inverse, then the maximal group He can be −1 −1 written as He = {x ∈ S : xx = e = x x}. Two elements x, y of an inverse semigroup S are called conjugated if there exists an element s ∈ S such that x = sys−1 and y = s−1xs. For an element x ∈ S let

S x = {y ∈ S : ∃s ∈ S y = sxs−1, x = s−1ys} be the conjugacy class of x in S. If x is an idempotent, then xS ⊂ E(S). Observe that for any element s ∈ S the idempotents ss−1 and s−1s are conjugated as s−1(ss−1)s = s−1s and s(s−1s)s−1 = ss−1. A homomorphism between semigroups X, Y is a function h : X → Y preserving the operation in the sense that h(x·y) = h(x)·h(y) for all x, y ∈ X. The uniqueness of the inverse element in an inverse semigroup implies that each homomorphism h : X → Y between inverse semigroups preserves the inversion in the sense that h(x−1) = h(x)−1 for all x ∈ X. More information on inverse semigroups can be found in [8]. A topological semigroup is a semigroup S endowed with a topology making the semigroup operation · : S × S → S continuous. A topological inverse semigroup is an inverse semigroup S endowed with a topology making the multiplication · : S × S → S and the inversion ()−1 : S → S continuous. A topological semilattice E is Lawson if open subsemilattices form a base of the topology of E. For an element e of a topological semilattice E by ⇑e we denote the interior of the upper cone ↑e in E. An element e ∈ E of a topological semilattice E is called locally minimal if its upper cone ↑e is open in E. In this case ⇑e = ↑e. More generally, we shall say that an idempotent e ∈ E is locally minimal in a subset A ⊂ E containing e if e has a neighborhood Ve ⊂ E such that Ve ∩ A ⊂ ↑e. A topological semilattice E is called a – a U-semilattice if for each point x ∈ E and its neighborhood U ⊂ E there exists an idempotent e ∈ U such that x ∈ ⇑e; – a U0-semilattice if for each point x ∈ E and its neighborhood U ⊂ E there exists a locally minimal idempotent e ∈ U such that x ∈ ↑e.

It is clear that each U0-semilattice is a U-semilattice. By [5, 2.12], each locally compact Lawson semilattice is a U-semilattice and each compact U-semilattice is Lawson. On the other hand, each locally compact zero- dimensional topological semilattice is a U0-semilattice, see [5, 2.6]. A topological inverse semigroup S is called a – a U-semigroup if its maximal semilattice E(S) is a U-semilattice; – a U0-semigroup if its maximal semilattice E(S) is a U0-semilattice; * – a U0-semigroup if for any idempotent x ∈ E(S) and any neighborhood U ⊂ E(S) of x there exists an idempotent e ∈ E(S) such that x ∈ ⇑e and e is locally minimal in its conjugacy class eS = {f ∈ E(S): −1 −1 S ∃s ∈ S f = ses , e = s fs} in the sense that for some neighborhood Ve ⊂ E(S) of e we get Ve ∩ e ⊂ ↑e. 62 Ë Iryna Pastukhova

It is clear that for each topological inverse semigroup S we get the implications

* U0-semigroup ⇒ U0-semigroup ⇒ U-semigroup.

* A topological inverse Cliord semigroup S is a U0-semigroup if and only if it is a U-semigroup. This follows from the observation that for any idempotent e ∈ E(S) the conjugacy class eS of e is the singleton {e} (as e commutes with all elements of the inverse Cliord semigroup S).

3 (Weakly) ditopological inverse semigroups

For two subsets A, B of a semigroup S consider the subsets

B[−1]A = {y ∈ S : ∃b ∈ B ∃a ∈ A by = a} and AB[−1] = {x ∈ S : ∃a ∈ A ∃b ∈ B a = xb} which can be thought as the results of left and right division of A by B in the semigroup S. For an inverse semigroup S consider the unary operations

−1 −1 λS : S → E(S), λ : x 7→ xx , and ρS : S → E(S), ρ : x 7→ x x, and observe that λS(x) · x = x = x · ρS(x) for all x ∈ S. The equality −1 [−1] −1 −1 −1 [−1] −1 −1 ({xx } {x}) ∩ λS (xx ) = {x} = ({x}{x x} ) ∩ ρS (x x) motivates the following denition, which was rst introduced in [1].

