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 hjH and hjE are continuous. Let us dene a homomorphism h : X ! Y between topological semigroups to be EH-continuous if – the restriction hjEX to the set of idempotents of X is continuous; – for every subgroup H ⊂ X the restriction hjH 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 = fx 2 X : xx = xg 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 2 S there is a unique element x−1 2 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 2 S; – a semilattice if it is commutative and every element x 2 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 2 E let #x = fy 2 E : y ≤ xg and "x = fy 2 E : x ≤ yg be the lower and upper cones of x, respectively. For a semigroup S by E(S) = fe 2 S : ee = eg we denote the set of idempotents of S and for each idempotent e 2 E(S) let He = fx 2 S : 9y 2 S xy = e = yx, xe = x = ex, ye = y = eyg 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 = fx 2 S : xx = e = x xg. Two elements x, y of an inverse semigroup S are called conjugated if there exists an element s 2 S such that x = sys−1 and y = s−1xs. For an element x 2 S let S x = fy 2 S : 9s 2 S y = sxs−1, x = s−1ysg be the conjugacy class of x in S. If x is an idempotent, then xS ⊂ E(S). Observe that for any element s 2 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 2 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 2 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 2 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 2 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 2 E and its neighborhood U ⊂ E there exists an idempotent e 2 U such that x 2 *e; – a U0-semilattice if for each point x 2 E and its neighborhood U ⊂ E there exists a locally minimal idempotent e 2 U such that x 2 "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 2 E(S) and any neighborhood U ⊂ E(S) of x there exists an idempotent e 2 E(S) such that x 2 *e and e is locally minimal in its conjugacy class eS = ff 2 E(S): −1 −1 S 9s 2 S f = ses , e = s fsg 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 2 E(S) the conjugacy class eS of e is the singleton feg (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 = fy 2 S : 9b 2 B 9a 2 A by = ag and AB[−1] = fx 2 S : 9a 2 A 9b 2 B a = xbg 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 2 S. The equality −1 [−1] −1 −1 −1 [−1] −1 −1 (fxx g fxg) \ λS (xx ) = fxg = (fxgfx xg ) \ 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 2 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 2 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 ditopo- logical 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.