**INTRODUCTION**

Topological semigroups, their actions, their representations and topological
congruences over them have been studied by many mathematicians and have a very
wide usage in many fields like topological vector spaces, geometry or analysis.
More specially, there are many works about topological Rees congruences on semigroups
and there are some works about the congruences on S-flows which are compact
topological S-acts (Berglund* et al*., 1989; Gonzalez,
2001; Gutik and Pavlyk, 2006; Hryniv,
2005; Lawson and Lisan, 1994; Lawson
and Madison, 1971; Normak, 1993, 2006).

As the first step in the study of the category of topological S-acts, in this
note, we study the notion of topological congruences on topological S-acts.
This study can be used in the study of universal objects like push out or coequalizer.
Here, we briefly state the notions and definitions which will be used in present
study about topological S-acts and their congruences. Next, we will state a
criterion for detecting topological congruences on topological S-acts and as
well, on topological semigroups. As an application of the results, we give a
necessary and sufficient condition for a longstanding problem raised by Wallce
(1955). Later we will give a better criterion for Rees congruence ρ_{B}
on a topological S-act (A, τ_{A}) where, B is a closed subact of
A. As a part of this section, we will prove that if B is a closed compact or
clopen subact of (A, τ_{A}), then the Rees congruence ρ_{B
} is a topological congruence on (A, τ_{A}).

Recall that, for a semigroup S, a set A is a left S-act if there is, so called,
an action λ: SxA→A such that, denoting λ (s, a): = sa, (st) a
= s (ta) and, if S is a monoid with 1, 1a = a (for more information about S-acts,
(Ebrahimi and Mahmoudi, 2001; Kilp
*et al*., 2000).

A semigroup S with topology τ_{S }is a topological semigroup if the multiplication μ: SxS→S is (jointly) continuous, where SxS has product topology. We denote such a topological semigroup by (S, τ_{S}).

For a topological semigroup (S, τ_{S}), a (left) topological S-act is an S-act A with a topology τ_{A }on it such that the action λ: SxA→A is jointly continuous. (Note that SxA is considered with product topology). If A with a topology τ_{A }is a topological S-act, we simply denote it by (A, τ_{A}).

Since, in our study, we need to discuss on one S-act or one semigroup with different topologies, we use this terminology to prevent misunderstanding. We denote the category of all left topological S-acts with continuous S-homomorphisms between them with S-Top.

Recall that for a semigroup S and an S-act A, the functions λ_{s}: A→A is defined by y→sy for any yεY. Similarly for any aεA the function ρ_{a}: S→A is defined by y→sa. Since, in some of our results we need to talk about λ_{s}: A→A and λ_{s}: A/θ→A/θ at the same time, to prevent misunderstanding, we show the latter by Λ_{s}: A/θ→A/θ.

Now, if S has a topology τ_{S }for which its multiplication SxS→S
is (separately) continuous, that is, λ_{s }and ρ_{s }are
continuous for all sεS, then S with this topology is called a semitopological
semigroup (for more information about semitopological semigroups (Berglund
and Hofmann, 1967; Wolfgang, 1984).

Similarly, one can define a semitopological S-act by taking λ_{s}:
A→A and ρ_{a}: S→A to be continuous for each sεS
and aεA (Khosravi, 2009).

We say a topological semigroup (S, τ) has (left) ideal topology, if any
open set in S is a (left) ideal of S (this definition is more general than what
is implied by Normak (1993). Similarly, a topological
S-act (A, τ_{A}) has subact topology, if any open set in A is a
subact of A.

A semigroup topological congruence on a topological semigroup (S, τ_{S}) is a semigroup congruence θ (that is, if sθs^{i }where, s, s^{i}εS, then for any tεS, tsθts^{i} and stθs^{i}t) such that the semigroup S/θ with the quotient topology is a topological semigroup.

For an S-act A and a congruence θ on it, we denote the usual quotient map from A to A/θ by π. By a zero in an S-act A, we mean an element aεA such that for any sεS, sa = a. For an S-act A, by P (A) we mean the powerset of the underling set of A.

By a clopen subset in a topological space, we mean a closed and open subset in that space. In this study, for any subset C of a topological space X, by cl(C) and int(C), we mean the closure of the set C in X and the interior of C in X, respectively. By a descending chain of open subsets in a topological space, we mean a collection of open sets which is induced by a totally ordered set J such that for all α, βεJ, if α≤β, then O_{α}⊇O_{β}.

