Perfectly supportable semigroups are \sigma-discrete in each Hausdorff shift-invariant topology

In this paper we introduce perfectly supportable semigroups and prove that they are \sigma-discrete in each Hausdorff shift-invariant topology. The class of perfectly supportable semigroups includes each subsemigroup S of the semigroup FRel(X) of finitely supported relations on an infinite set X such that S contains the group FSym(X) of finitely supported permutations of X.


Introduction
The problem considered in this paper traces its history back to S.Ulam who asked in [5, p.178] and [6] if for some infinite set X the group Sym(X) of bijections of X carries a non-discrete locally compact group topology. The Ulam's problem was solved in negative in 1967 by E.Gaughan [3]. This result motivated the following problem posed in [4]: Problem 1.1. Let X be a set of cardinality |X| = c and let FSym(X) be the group of bijections f : X → X having finite support supt(f ) = {x ∈ X : f (x) = x}. Does FSym(X) admit a non-discrete locally compact Hausdorff group topology?
In [4] it was shown that for any infinite set X the group FSym(X) does not admit a compact Hausdorff group topology. A negative answer to Problem 1.1 was given in [1], where the following theorem was proved: Theorem 1.2 (Banakh-Guran-Protasov). For any set X the group FSym(X) is σ-discrete in each Hausdorff shift-invariant topology on FSym(X). Consequently, each Baire Hausdorff shift-invariant topology on FSym(X) is discrete.
In this paper we generalize Theorem 1.2 to the class of perfectly supportable semigroups. Such semigroups are defined in Section 2. Our main result is Theorem 3.2, proved in Section 3. It implies Corollary 3.5 saying that each perfectly supportable semigroup is σ-discrete in each Hausdorff shift-invariant topology. In Section 4 we present an example of perfectly supportable semigroup FRel(X), which contains many other perfectly supportable (semi)groups as sub(semi)groups.

supt-Semigroups and supt-perfect semigroups
In this section we define the classes of semigroups called supt-semigroups and supt-perfect semigroups.
Definition 2.1. A supt-semigroup is a pair (S, supt) consisting of a semigroup S and a function supt : S → 2 X with values in the power-set 2 X of some set X, such that for each elements f, g ∈ S we get: The function supt : S → 2 X is called the support map of the supt-semigroup (S, supt). The set supt(S) = Proposition 2.2. Let (S, supt) be a supt-semigroup with support X = supt(S).
This definition implies that each supt-finitary semigroup S coincides with its subsemigroup S <ω consisting of finitely supported elements of S.
A typical example of a supt-finitary supt-semigroup is the group FSym(X) = Sym(X) <ω of finitely supported bijections of a set X.
To define supt-perfect semigroups, we need to recall some information on centralizers. For a semigroup S, an element f ∈ S, and a subset T ⊂ S by we denote the centralizers of f and T , respectively. The condition (2) of Definition 2.1 implies that S(F ) ⊂ C(S(X\F )) for any finite subset F ⊂ X. For supt-perfect semigroups the converse implication is also true.
is called a supt-perfect semigroup if it is supt-finitary and for each finite subset F ⊂ X we get S(F ) = C(S(X\F )).
Definition 2.5. A semigroup S is called perfectly supportable if for some function supt : S → 2 X the pair (S, supt) is a supt-perfect semigroup.
Let us discuss the relation of perfectly supportable semigroups with some other classes of semigroups. Definition 2.6. We shall say that a semigroup S • is locally finite if each finite subset F ⊂ S lies in a finite subsemigroup T ⊂ S; • has finite double centralizers if for any finite subset F ⊂ S its double centralizer C(C(F )) is finite.
Since the double centralizer C(C(F )) is a subsemigroup that contains F , each semigroup with finite double centralizers is locally finite. Proposition 2.7. Each perfectly supportable semigroup has finite double centralizers and hence is locally finite.
Proof. Let supt : S → 2 X be a support map that turns S into a supt-perfect semigroup. We claim that for each finite subset T ⊂ S its double centralizer C(C(T )) lies in the finite subsemigroup S(F ) where F = f ∈T supt(f ). Assuming that C(C(T )) ⊂ S(F ) and taking into account that S(F ) = C(S(X\F )), we can find an element f ∈ C(C(T )) \ C(S(X\F )). Then for some g ∈ S(X\F ) we get f g = gf . On the other hand, the condition (2) of Definition 2.1 guarantees that g ∈ S(X\F ) ⊂ C(S(F )) ⊂ C(T ) and hence f / ∈ C(C(T )), which is a desired contradiction.