Denition 3.1. A topological inverse semigroup S is called ditopological if it satises one of the following equivalent conditions: – for any point x ∈ S and neighborhood Ox ⊂ S of x there exist a neighborhood Ux ⊂ S of x and a −1 neighborhood Wxx−1 ⊂ E(S) of the idempotent λS(x) = xx such that

[−1] ∩ −1 ⊂ (Wxx−1 Ux) λS (Wxx−1 ) Ox;

– for any point x ∈ S and neighborhood Ox ⊂ S of x there exist a neighborhood Ux ⊂ S of x and a −1 neighborhood Wx−1 x ⊂ E(S) of the idempotent ρS(x) = x x such that

[−1] ∩ −1 ⊂ (Ux Wx−1 x) ρS (Wx−1 x) Ox .

Ditopological inverse semigroups were introduced in [1] and studied in [1], [2], [3]. By [1], the class of ditopological inverse semigroups contains all compact topological inverse semigroups, all topological groups, all topological semilattices, and is closed under taking Cliord subsemigroups, Tychono products, reduced products and semidirect products. Next, we introduce a new notion of a weakly ditopological inverse semigroup.

Denition 3.2. A topological inverse semigroup S is dened to be weakly ditopological if it satises one of the following equivalent conditions: – for any point x ∈ S and neighborhood Ox ⊂ S of x there exist a neighborhood Ux ⊂ S of x and −1 −1 neighborhoods Wxx−1 , Wx−1 x ⊂ ES of the idempotents xx , x x such that

[−1] ∩ −1 ∩ −1 ⊂ (Wxx−1 Ux) λS (Wxx−1 ) ρS (Wx−1 x) Ox;

– for any point x ∈ S and a neighborhood Ox ⊂ S of x there exist a neighborhood Ux ⊂ S of x and −1 −1 neighborhoods Wxx−1 , Wx−1 x ⊂ ES of the idempotents xx , x x such that

[−1] ∩ −1 ∩ −1 ⊂ (Ux Wx−1 x) λS (Wxx−1 ) ρS (Wx−1 x) Ox . Automatic continuity of homomorphisms between topological inverse semigroups Ë 63

The equivalence of these two conditions follows from the continuity of the inversion in topological inverse semigroups. It is clear that each ditopological inverse semigroup S is weakly ditopological. In Example 3.4 we shall construct a weakly ditopological inverse semigroup which is not ditopological. If a topological inverse [−1] [−1] semigroup S is Cliord, then λS = ρS and A B = BA for any sets A ⊂ S and B ⊂ E(S), which implies the following characterization.

Proposition 3.3. A topological inverse Cliord semigroup is weakly ditopological if and only if it is ditopological.

Denition 3.2 implies that the class of weakly ditopological inverse semigroups is closed under taking inverse subsemigroups and Tychono products.

Example 3.4. There exists a metrizable topological semigroup S which is weakly ditopological but not ditopological.

Proof. The semigroup S is an inverse subsemigroup of the inverse semigroup HIk(R) of homeomorphisms f : dom(f ) → ran(f ) between compact subsets dom(f ) and ran(f) of the real line. For two elements f , g ∈ HIk(R) their composition f ◦g is the homeomorphism with dom(f ◦g) = g−1(dom(f)) and ran(f ◦g) = f(dom(f )∩ran(g)) dened by f ◦ g(x) = f (g(x)) for x ∈ dom(f ◦ g). We identify each homeomorphism f ∈ HIk(R) with its graph {(x, f(x)) : x ∈ dom(f )} ⊂ R × R and endow the semigroup HIk(R) with the Vietoris topology, which is generated by the subbase consisting of the sets {f ∈ HIk(R): f ⊂ U} and {f ∈ HIk(R): f ∩ U ≠ ∅} where U runs over open subsets of the plane R × R. It is well-known [7, 4.5.23] that the Vietoris topology on HIk(R) is metrizable. C P∞ 2xi x ∞ ∈ { }N ⊂ n ∈ ω Let = i=1 3i :( i)i=1 0, 1 [0, 1] be the standard Cantor set. For every consider the C {x ∈ C x 1 } C f C → closed-and-open subset n = : < 1 − 3n of the Cantor set and let n : R be the map dened by ( x + 2, if x ∈ Cn , fn(x) = x + n + 2, if x ∈ C \ Cn .

