Download O. Frink and G. Grätzer, The closed subalgebras of a topological

154
ARCH. MATH.
The Closed SubaJgebras or a Topological Algebra
By
ORRIN FRINK *) and GEORGE GRATZER
By an algebra we mean an abstract algebra <X, {fJ}) with finitary operations {fJ},
in the sense of GARRETT BIRKHOFF [lJ. Such an algebra may also be a topological
space <X, Y), where Y is the family of all closed sets of the space X. The topology
mayor may not be related to the algebraic operations {fj}. It is easily seen that the
family ~ of all subalgebras of X which are also closed sets in the topology has the two
properties: I. X E~, and II. ~ is closed under arbitrary non-empty intersection. We
call any family of sets with properties I and II a Moore ff!mily (after E. H. MOORE),
following the terminology of GARRETT BIRKHOFF. The term closure family is also used.
It might be supposed that every Moore family of sets is the family of all closed
subalgebras of some algebra with topology, but we show this is not the case. In this
note, we derive conditions which are necessary and sufficient in order that a family of
sets shall be the family of all closed subalgebras of an algebra with topology, both in
she general case where the algebra and topology are unrelated, and in the case of a
topological algebra, where it is assumed that the closure of every subalgebra is a
tubalgebra. This includes the case where the operations {Ii} are continuous in the
topology. We investigate also the semilattice of all finitely generated closed subalgebras of a topological algebra.
U
Definitions and terminology. A family ~ of subsets of a set X =
~ is a Moore
family if X E C(j' and C(j' is closed under arbitrary non-empty intersection. It is well
known [2] that a family d of subsets of a set X is the family of all subalgebras of an
algebra with finitary operations if and only if d is a Moore family, and in addition d
contains the union of each directed subfamily of d (d is directed-union closed). An
equivalent condition is that d be chain-union closed, that is, that d contain the
union of each chain of its sets. The equivalence of the conditions directed-union closed
and chain-union closed was proved by J. SCHMIDT and also by J. MAYER-KALKSCHMIDT and E. STEINER [4]. A Moore family which is directed-union closed (or
chain-union closed) will be called an algebraic family.
If C(j' is any Moore family, we shall denote by d (~) the family of all unions of
directed subfamilies of ~. It is easily verified (see [4J) that d (C(j') is an algebraic
family; hence it is the smallest algebraic family containing C(j'. We call it the algebraic
family generated by C(j'.
*) Supported by National Science Foundation Research Grant GP-3132.
Vol. XVII, 1966
The Closed Subalgebras of a Topological Algehra
155
By a topological family :T we mean a Moore family of subsets of a set X which is
closed uuder finite union. The pair <X,:T) is then a topological space, and:T is the
family of all closed sets of the space. If'1J is any Moore family, we denote by fF ('1J)
the family of all finite unions of members of'1J, and by :T ('1J) the family of all nonempty intersections of members of fF ('1J). It is well known that :T ('1J) is a topological
family; it is the smallest topological family containing '1J, and we call it the topological
family generated by '1J. The family fF ('1J) is a base for the closed sets of the topology,
and the family '1J is a sub-base.
Given a Moore family '1J and any subset A of X = U'1J, we define the closure A
of A to be the closure of A in the topology :T('1J) generated by '1J, that is,
A = n{FEfF('1J): FJA}.
Families or closed subalgebras. Suppose X is an algebra, and sf is the algebraic
family of all subalgebras of X. If X is also a topological space, and:T is the family of
all closed subsets of X, then '1J = sf n :T is the family of all closed subalgebras of X.
Clearly '1J is a Moore family, since sf and:T are Moore families. The family '1J need be
neither chaiu-union closed nOr finite-union closed, as can be seen from the case of the
closed subgroups of a tElpologieal group. We now derive a necessary and sufficient
condition for a Moore family of sets to be the family of all closed subalgebras of an
algebra which is also a topological space.
Theorem 1. A Moore family'1J is the family of all dosed suhalgebras of some algebra X
which is also a topological space if and only if the condition:
(i)
holds, that is, if and only if '1J is the family of all members of the algebraic family generated
by '1J, which are closed in the topology generated by '1J.
Proof. The condition (i) is clearly sufficient, since sf('1J), as an algebraic family, is
the family of all subalgebras of an algebra constructed on the set X = U'1J, and:T ('1J)
is the family of all closed sets of a topology for X. To show that (i) is necessary, let'1J
be the family of all closed subalgebras of an algebra X with topology, and let B be any
subset of X which is a member of sf ('1J) and of :T('1J). Then B is a subalgebra of X,
since it is the directed union of subalgebras of X, by definition of the family sf ('1J).