The topological structure of supt-perfect and perfectly supportable semigroups
In this section we shall study the topological structure of supt-perfect and perfectly supportable semigroups endowed with shift-invariant topologies.
A topology τ on a semigroup S is called shift-invariant if for every a ∈ A the left and right shifts l a : S → l a : x → ax and r a : S → S, r a : x → xa, are continuous. This is equivalent to saying that the semigroup operation S × S → S is separately continuous. Now on each semigroup S we define a shift-invariant T 1 -topology (called the semi-Zariski topology) which is weaker than each Hausdorff shift-invariant topology on S.
The semi-Zariski topology Z s on a semigroup S is the topology generated by the sub-base consisting of the sets {x ∈ S : axb = c} and {x ∈ S : axb = cxd} where a, b, c, d ∈ S 1 and S 1 = S ∪ {1} stands for the semigroup S with attached external unit 1 / ∈ S (i.e., an element 1 such that 1x = x1 = x for all x ∈ S 1 ). The semi-Zariski topology Z s is a particular case of algebraically determined topologies on G-acts, considered in [2].
The definition of the semi-Zariski topology implies the following simple (but important) fact.
Proposition 3.1. The semi-Zariski topology Z s on a semigroup S is weaker than each shift-invariant Hausdorff topology on S.
Now, we shall study the semi-Zariski topology on supt-perfect semigroups. The following theorem is the main result of this paper.
Theorem 3.2. Let (S, supt) be a supt-perfect semigroup endowed with the semi-Zariski topology Z s . For every Proof. First, we prove two lemmas.
By induction we shall construct a sequence of elements (g i ) i∈ω of the semigroup S and an increasing sequence (F i ) i∈ω of finite subsets of X such that F 0 = F and for every i ∈ ω the following conditions hold: Assume that for some i ≥ 0 the set F i has been constructed. Since x ∈ supt(f ) and x / ∈ F i , we get f / ∈ S(F i ) = C(S(X\F i )) and hence there is an element This completes the inductive construction.
It follows that for any number i < j we get which implies that the non-empty sets supt(g i )\{x} and supt(g j )\{x} are disjoint. Then g i = g j and supt( Proof. Let m = |supt(f )| and supt(f ) = {x 1 , . . . , x m } be an enumeration of the finite set supt(f ). For every i ≤ m by induction we shall construct an increasing sequence of finite sets F 0 ⊂ F 1 ⊂ · · · ⊂ F m in X and a sequence F 1 , . . . , F m of non-empty finite subsets of S \ C(f ) such that for every positive i ≤ m the following conditions are satisfied: We start the inductive construction letting F 0 = supt(f ). Assume that for some i < m the finite set F i−1 has been constructed. Let F = F i−1 \ {x i } and apply Lemma 3.3 to find a non-empty family F i ⊂ S(X\F ) \ C(f ) which satisfies the conditions (1) and (3) of the inductive construction. It follows from F i ⊂ S(X\F ) that the condition (2) is satisfies. Finally, define the finite set F i by the condition (4). This completes the inductive construction.
for some i ≤ m and consider two cases. (i) If F i ⊂ S({x i }), then for each element g ∈ F i we get supt(g) ∩ supt(h) ⊂ {x i } ∩ supt(h) = ∅ and hence gh = hg by the condition (2) of Definition 1.
(ii) If F i ⊂ S({x i }), then |F i | = n + 1 and by the conditions (3) of the inductive construction, the family supt(g) \ {x i } g∈Fi is disjoint and consists of non-empty sets. Since |F i | = n + 1 > |supt(h)|, there is an element g ∈ F i such that supt(g) \ {x i } is disjoint with supt(h). Since x i / ∈ supt(h), the supports supt(g) and supt(h) are disjoint and hence gh = hg by the condition (2) of Definition 2.1.
In both cases, we get an element g ∈ F with gh = hg, which contradicts the choice of h ∈ O f . Now we can finish the proof Theorem 3.2. Let n ∈ ω.
1. To show that the set S ≤n = {f ∈ S : |supt(f )| ≤ n} is closed in S, fix any element f ∈ S \ S ≤n . By Lemma 3.4, the element f has a neighborood O f ⊂ S in the semi-Zariski topology Z s such that each g ∈ O f ∩S ≤n has support supt(g) ⊃ supt(f ), which implies that n ≥ |supt(g)| ≥ |supt(f )| > n and O f ∩ S ≤n = ∅. So, S ≤n is closed in S.

The second item of Theorem 3.2 follows directly from Lemma 3.4.
3. Finally we show that the set S =n = {f ∈ S : |supt(f )| = n} is discrete in (S, Z s ). Fix any element f ∈ S =n and using Lemma 3.4, find a neighborhood O f ∈ Z s of f such that supt(f ) ⊂ supt(h) for each h ∈ O f ∩ S =n .
Since |supt(f )| = n = |supt(h)|, we conclude that supt(f ) = supt(g) and hence O f ∩ S =n lies in the semigroup S(supt(f )), which is finite by the condition (2) of Definition 2.4. Since the semi-Zariski topology Z s satisfies the separation axiom T 1 , the open finite subspace O f ∩ S =n of S = n is discrete and hence f is an isolated point of S =n , which means that the space S =n is discrete.
Let us recall that topological space X is σ-discrete if it can be written as a countable union of discrete subspaces. A topology τ on a set X is called Baire if for any sequence (U n ) n∈ω of open dense subsets of the topological space (X, τ ) the intersection n∈ω U n is dense in X. It is well-known that each σ-discrete Baire T 1 -space has an isolated point. Theorem 3.2 implies the main corollary of this paper: Corollary 3.5. Each perfectly supportable semigroup S is σ-discrete in its semi-Zariski topology and hence is σ-discrete in each shift-invariant Hausdorff topology on S.
Another corollary concerns perfectly supportable groups. A group G is called perfectly supportable if it is perfectly supportable as a semigroup.
Corollary 3.6. Each perfectly supportable group G is discrete in each Baire shift-invariant Hausdorff topology on G.
Proof. Let τ be a Baire Hausdorff shift-invariant topology on a perfectly supportable group G. Since G is a group, the topological space (G, τ ) is topologically invariant.
By Corollary 3.5, the topological space (G, τ ) is σ-discrete and being Baire, has an isolated point. The topological homogeneity of (G, τ ) guarantees that each point of G is isolated and hence the space (G, τ ) is discrete.