Let S be the inverse subsemigroup of HIk(R), generated by the set {fn : n ∈ ω}. Observe that for any n, m ∈ ω the composition fn ◦ fm is the empty homeomorphism (as ran(fn) ∩ dom(fm) = ∅). Also −1 −1 observe that fn fn is the identify homeomorphism eω of C. If n ≠ m, then fn fm coincides with the identity homeomorphism ek : Ck → Ck of the set Ck for k = min{n, m}. These observations imply that

−1 −1 S = ek , fn ek , ek fm , fn ek fm : n, m ∈ ω, k ∈ ω ∪ {ω} .

−1 −1 Next, observe that for n ≥ k we get fn ek = f0ek and ek fn = ek f0 . This implies that

−1 −1 S = ek , fn ek , ek fm , fn ek fm : n, m ≤ k ≤ ω .

It is easy to see that the sequence of idempotents (ek)k∈ω converges to the idempotent eω in the space S. −1 −1 −1 −1 Moreover, for every n, m ∈ ω the sequences (fn ek)k∈ω, (ek fm )k∈ω, (fn ek fm )k∈ω converge to fn, fm , fn fm , respectively. Using these facts it can be shown that S is a topological inverse semigroup. −1 To see that S is not ditopological, observe that the sequence (fn)n∈ω does not converge to f0. Yet fn fn = eω = limn→∞ en and fn en = f0en → f0 as n → ∞. To see that S is weakly ditopological, it suces to show that a sequence {xn}n∈ω ⊂ S contains a −1 −1 −1 −1 subsequence convergent to a point x ∈ S if limn→∞ xn xn = xx , limn→∞ xn xn = x x and limn→∞ zn xn = x −1 for some sequence of idempotents (zn)n∈ω, convergent to xx . Replacing (xn)n∈ω by a suitable subsequence, −1 we can assume that it is contained in one of the sets {ek : k ∈ ω}, {fn ek : n ≤ k}, {ek fm : m ≤ k}, −1 −1 −1 −1 {fn ek fm : n, m ≤ k}. If the idempotent xx is an isolated point of S, then zn = xx = xn xn for all suciently −1 −1 large numbers n and hence xn = xn xn xn = zn xn → x as n → ∞. So, we assume that the idempotent xx is −1 −1 −1 not isolated. Consequently, xx = eω or xx = fm fm for some m ∈ ω. 64 Ë Iryna Pastukhova

−1 −1 −1 −1 If xx = eω, then x = eω or x = fm for some m ∈ ω. If x = eω, then limn→∞ xn xn = xx = eω and −1 −1 −1 limn→∞ xn xn = x x = eω imply that xn ∈ {ek : k ∈ ω} for all suciently large n ∈ ω. Then xn = xn xn → xx−1 = x as n → ∞. −1 −1 −1 −1 −1 −1 −1 If x = fm for some m ∈ ω, then limn→∞ xn xn = xx = fm fm = eω and limn→∞ xn xn = x x = fm fm −1 −1 −1 imply that xn ∈ {ek fm : k ∈ ω} for all suciently large n ∈ ω. Then xn = ekn fm → fm = x as n → ∞. −1 −1 −1 Finally we assume that xx = fm fm for some m ∈ ω. Then x = fm or x = fm fl for some l ∈ ω. If x = fm, −1 −1 −1 −1 −1 −1 then limn→∞ xn xn = xx = fm fm and limn→∞ xn xn = x x = fm fm imply that {xn}n∈ω ⊂ {fm ek : k ≤ ω} for all suciently large n. Then xn = fm ekn → fm = x as n → ∞. −1 −1 −1 −1 So, x = fm fl for some l ∈ ω. In this case the convergences limn→∞ xn xn = xx = fm fm and −1 −1 −1 −1 limn→∞xn xn = x x = fl fl imply that {xn}n∈ω ⊂ {fm ek fl : k ≤ ω} for all suciently large n. Then −1 −1 xn = fm ekn fl → fm fl = x as n → ∞.

4 The continuity of EH-continuous homomorphisms with values in weakly ditopological inverse semigroups

In this section we prove the main result of the paper and derive some corollaries. Let us recall that by a topological inverse U0-semigroup we understand a topological inverse semigroup whose maximal semilattice is a U0-semilattice. By [3, 6.2], every EH-continuous homomorphism h : X → Y from a topological inverse U0-semigroup X to a ditopological inverse semigroup Y is continuous. The following theorem generalizes this result in two directions (weakening the requirements both on X and Y).