B is also a closed set, since by definition of the family :T('1J), it is the intersection of
finite unions of closed sets. Hence B is a closed subalgebra, and therefore it is a
member of '1J. It follows that '1J J sf ('1J) n:T ('1J). Conversely '1J c sf ('1J) n:T('1J),
since sf ('1J) and:T ('1J) are extensions of '1J. This gives condition (i), and completes the
proof.
A counter-example. To show that not every Moore family of sets satisfies condition
(i), we consider the following example: Let X be the real closed interval [0, 1J, and
let the family '1Jof subsets of X consist of (1) all finite subsets of X, and (2) the three
closed intervals [0, 1J, [0,1/ 4J, and [1/4, 1/2J. It is easily verified that '1J is a Moore
family. The family sf('1J) consists of all subsets of X, since every subset is the union
of the directed family of its finite subsets. Hence the closed interval [O,l/2J is in
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O. FRINK and
G. GUATZER
ARCH. MATH.
d ('6'). The interval [0, 1/2 ] is also in.r (et'), since it is the finite union of the intervals
(0,1/ 4] and [1/ 4, 1/ 2]. But the interval [0, 1/ 2J is not in et'. Hence et' *d(et') n .r(et'),
and condition (i) is not satisfied. It follows that not every Moore family is the family
of all closed subalgebras of an algebra with topology.
Topological algebras. If an algebra X is also a topological space, the algebra may
be related to the topology in some way. For example, the operations of the algebra
may be continuous in the topology. In this case it may be verified that the topological
closure of every subalgebra is also a subalgebra. If the family d of all subalgebras
of X is given, the operations {fi} of the algebra are not determined uniquely. Hence
the question of whether these operations are continuous in the topology becomes rather
indefinite, if only the family d is known. It therefore seems natural to adopt the
following definition as a generalization of the notion of a topological algebra.
Definition. An algebra X with finitary operations which is also a topological space
is called a topological algebra if the closure of every subalgebra of X is also a sub·
algebra.
It is interesting to note that if we reverse the role of the families d of subalgebras
and .r of closed subsets in this definition, the result is less useful. In a topological
algebra it is not always true that the subalgebra generated by a closed set is closed.
Consider the following example. Let X be the additive group of the real numbers
with the usual topology, and let the set A consist of the numbers 0 and {1/n}, with
n = 1,2, .... The set A is closed. The subalgebra generated by A consists of all the
rational numbers, and is not closed.
Having defined the notion of topological algebra, we now give a characterization of
all closed subalgebras of a topological algebra.
Theorem 2. A family et' of subsets of a set is the family of all closed subalgebras of a
topological algebra if and only if r'(j' is a Moore family which satisfies the condition:
A Ed(et')
(ii)
A ECC,
that is if the closure A in the topology generated by CC, of any subalgebra in the algebraic
family d (et') generated by CC, is also a member of CC.
Proof. The condition is sufficient, for if CC is a Moore family which satisfies (ii), and
X =
et', then CC is the family of all subalgebras of the algebra generated by CC
which are closed in the topology generated by CC, that is, condition (i) holds. Hence
CC is the family of all closed subalgebras of an algebra with topology. It follows from
condition (ii) that the closure of every subalgebra is a member ofCC, and is therefore a
subalgebra. Hence by definition, X is a topological algebra.
To prove the necessity of the condition, suppose that CC is the family of all closed
subalgebras of a topological algebra X, and A is any member of d (CC). The set A is a
member of.r (CC), and hence it is the intersection of finite unions of closed sets. Hence
A is a closed set. Since by assumption X is a topological algebra, and A is a subalgebra
of X, A is also a subalgebra. Since A is a closed subalgebra, it is a member of CC, and
condition (ii) is verified. Since CC is also a Moore family, this completes the proof of
Theorem 2.
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VoLXVU.196{'
The Closed Suhalgehras of a Topological Algehra
157
Another counter-example. It has been shown that condition (ii) implies condition
(i). To sbow that (ii) is actually stronger than (i), we give an example of a Moore
family which satisfies one but not the other.
Let X = N u {a} U {b} be a set consisting of the positive integers N and the two
special elements a and b. Let E be the set consisting of the even positive integers and
the element a, and 0 be the set consisting of the odd positive integers and a. Then
E (\ 0
{a},andE U 0 = N u {a}.
Define the family'(f as follows: '?! consists of (1) all finite subsets of N (including the
empty set), and (2) the sets X, 0, E, and {a}. It is easily verified that'?! is a Moore
family. The algebraic family d ('?!) generated by '?! is made up of unions of directed
subfamilies of '?!. Hence it consists of the sets X, 0, E, {a}, and all subsets of N. In
particular it contains the set N itself, but not the set N U {a}.
The topological family .r ('?!) consists of intersections of finite unions of members
of'?!. It contains all sets ofthe form E U F and 0 U F, where F is a finite subset of N.
