Closed Congruences on Semigroups
Congruencias Cerradas en Semigrupos
Genaro Gonz´alez (
[email protected]
)
Departamento de Matem´atica Facultad Experimental de Ciencias Universidad del Zulia, Apartado Postal 526Maracaibo 4001, Venezuela.
Abstract
A compatible equivalence relation on a semigroupSis called a con-gruence. Wallace proved that if R is a closed congruence defined on a compact topological semigroupS thenS/Rwith the quotient topol-ogy and the standard multiplication is a topological semigroup. In this work Wallace’s result is proved for a more general class of topological semigroups; namely,σ-compact and locally compact semigroups. Key words and phrases: Congruence, topological semigroup, quo-tient map, saturated map,σ-compact, locally compact.
Resumen
Una una relaci´on de equivalencia compatible en un semigrupo es llamadacongruencia. Wallace prob´o que si R es una congruencia ce-rrada definida en un semigrupo topol´ogico compacto S entonces S/R
con la topolog´ıa cociente y la multiplicaci´on est´andar es un semigrupo topol´ogico. En este trabajo se prueba el resultado de Wallace para una clase m´as general de semigrupos topol´ogicos, a saber losσ-compactos y localmente compactos.
Palabras y frases clave: Congruencia, semigrupo topol´ogico, aplica-ci´on cociente, aplicaci´on saturada,σ-compacto, localmente compacto.
1
Introduction
A relationRon a semigroupSis said to beleft[Right]compatibleif (a, b)∈R
and x∈S imply (xa, xb)∈R [(ax, bx)∈R], andcompatibleif it is both left and right compatible.
A compatible equivalence relation on a semigroupSis called acongruence. Observe that an equivalence R on a semigroup S is a congruence if and only if (a, b)∈Rand (c, d)∈Rimply (ac, bd)∈R.
Theorem 1. Let S be a semigroup and let R be a congruence on S. Then
S/Ris a semigroup under the multiplication defined byπ(x)π(y) =π(xy)and
π:S→S/Ris a surmorphism.
It is useful to observe that ifRis a congruence on a semigroupS andm,
m′ are the multiplication on S and S/R respectively then m′ is the unique multiplication onS/Rsuch that the following diagram conmutes:
(S/R)×(S/R) −−−−→m′ S/R
π×π
x
x
π
S×S −−−−→m S
IfRis a congruence onS, which is closed onS×S we will call it aclosed congruence.
Iff:X →Y is a function from a spaceX to a spaceY and A⊂X, then
A is said to bef-saturatediff−1(f(A)) =A.
Iff:X →Y is surjective, thenf is said to be aquotient map ifW being open [closed] inY is equivalent tof−1(W) being open [closed] inX. Observe
that a quotient map is necessarly a continuous surjection. It is clear that each open continuous surjection fromX ontoY is a quotient map.
If X is a space, Y is a set, and f: X →Y is a surjection, then we can define a topology on Y by declaring a subsetW ⊂Y to be open if and only if f−1(W) is open inX. This topology is called thequotient topology on Y
induced byf. Note that ifY is given the quotient topology induced byf then
f is a quotient map.
IfRis a congruence on a semigroupS, we giveS/Rthe quotient topology induced by the natural mapπ:S→S/R.
We want now to present an example of a topological semigroup S and a closed congruenceRonS such thatS/Ris not a topological semigroup, since
S/R fails to be Hausdorff.
2
Results
We proceed to establish a sequence of lemmas which will be used to demon-strate that ifS is a locally compact,σ-compact semigroup andR is a closed congruence onS, thenS/Ris a topological semigroup. This result was estab-lished for compact semigroups in [1].
Lema 1. Let S be a topological semigroup and let R be a closed congruence on S such thatπ×π:S×S→S/R×S/R ia a quotient map. ThenS/Ris a topological semigroup.
Proof. Letmdenote multiplication onSandm′multiplication onS/R. Then the diagram,
(S/R)×(S/R) −−−−→m′ S/R
π×π
x
x
π
S×S −−−−→
m S
commutes. Sinceπ◦mis continuous andπ×πis a quotient map we get that
m′
is continuous. SinceR=π×π−1(∆(S/R)) is closed andπ×πis a quotient
map we have that ∆(S/R) is closed. Hence S/R is a Hausdorff topological space.
Lema 2. Let Rbe a closed equivalence on a spaceX and let K be a compact subset ofX. Thenπ−1(π(X))is closed.
Proof. Let π1: X×K →X denote the first projection. Thenπ−1(π(K)) =
π1((X×K)∩R) is closed.
Lema 3. LetX be a compact Hausdorff space and letRbe a closed equivalence on X. Then:
a) π:X →X/Ris closed.
b) if V is an open subset ofX, thenVR={x∈X :π−1(π(x))⊂V} is open
andπ-saturated.
c) If A is a closed π-saturated subset of X and W is an open subset of X
such thatA⊂W, then there exists an openπ-saturated subsetU and a closedπ-saturated subset K such that A⊂U ⊂K⊂W.
Proof. To prove a) let C be a closed subset of X. Then C is compact, and henceπ−1(π(C)) is closed. Sinceπis a quotient,π(C) is closed, soπis closed.