* Theorem 4.1. Each EH-continuous homomorphism h : X → Y from a topological inverse U0-semigroup X to a weakly ditopological inverse semigroup Y is continuous.

Proof. Given any point x ∈ X and neighborhood Oy ⊂ Y of the point y = h(x), we need to nd a neighborhood Ox ⊂ X of x such that h(Ox) ⊂ Oy. Since the topological inverse semigroup Y is weakly ditopological, there −1 −1 exist an open neighborhood Vy ⊂ Y of y and neighborhoods Vyy−1 , Vy−1 y ⊂ ES of the idempotents yy , y y such that [−1] ∩ −1 ∩ −1 ⊂ (Vy Vy−1 y) λY (Vyy−1 ) ρY (Vy−1 y) Oy . Applying the denition of a weakly ditopological inverse semigroup once more, we can nd an open neighborhood Wy ⊂ Vy of y, and neighborhoods Wyy−1 ⊂ Vyy−1 and Wy−1 y ⊂ Vy−1 y of the idempotents yy−1, y−1y such that [−1] ∩ −1 ∩ −1 ⊂ (Wyy−1 Wy) λY (Wyy−1 ) ρY (Wy−1 y) Vy .

Replacing Wyy−1 by a smaller neighborhood, we can additionally assume that Wyy−1 y ⊂ Wy. By the continuity of the restriction h|E(X), there are open neighborhoods Wxx−1 , Wx−1 x ⊂ E(X) of the −1 −1 idempotents xx , x x such that h(Wxx−1 ) ⊂ Wyy−1 and h(Wx−1 x) ⊂ Wy−1 y. * Since X is an inverse U0-semigroup, there is an idempotent e ∈ Wxx−1 such that x ∈ ⇑e and e is locally X X minimal in its conjugacy class e . Consequently, e has an open neighborhood Ve ⊂ X such that Ve ∩ e ⊂ ↑e. It follows that the element ex lies in the shift He x of the maximal subgroup He. The choice of the neighborhood Wyy−1 guarantees that h(ex) = h(e)y ∈ Wyy−1 y ⊂ Wy. −1 Since the map He → He x, z 7→ zx, is a homeomorphism (with inverse He x → He, z 7→ zx ), the continuity of the restriction h|He implies the continuity of the restriction h|He x. Then we can nd an open neighborhood Wex ⊂ X of the point ex such that h(Wex ∩ He x) ⊂ Wy. We claim that the open neighborhood

−1 −1 −1 −1 −1 Ox = {z ∈ X : zz ∈ Wxx−1 , z z ∈ Wx−1 x , ezx x ∈ Wex , λX(zx x) ∈ Wxx−1 , ρS(zx x) ∈ Wx−1 x , −1 −1 −1 −1 −1 −1 −1 −1 zx xz ∈ ⇑e, xz zx ∈ ⇑e, xz ezx ∈ Ve , zx exz ∈ Ve} of x has the desired property: h(Ox) ⊂ Oy. Automatic continuity of homomorphisms between topological inverse semigroups Ë 65

−1 Take any element z ∈ Ox. We claim that the element ezx belongs to the maximal group He. Indeed, ezx−1(ezx−1)−1 = ezx−1xz−1e = e as zx−1xz−1 ∈ ⇑e. Consequently, the idempotent (ezx−1)−1ezx−1 = −1 −1 −1 −1 X xz ezx is conjugated to e and the choice of the neighborhood Ve guarantees that xz ezx ∈ Ve∩e ⊂ ↑e. So, e ≤ xz−1ezx−1. By analogy we can prove that e ≤ zx−1exz−1 and hence xz−1ezx−1 ≤ xz−1z(x−1exz−1)zx−1 = e (as xz−1zx−1 ∈ ⇑e). The inequalities e ≤ xz−1ezx−1 ≤ e imply that (ezx−1)−1ezx−1 = xz−1ezx−1 = e and −1 −1 −1 hence ezx ∈ He. Then ezx x ∈ He x ∩ Wex and hence h(ezx x) ∈ h(He x ∩ Wex) ⊂ Wy. −1 −1 The denition of the neighborhood Ox 3 z guarantees that λX(zx x) ∈ Wxx−1 and ρX(zx x) ∈ Wx−1 x. −1 −1 −1 −1 Then λY (h(zx x)) = h(λX(zx x)) ∈ h(Wxx−1 ) ⊂ Wyy−1 and ρY (h(zx x)) = h(ρX(zx x)) ∈ h(Wx−1 x) ⊂ Wy−1 y. −1 −1 Taking into account that h(e) ∈ h(Wxx−1 ) ⊂ Wyy−1 and h(e)h(zx x) = h(ezx x) ∈ Wy, we conclude that