**S-ACT TOPOLOGICAL CONGRUENCE**

An S-act topological congruence or when there is no ambiguity about S-act, briefly, a topological congruence on a topological S-act (A, τ_{A}) is an S-act congruence θ (that is, if aθa^{i }for a, a^{i}εA, then saθsa^{i }for all sεS) such that the S-act A/θ with the quotient topology is a topological S-act.

**Remark 1:** Considering a semigroup S as an S-act, any semigroup congruence is an S-act congruence, but the converse is not true in general. However, there are topological semigroups, namely S with some congruence θ such that θ is not a topological semigroup congruence while θ is a topological S-act congruence.

Let, (A, τ_{A}) be a topological S-act and θ be a congruence
on A. The following is a closure operator on A (Burris and
Sankappanavar, 1981; Dikranjan and Tholen, 1995).

A subset X of A is called a closed subset, (relative to C_{θ}),
if C_{θ }(X) = X. Consider the lattice of closed subsets relative
to C_{θ }and denote it by L_{θ }(L_{θ}
= {B ⊆ A|C_{θ }(B) = B}). Since, L_{θ }is a sub-Boolean
algebra of P (A), it obviously forms a topology on A.

**Remark 2:** Note that for a topological S-act (A, τ_{A}) and a congruence θ on it, for any open set Oετ_{A }∩ L_{θ}, the image O under the map π: A→A/θ is open in A/θ with the quotient topology.

For any topological S-act (A, τ_{A}), the study of a topological congruence θ on (A, τ_{A}), depends essentially on the behavior of the original topology τ_{A }and the topology L_{θ}. The following proposition shows this relation.

**Proposition 3:** Let, (A, τ) be a topological S-act and θ be a congruence on A, then θ is a topological congruence if and only if (A, τ∩ L_{θ}) is a topological S-act.

**Proof:** (⇒) Let θ be a topological congruence. Let sεS, aεA and Uετ∩ L_{θ }with saεU. By Remark 2, π(U) is an open set which contains s[a]. So, there exist open sets W_{s }and V_{[a]} in S and A/θ, respectively such that sεW_{s}, [a]εV_{[a] }and s[a]εW_{s}·E V_{[a]} ⊆ π(U). Note that the open set π^{-1}(V_{[a]}) contains a and belongs to τ ∩ L_{θ }and further satisfies sa ε W_{s}·π^{-1}(V_{[a]}) ⊆ π^{-1}(π(U)) = U, (since UεL_{θ}). So (A, τ∩L_{θ}) is a topological S-act.

(Z) Let (A, τ∩ L_{θ}) be a topological S-act for some congruence θ on A. We show that A/θ with the quotient topology is a topological S-act.

Let sεS, [a]εA/θ and U in A/θ are given such that s[a]εU. So saεπ^{-1}(U) and we know that π^{-1}(U)ετ∩ L_{θ}, so by the assumption, there exist open sets V_{a}ετ∩L_{θ }and W_{s }such that aεV_{a}, sεW_{s }and saεW_{s}· V_{a }⊆ π^{-1 }(U).

Since V_{a}ετ _∩ L_{θ}, by Remark 2, we have π(V_{a}) is open in A/θ. So s[a]εW_{s }· π(V_{a}) ⊆ π(π^{-1}(U)) = U, (since π^{-1}(U)ετ∩ L_{θ}).

By a similar argument as in the proof of Proposition 3, (1) and (2) we get
the semigroup congruence version of the above proposition which is a necessary
and sufficient condition for this question raised by Wallce
(1955) that when the quotient semigroup S/θ is topological.

**Proposition 4:** Let (S, τ_{S}) be a topological semigroup
and θ be a semigroup congruence on it. Then the following are equivalent:

• |
θ is a semigroup topological congruence |

• |
(S, τ_{S} ∩ L_{θ}) is a topological semigroup |

Since, the next proposition can be proved easily, we state it without proof.

**Proposition 5:** Suppose that (A, τ_{A}) is a topological S-act and θ is a congruence on A. If π: A→A/θ is an open map, then θ is a topological congruence.

Note that we have the following property for any congruence θ on A.

**Proposition 6: (Khosravi, 2009):** Let (A, τ_{A})
be a topological S-act. Then for every congruence θ on A, we have:

• |
For all sεS, the map Λ_{s}: A/θ→A/θ defined
by [a]→s[a] is continuous |

• |
For all aεA, the map ρ_{[a]}: S→A/θ s→s[a]
is continuous |

As a quick consequence of the above proposition, we have:

**Corollary 7:** For any topological S-act (A, τ_{A}) and any
congruence θ on A, A/θ with the quotient topology is a semitopological
S-act.