It contains the set N U {a}
E U 0, but not the set N. We now verify that condition (ii) does not hold, by taking the set A of the condition to be N. The closure N is
the set N V {a}, which is the smallest set of .r ('?!) which contains N. But N u {a} is
not a member of '?!, although N is a member of d('?!). Hence condition (ii) is not
satisfied. It can be verified, however, that condition (i) holds for this family'?!, since
'{f = d ('?!) (\ .r ('?!).
It follows that it is possible to have a family of sets which is·the family of aU closed
subalgebras of an algebra with topology, but not of any topological algebra.
Finitely generated closed subalgebras. If X is an algebra with topology or a topological algebra and F is any finite subset of X, then the intersection of all closed subalgebras of X which contain F is a closed subalgebra, and is the smallest closed subalgebra containing F. It is said to be the closed subalgebra generated by F, and it is
also said to be finitely generated.
The family of all closed subalgebras of X is a Moore family and hence a complete
lattice. If A and B are two finitely generated closed subalgebras of X, then the join
A U B of A and B in this lattice is also a finitely generated closed subalgebra.
However, the meet A (\ B, which is the intersection of A and B, is in general not
finitely generated, although it is a closed subalgebra. Hence the family of aU finitely
generated closed subalgebras of X is an upper semilattice, but not in general a lattice.
If '?! is the family of all closed subalgebras of X, and S is the semilattiee of all
finitely generated closed subalgebras, then every member 0 of'?! is the set union of
the directed family D(O) consisting of all elements of S which are contained in O. It
is easily verified that the family D (0) is an ideal of the semilattice S. A subset D of a
semilattice S is caned an ideal if the join A U B of two elements of S is in D if and
only if both A and B are in D.
The correspondence 0 -+ D{O) is thus a one-to-one order-preserving mapping of
the complete lattice'?! into the lattice of all ideals of a semilattice. Conversely, the
lattice of an ideals of any semilattice is the family of all subalgebras of an algebra
with finitary operations, and with the discrete topology is the family of all closed
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O. FRINK and
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ARelI. MATlI.
subalgebras of a topological algebra. The finitely generated closed subalgebras in this
case are just the principal ideals. We state these facts in the form of a theorem.
Theorem 3. The family of all closed subalgebras of an algebra with topology is lattice
isomorphic to a complete lattice which is a family of ideals of a semilattice. The semilattice
may be taken to be the family of all finitely generated closed subalgebras of the algebra. In
this isomorphism, the finitely generated closed subalgebras correspond to the principal
ideals. Conversely, the family of all ideals of any semilattice is the family of all closed
subalgebras of a topological algebra.
In this representation, the lattice of ideals corresponding to the closed subalgebras
will not usually include all the ideals; in general it is not even a sublattice of the
lattice of all ideals.
Conclusion. There are some interesting related problems which we have not solved.
One is to characterize the family of all closed snbalgebras in the case where it is
assumed that the algebraic operations are continuous in the topology. If the operations are not determined uniquely by the family of closed subalgebras, is there at
least one determination in which they are continuous? Another problem is to characterize the families of finitely generated closed subalgebras of a topological algebra.
Added in proof: The last problem is solved in the paper "On the family of certain subalgebras
of a universal algebra". ~ndagationes Math. 27,790-802 (1965). See esp. Cor. 2 to Theorem 1.
References
[1] G. BmKHoFF, Lattice Theory. Revised edition, Providence 1948.
[2] G. BIRKHOFF and O. FRINK, Representations of lattices by sets. Trans. Amer. Math. Soc. 64,
299-316 (1948).
[3] R. P. DILWORTH and P. CRAWLEY, Decomposition theory for lattices without chain condition.
Trans. Amer. Math. Soc. 96,1-22 (1960).
[4] J. MAYER-KALKSCHMIDT and E. STEINER, Some theorems in set theory and applications in the
ideal theory of partially ordered sets. Duke Math. J. 31, 287 -289 (1964).
[5] L. NACH:BIN, On a characterization of the lattice of all ideals of a Boolean ring. Fundamenta
Math. 36,137-142 (1949).
[6] J. ScmuDT, tJber die Rolle der transfiniten Schlullweisen in einer allgemeinen Idealtheorie.
Math. Nachr. 7, 165-182 (1952).
[7] J. SCHMIDT, Einige grundlegende Begriffe und Satze aus der Theorie der Hiillenoperatoren.
Ber. Math. Tagung Berlin 21-48, Berlin 1953.
Eingegangen am 2.10.1964
Anschrift der Autoren:
G. Gratzer
O. Frink
College of the Liberal Arts
Department of Mathematics
The Pennsylvania State University
230 McAllister Building
University Park (Pa.), USA