−1 ∈ [−1] ∩ −1 ∩ −1 ⊂ h(zx x) (Wyy−1 Wy) λY (Wyy−1 ) ρY (Wy−1 y) Vy .

The denition of the neighborhood Ox guarantees that λX(z) ∈ Wxx−1 and ρX(z) ∈ Wx−1 x, which implies λY (h(z)) ∈ h(Wxx−1 ) ⊂ Wyy−1 ⊂ Vyy−1 and λY (h(z)) ∈ h(Wx−1 x) ⊂ Wy−1 y ⊂ Vy−1 y. Taking into account that −1 −1 h(z)y y = h(zx x) ∈ Vy, we conclude that

∈ [−1] ∩ ∩ ⊂ h(z) (Vy Vy−1 y) λY (Vyy−1 ) ρY (Vy−1 y) Oy .

* Since topological inverse U0-semigroups and topological inverse Cliord U-semigroups are U0-semigroups, Theorem 4.1 implies the following two corollaries.

Corollary 4.2. Each EH-continuous homomorphism h : X → Y from a topological inverse U0-semigroup X to a weakly ditopological inverse semigroup Y is continuous.

Corollary 4.3. Each EH-continuous homomorphism h : X → Y from a topological inverse Cliord U-semigroup X to a weakly ditopological inverse semigroup Y is continuous.

Since each locally compact Lawson topological semilattice is a U-semilattice [5, 2.12] and each locally compact zero-dimensional topological semilattice is a U0-semilattice [5, 2.6], Corollaries 4.2, 4.3 imply two other corollaries, generalizing Corollaries 6.3 and 7.2 in [3].

Corollary 4.4. Each EH-continuous homomorphism h : X → Y from a topological inverse semigroup X with zero-dimensional locally compact semilattice E(X) to a weakly ditopological inverse semigroup Y is continuous.

Corollary 4.5. Each EH-continuous homomorphism h : X → Y from a topological inverse Cliord semigroup X with locally compact Lawson semilattice E(X) to a weakly ditopological inverse semigroup Y is continuous.

5 The continuity of Souslin-measurable homomorphisms between topological inverse semigroups

In this section we apply Theorem 4.1 to establish the continuity of Souslin-measurable homomorphisms between topological inverse semigroups. A function f : X → Y between topological spaces is called Souslin-measurable if for any open Fσ-set U ⊂ Y the preimage f −1(U) is a Souslin subset of X. A subset A of a topological space X is Souslin if A = S T s∈ωω n∈ω Fs|n for some family (Fs)s∈ω<ω of closed subsets of X. It is known [6] that each Borel subset of a metrizable space is Souslin. The following automatic continuity theorem can be found in [9].

Theorem 5.1 (Noll). Each Souslin-measurable homomorphism h : X → Y from a paracompact Čech-complete topological group X to a topological group Y is continuous. 66 Ë Iryna Pastukhova

Combining Theorems 4.1 and 5.1 we get the following corollary generalizing Theorem 5.1.

* Corollary 5.2. A homomorphism h : X → Y from a paracompact Čech-complete topological inverse U0- semigroup X to a weakly ditopological inverse semigroup Y is continuous if and only if h is Souslin-measurable and the restriction h|E(X) is continuous.

* Since each Cliord topological inverse U-semigroup is a topological inverse U0-semigroup, Corollary 5.2 implies another corollary.

Corollary 5.3. A homomorphism h : X → Y from a paracompact Čech-complete Cliord topological inverse U- semigroup X to a weakly ditopological inverse semigroup Y is continuous if and only if h is Souslin-measurable and the restriction h|E(X) is continuous.

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