To prove b), observe thatVR =V −π−1π(X−V). Sinceπ is closed, we
have thatVR is open. It is clear thatVR isπ-saturated.
To prove c), observe thatA⊂WR⊂W, sinceA isπ-saturated. SinceA
is normal and WR is open, there exists an open set V such that A ⊂ V ⊂ V ⊂WR ⊂W. Let U =VR and K =π−1π(V). ThenU andK satisfy the
required conditions.
To prove d), letπ(x) andπ(y) be distinct points ofX/R. Thenπ−1π(x)
and π−1π(y) are disjoint closed subsets ofX. Since X is normal, there exist
disjoint open sets M and N containing π−1π(x) and π−1π(y) respectively.
We obtain thatπ(MR) andπ(NR)are disjoint open subsets ofX/Rcontaining π(x) andπ(y) respectively.
Lema 4. Let X be a locally compact σ-compact space. Then there exists a sequence {Kn} of compact subsets of X such that X =Sn∈NKn andKn ⊂ K◦
n+1 (the interior ofKn+1 inX) for eachn∈N.
Proof. LetX =S
n∈NCn, whereCnis compact for eachn∈N. LetK1=C1
and {Ki ∈ I} a finite cover of K1 by open sets of finite closure. Let K2 =
S
i∈IKi. SinceK2∪C2 is compact there exists a finite cover {Ki2:i∈I} of K2∪C2by open sets with compact closures. LetK3=
S
i∈IK
2
i. Continuing
recursively, letKn+1 be a compact subset such that Kn∪Cn⊂Kn◦+1.
The next result is due to Jimmy Lawson (see [5]). It was communicated to me while I was his student at Louisiana State University.
Theorem 2. LetS be a locally compactσ-compact semigroup and letR be a closed congruence onS. Then S/Ris a topological semigroup.
Proof. We will establish thatS/Ris a topological semigroup by proving that
π×π:S×S→S/R×S/Ris a quotient map and then applying Lemma 1. LetQbe a subset ofS/R×S/Rsuch thatπ×π−1(Q) is open inS×S.
We want to show that Qis open in S/R×S/R. Let (π(a), π(b)) ∈Q. By lemma 4, there exists a sequence {Kn} of compact subsets of S such that S =S
n∈NKn andKn⊂Kn◦+1,∀n∈N.
Assume, without loss of generality that bothaandblie inK1. LetV0=
π−1(π(a))∩K
1 andW0=π−1(π(b))∩K1. It is easy to see that bothV0and
W0 are compact subsets ofK1. By Wallace theorem, there exist setsGand
T which are open inK1such thatV0×W0⊂G×T ⊂π×π−1(Q)∩K1×K1.
compact, and π|K
1-saturated such thatV1 is a K1-neigborhood of V0, W1 is
a K1-neigborhood ofW0,V1⊂G, andW1⊂T. By lemma 2π−1(π(V1)) and
π−1(π(W
1)) are closed. Again by Wallace’s theorem, there exist set M and
N which are open inK2 so that
[π−1
(π(V1))∩K2]×[π−1(π(W1))∩K2]⊂M×N ⊂π×π−1(Q)∩K2×K2.
As above, there are sets V2 and W2 which are closed in K2 (and hence
compact),π|K
2-saturated and such thatV2 is a K2-neigborhood of
π−1(π(W
1))∩K2,V2⊂M andW2⊂N.
Define recursively a pair of towered sequences{Vn} and{Wn} such that Vn and Wn are closed in Kn (and hence compact) and π|Kn-saturated such
thatVn is aKn-neigborhood ofπ−1(π(Vn−1))∩Kn,Wn is aKn-neigborhood
ofπ−1
(π(Wn−1))∩Kn andVn×Wn⊂π×π−1(Q) for eachn∈N.
LetV =S∞
n=1Vn andW =S
∞
n=1Wn. It is clear thatπ−1(π(a))⊂V and π−1(π(b))⊂W. Observe thatV isπ-saturated sinceV =S∞
n=1π
−1(π(Vn)),
and similarly W isπ-saturated.
To see thatV is open, letx∈V. Thenx∈Vn for somen∈N. NowVn+1
is a Kn+1-neighborhood ofVn and Vn⊂Kn◦+1. It follows thatVn+1∩Kn◦+1
is a neigborhood of Vn. Observing that Vn ⊂Vn+1∩Kn◦+1 ⊂Vn+1⊂V, we
see thatV is a neigboprhood ofx, and hence it is open.
A similar argument proves thatW is open. Finally, (π(a), π(b))∈π(V)×
π(W) ⊂Q and π(V)×π(W) is open since V and W are open π-saturated subsets ofS.
References
[1] Wallace, A.D., On the structure of topological semigroups, Bull. Amer. Math. Soc.,61(1955), 95–112.
[2] Lawson, J.D., Joint continuity in semitopological semigroups, Math. J., 18(1974), 275–285.
[3] Lawson, J. D., Williams, W. W.,Topological semilattices and their under-lying spaces, Semigroup Forum,1(1970), 209–223.
[4] Gonz´alez, G. Locally Generated Semigroups, A Dissertation Submitted to the Graduated Faculty of the Louisiana State University and Agricultural and Mechanical College, 1995.