**Remark 8:** Since, the lattice of closed subsets relative to a closure
operator is closed under arbitrary intersections, if (A, τ_{A})
is an Alexandroff topological S-act, then for any congruence θ on A, (A,
τ_{A }∩ L_{θ}) is an Alexandroff space. By the
definition of the quotient topology, this implies that A/θ with the quotient
topology is an Alexandroff space.

**Corollary 9:** Let (A, τ) be an Alexandroff topological S-act and
θ be an arbitrary congruence on A. Then θ is a topological congruence
and A/θ with the quotient topology is an Alexandroff topological S-act.

**Proof:** Since for an Alexandroff topological S-acts, the joint continuity
and separately continuity of the action is equivalent (Khosravi,
2009) θ is a topological congruence by Corollary 8 and A/θ is
Alexandroff. As a quick result of the above proposition, we have:

**Corollary 10:** Let S-Alex be the category of all Alexandroff topological S-acts with continuous S-homomorphisms between them. Then S-Alex is closed under quotient. Similar to Proposition 9, we have:

**Proposition 11:** Let (S, τ_{S}) be an Alexandroff topological semigroup and (A, τ_{A}) be a topological S-act and θ be an arbitrary congruence on A. Then θ is a topological congruence.

**Corollary 12:** For an Alexandroff topological semigroup (S, τ_{S}), S-Top is closed under quotient and it is complete and cocomplete.

**REES TOPOLOGICAL CONGRUENCE**

Here, we study the Rees congruences on topological S-acts and on topological semigroups. In fact, in this section, we consider this question: for which subact Y of a topological S-act (A, τ_{A}), the Rees congruence ρ_{Y }is a topological congruence?.

In the rest of this note, suppose that (S, τ_{S}) is a topological semigroup. For a topological S-act (A, τ_{A}) and its closed subact Y, first we give necessary and sufficient conditions for this question. Then by using these conditions, in some cases like when the lattice of open sets which contain Y, has a minimum element, or when (S, τ_{S}) is locally compact or Alexandroff, we answer to this question.

**Notation 13:** From now on, we denote the lattice L_{ρY }by L_{Y }for simplicity. Similarly, we denote the operator C_{ρY }by C_{Y}. Also, for a topological S-act (A, τ_{A}), by a Rees quotient space, we mean the quotient space A/Y, for some subact Y.

**Remark 14:** As a quick consequence of Corollary 9, we can easily conclude that every Rees congruence on a topological S-act (A, τ_{A}) where (S, τ_{S}) is an Alexandroff topological semigroup, is topological.

Since by Proposition 3, for any topological S-act (A, τ_{A}), the study of congruence ρ_{Y }on A depends on the structure of the lattice τ_{A}∩ L_{Y} before we continue present study in this section, we explain this structure in the following.

**Remark 15:** For a topological S-act (A, τ_{A}) and a subact
Y of it, the elements of τ ∩ L_{Y }are:

• |
Open sets which contain Y |

• |
Open sets which are disjoint from Y |

We are going to study the S-act Rees congruence ρ_{Y }where Y
is a closed subact. For this purpose, we divide our discussion to the following
cases:

• |
Y is a closed compact subact |

• |
Y is a closed subact and the lattice of open sets in τ_{A}∩L_{Y
}which contain Y, has a minimum element |

• |
Y is a closed subact and there exists a chain of open sets in τ_{A}∩
L_{Y }around Y which has no minimal element |

Then, we state and prove an equivalence condition which can be used in all
of these cases. Then, by using this tool, we characterize topological Rees congruences
in the first and the second case exactly. For the third case, we answer the
question in some cases with extra assumptions.

Before we state the next proposition in this section, we need the following notation.

**Notation 16:** For an S-act A and two subsets U and V of A, by (U: V), we mean the following set:

Now, we state the Rees congruence version of Proposition 3 for a closed subact,
which simplifies present study in future.

**Proposition 17:** Suppose that Y is a closed subact of a topological S-act (A, τ_{A}). Then the following are equivalent:

(i) |
ρ_{Y }is a topological congruence. |

(ii) |
For each open set U in A which contains Y, there exists a family of
open subsets of U like {V_{α }}_{αεJ }such
that each V_{α }contains Y and {int((U: V_{α}))}_{αεJ}
is an open covering for S. |

(iii) |
Considering A with topology τ_{A}∩L_{Y}, the
action λ: SxA→A, is continuous at any point (s, y) where, yεY
and sεS. |

(iv) |
The action of A/Y is continuous at any point (s, [y]_{ρY})
where yεY and sεS. |