An example of a supt-perfect semigroup
In this section we consider an important example of a supt-semigroup (which contains many other suptsemigroups as supt-subsemigroups).
Given a set X, consider the semigroup Rel(X) = 2 X×X of all relations f ⊂ X × X, endowed with the operation f • g = {(x, z) ∈ X × X : ∃y ∈ X (x, y) ∈ f, (y, z) ∈ g} of composition of relations. For a relation f ⊂ X × X by f −1 we denote the inverse relation Each relation f ⊂ X × X can be considered as a multi-valued function assigning to each point x ∈ X the subset f (x) = {y ∈ X : (x, y) ∈ f } and to each subset A ⊂ X the subset f (A) = a∈A f (a) of X. Observe that two relations f, g ⊂ X × X are equal if and only if f (x) = g(x) for all x ∈ X.
Each function f : X → X can be identified with the relation {(x, f (x)) : x ∈ X}. So, the semigroup End(X) of all self-maps of X is a subsemigroup of the semigroup Rel(X). By the same reason, the symmetric group Sym(X) is a subgroup of the semigroup Rel(X).
For a relation f ⊂ X × X its support is defined as the subset Proof. Fix any relations f, g ⊂ X × X.
To show that supt(f • g) ⊂ supt(f ) ∪ supt(g), take any point x ∈ X that does not belong to the union ∈ supt(f • g). Next, assuming that supt(f ) ∩ supt(g) = ∅, we shall show that f • g(x) = g • f (x) for each point x ∈ X. The inclusion f • g(x) ⊂ g • f (x) will follow as soon as we prove that each point y ∈ f • g(x) belongs to the set g • f (x). Find a point z ∈ g(x) such that y ∈ f (z).
This completes the proof of the inclusion y ∈ g • f (x), which implies that f • g(x) ⊂ g • f (x). By analogy we can prove that g • f (x) ⊂ f • g(x).
In the supt-semigroup Rel(X) consider the supt-subsemigroup Observe that for each (finite) subset F ⊂ X the subsemigroup {f ∈ Rel(X) : supt(f ) ⊂ F } can be identified with the (finite) semigroup Rel(F ), which has cardinality 2 |F ×F | . This observaion implies: The semigroup FRel(X) endowed with the support map supt : FRel(X) → 2 X is supt-finitary.
Finally we check that the supt-semigroup FRel(X) and some its subsemigroups are supt-perfect. For an infinite set X, a subsemigroup S ⊂ FRel(X) is supt-perfect if for each finite subset E ⊂ X and each point x ∈ X \ E there is a relation g ∈ S(X\E) such that x / ∈ g(x) = ∅.
Proof. Given a finite subset F ⊂ X we should check that S(F ) = C(S(X\F )). The inclusion S(F ) ⊂ C(S(X\F )) holds because S is a supt-semigroup. To prove that C(S(X\F )) ⊂ S(F ), take any element f ∈ C(S(X\F )) and assume that f / ∈ S(F ). Then supt(f ) ⊂ F and we can choose a point x ∈ supt(f ) \ F . By our assumption, for the finite set E = F ∪ (supt(f ) \ {x}) there is a relation g ∈ S(X\E) ⊂ S(X\F ) such that x / ∈ g(x) = ∅. Then {x} ∪ g(x) ⊂ supt(g).

Proposition 4.3 and Corollary 3.5 imply:
Corollary 4.4. Let X be an infinite set and S ⊂ FRel(X) be a subsemigroup such that for each finite subset E ⊂ X and each point x ∈ X \ E there is a relation g ∈ S such that x / ∈ g(x) = ∅ and supt(g) ∩ E = ∅. Then the semigroup S is σ-discrete is its semi-Zariski topology and consequently is σ-discrete in each Hausdorff shift-invariant topology on S.
This corollary implies the following theorem announced in Abstract.
Theorem 4.5. Let X be an infinite set X and S be a semigroup such that FSym(X) ⊂ S ⊂ FRel(X). Then (1) S is perfectly supportable; (2) S is σ-discrete is its semi-Zariski topology Z s ; (3) S is σ-discrete in each Hausdorff shift-invariant topology on S.

Some Open Problems
The second statement of Theorem 3.2 suggests the following question. By Proposition 2.7, each perfectly supportable semigroup has finite double centralizers. We do not know if the converse is also true.