**Proof:** (1)⇒(2) Suppose that ρ_{Y }is a topological
congruence. By Proposition 3, (A, τ_{A}∩L_{Y}) is a
topological S-act, so for any arbitrary aεY, sεS and Uετ_{A}∩L_{Y
}such that saεU, there exist open sets V_{a,s}ετ_{A}∩L_{Y
}and W_{s}ετ_{S }which contain a and s, respectively
such that saεW_{s}·V_{a,s}⊆U. Now note that
since Y is an S-act and aεY⊆V_{a,s}, then saεV_{a,s}.
Therefore without lose of generality, we can assume that V_{a,s }⊆
U. Consider the family {V_{a,s }}_{sεS }which is found
by the above discussion. Since aεV_{a,s }for any sεS and aεY,
by Proposition 15, Y⊆V_{a,s }for any sεS. Note that for any
sεS, int((U: V_{a,s})) contains W_{s}. So, obviously {int((U:
V_{a,s}))}_{sεS }is an open covering for S.

(2)⇒(3) We prove this part by Proposition 3. Suppose that we are given aεA and sεS and Uετ_{A}∩L_{Y }such that saεU. By hypotheses, there exists an open covering {int((U: V_{α}))}_{αεJ }for S such that for all αεJ, V_{α }contains Y. So, there exists a β such that sεint((U: V_{β})). We have obviously V_{β}ετ_{A}∩L_{Y }and saεint((U: V_{β}))·V_{β}⊆U.

(3)⇒(4) Suppose that we are given yεY, sεS and an open set U
in A/Y such that s[y]εU. Obviously π^{-1}(U)ετ_{A}∩L_{Y
}and syεπ^{-1} (U). By hypothesis, there exist W_{s
}and V_{y}ετ_{A}∩L_{Y} which contain
s and y, respectively such that syεW_{s }·V_{y }⊆
π^{-1} (U). By Remark 2, π(V_{y}) is open in A/Y and
syεW_{s }· π(V_{y}) ⊆ U.

(4)⇒(1) According to Proposition 3, we need to prove that (A, τ_{A}∩L_{Y})
is a topological S-act. Suppose that aεA, sεS and Uετ_{A}∩L_{Y}
are given such that saεU. Since (A, τ_{A}) is a topological
S-act, there exist open sets V_{a }and W_{s} which contain a
and s, respectively such that W_{s}·V_{a}⊆U. If
a∉Y, then define O = V_{a}∩(A-Y). By proposition 3, O belongs
to τ_{A}∩L_{Y }which contains a and satisfies:

So the action of A is continuous at every point (s, a) s.t. a is not in Y and
sεS. Now suppose that aεY. Since Uετ_{A}∩L_{Y}
is a nonempty open set, the set U contains Y and we have obviously π^{-1}(π(U))
= U. Hence s[a]επ(U) where, π(U) is an open set in the quotient
topology. By the hypothesis, there exist open sets W_{s }and V_{[a]}
which contains s and [a], respectively and s[a]εW_{s}·V_{[a]}⊆π(U).

**Remark 18:** Note that if for some topological S-act (A, τ_{A}) and two subacts of it, namely Y and Z, the lattices L_{Y}∩τ_{A} and L_{Z}∩τ_{A} are the same, then ρ_{Y }is topological if and only if ρ_{Z} is topological. This fact can be used as a method for studying Rees congruences by relating them to some known topological Rees congruences. We explain this method by the next example.

**Example 19:** Consider (N^{∞}, min) with topology τ = {Ψ, {N^{∞}}}∪{{1, ..., n}|n≥4}. It is obviously a topological semigroup and for ideals Y: = {1, 2} and Z: = {1, 2, 3}, we have τ∩L_{Y }= τ∩L_{Z}. (Since Z is closed and τ∩L_{Z }has minimum element, ρ_{Z }is topological by Proposition 28, therefore, by the above remark, ρ_{Y} is topological).

**Proposition 20:** Let (S, τ_{S}) be a topological monoid with some topology on it such that 1 just belongs to one open set in S. If (A, τ_{A}) is a topological S-act and Y is its closed subact, then ρ_{Y }is topological.

**Proof:** First note that all of the topological S-acts on this monoid have subact topology. Since (A, τ_{A}) is a topological S-act and we have obviously 1a = aεU for any arbitrary Uετ_{A }and aεU, there exist open sets W_{1 }and V_{a }such that 1aεW_{1}·V_{a}⊆U. But by the assumption, we have W_{1} = S. So Sa⊆U for all every point (s, a) s.t. a is not in Y and aεU, therefore U is a subact. By Proposition 17 since for any open set U in τ_{A }we have (U: U) = S, the congruence ρ_{Y }is topological.

Similarly, by Proposition 17. one can easily prove that:

**Proposition 21:** Let (A, τ_{A}) be a topological S-act with subact topology. Then for any closed subact Y, the Rees congruence ρ_{Y }is topological.

As a quick consequence of proposition 20, we have:

**Corollary 22:** Let (S, τ_{S}) be a topological monoid with some ideal topology. Then any Rees congruence ρ_{Y }for any closed subact Y of a topological S-act is topological.

**Proposition 23:** Let Y be a compact closed subact of (A, τ_{A}). Then ρ_{Y }is a topological congruence.

**Proof:** To prove this assertion, we use Proposition 17. In fact we show
that if we consider A with topology τ_{A }∩L_{Y}, then
the action λ: SxA→A is continuous at every point (s, y) where, yεY
and sεS. Suppose that we are given sεS, yεY and an open set U
which contains Y such that syεU. Since, (A, τ_{A}) is a topological
S-act, there exist open sets V(y) ετ_{A }and W (s, y) ετ_{S
}which contain y and s, respectively such that:

Fixing s and repeating the above argument for any yεY, we reach to the
family {V (y): yεY } which is clearly an open covering for Y. Since, Y
is a compact subact, there exist open sets V (y_{1}),...,V(y_{k })
for some kεN such that:

Y⊆ V (y_{1})∪...∪ V (y_{k
}) |

Let V be the union of V (y_{i }), for 1≤i≤k and W be the intersection
of W (s, y_{i}), for 1≤i≤ k. We have saεW· V⊆U.

Since any closed subset of a compact space is compact, as a quick result of the above proposition, we have:

**Corollary 24:** Let (A, τ_{A }) be an S- flow. Then for any Rees congruence ρ_{Y }on A such that Y is closed, the Rees congruence ρ_{Y }is topological.

**Remark 25:** Note that if π : X→X/ρ is a quotient map and Z is a locally compact space, then the function πxid_{Z }is a quotient map, too.

**Proposition 26:** Let (S, τ_{S}) be a locally compact topological semigroup and (A, τ_{A}) be a topological S-act. Then any congruence θ on (A, τ_{A}) is topological.

**Proof:** Let λ be the action of S on A and θ be a congruence on A. First, note that the following diagram is commutative:

where, π: A→A/θ is the natural quotient map. Let O be an open
set in A/θ. Since π and λ are continuous, the inverse image of
O under π○λ is open in SxA. Define V := (π○λ)^{-1
}(O). Since, S is locally compact, by Remark 25 the function id_{S}xπ
is a quotient map and (id_{S }xπ)(V ) is open in SxA/θ. Hence
λ_{A/θ }is continuous and θ is topological.

**Example 27:** Hryniv (2005) presented an example
of a locally compact topological semigroup (S, τ_{S}) and a closed
ideal I such that S/I is not a topological semigroup. Therefore by the above
proposition, for the topological semigroup (S, τ_{S }) which is
presented (Hryniv, 2005), ρ_{I }is a topological
S-act congruence however it is not a topological semigroup congruence.

Up to now, for a topological S-act (A, τ_{A }), we prove present results by putting some conditions on (A, τ_{A }) or (S, τ_{S}), or by putting some conditions on Y. In the next proposition, we put a condition on the lattice τ_{A} ∩ L_{Y}.

**Proposition 28:** Let Y≤A be a closed subact of (A, τ_{A}) such that there exists a minimum open subset which contains Y. Then ρ_{Y} is a topological congruence if and only if the minimum open set which contains Y is a subact.

**Proof:** (Z) suppose that ρ_{Y }is a topological congruence.
So by Proposition 17, for any arbitrary aεY, sεS and minimum open
set O which contains Y, there is an open set V_{a}⊆ O which contains
Y and an open set W_{s }around s such that saεW_{s }·V_{a}⊆
O. Since O is the minimum open set, V_{a}= O. Therefore, we have:

(⇒) Again, we use Proposition 17. Suppose that we are given aεY,
sεS and an open set U which contains Y and saεU. Since, there exists
a minimum open set O⊆ U which is a subact of A and contains Y, we have
saεS· O⊆ O⊆ U. So ρ_{Y }is a topological
congruence.

**Corollary 29:** Let (A, τ_{A }) be a topological S-act and Y be a clopen subact. Then ρ_{Y }is a topological congruence.

**ACKNOWLEDGMENTS**

The author gratefully acknowledges his supervisor, Professor Ebrahimi and his advisor Professor Mahmoudi for their guidance and kindness. Also he is grateful to Professor Gutik, for his fruitful communications and helps.