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arXiv:1901.07364v1 [math.DS] 22 Jan 2019
A STUDY OF HOLOMORPHIC SEMIGROUPS
BISHNU HARI SUBEDI
Abstract. In this paper, we investigate some characteristic features of holo-
morphic semigroups. In particular, we investigate nice examples of holomor-
phic semigroups whose every left or right ideal includes minimal ideal. These
examples are compact topological holomorphic semigroups and examples of
compact topological holomorphic semigroups are the spaces of ultrafilters of
semigroups.
1. Introduction
Semigroups are very classical algebraic structures with a binary composition
that satisfies associative law. It naturally arose from the general mapping of a set
into itself. So, a set of holomorphic functions on the complex plane Cor Riemann
sphere C∞or certain subsets thereof naturally forms a semigroup. We call that any
analytic map on Cor C∞or on certain subsets thereof by a holomorphic function.
Definition 1.1 (Holomorphic semigroup and subsemigroup).Aholomorphic
semigroup Sis a semigroup of holomorphic functions defined on Cor C∞or certain
subsets thereof with the semigroup operation being functional composition. A non-
empty subset T⊆Sis a subsemigroup of Sif f◦g∈Tfor all f, g ∈S.
Let
(1.1) F={fα:fαis a holomorphic function for all α∈∆},
where index set ∆ is allowed to be infinite in the case of discrete semigroups and un-
countable in the case of continuous semigroups. In this sense, a holomorphic semi-
group is a set Sof holomorphic functions from Fsuch that fα+β(z) = fα(fβ(z))
for all z∈Cor C∞or certain subsets thereof and for all α, β, α +β∈∆. If ∆ ⊆N
then Sis called one parameter discrete semigroup and such semigroup consists all
iterates fn, n ∈Nof the function f. If ∆ = (0, T ), T > 0, an interval of the real
line R, then Sis called one parameter continuous semigroup and the iterates ft
with t∈(0, T ), T > 0 is a fractional iterates of f.
We are more interested in special holomorphic semigroups whose each element
can be expressed as a finite composition of certain holomorphic functions. More
formally, such a semigroup is defined as follows:
Definition 1.2 (Holomorphic semigroup generated by holomorphic func-
tions).Let Fbe a family of holomorphic functions as defined in 1.1. A holo-
morphic semigroup Sgenerated by Fis a semigroup of all elements that can be
1991 Mathematics Subject Classification. Primary 30D05, Secondary 58F23.
Key words and phrases. Holomorphic semigroup, ideals, abundant semigroup, compact topo-
logical semigroup, filter and ultra filter.
This research work is supported by a Ph.D Faculty Fellowship of the University Grants Com-
mission, Nepal and NMS-Nick Simons Fellowship of the Nepal Mathematical Society, Nepal.
1
2 BISHNU HARI SUBEDI
expressed as a finite composition of elements in F. We denote such holomorphic
semigroup by
S=hfαiα∈∆or simply S=hfαi.
Holomorphic semigroup Sis said to be a rational semigroup or a transcendental
semigroup depending on whether Fis a collection of rational functions or tran-
scendental entire functions. In particular, Sis said to be a polynomial semigroup if
Fis a collection of polynomial functions. The transcendental semigroups or poly-
nomial semigroups are also called entire semigroups. The holomorphic semigroup
Sis abelian if
fα◦fβ=fβ◦fα
for all generators fα,fβof S. Indeed, the semigroup S=hfαiis the intersection of
all semigroups containing Fand it is a least semigroup containing F. Note that
each f∈Sis constructed through the composition of finite number of functions fα
for α∈∆. That is,
f=fα1◦fα2◦ · · · ◦ fαm
for some m∈Nand α1, α2,...αm∈ {1,2,...,m}. A semigroup generated by
finitely many holomorphic functions fj,(j= 1,2,...,n) is called a finitely generated
holomorphic semigroup and we write
S=hf1, f2,...,fni.
If Sis generated by only one holomorphic function f, then Sis called a cyclic
holomorphic semigroup, and we write
S=hfi.
In this case, each g∈Scan be written as g=fn, where fnis the nth iterate of f
with itself. We say that S=hfiis a trivial semigroup.
Example 1.1. Let Sbe a set consisting all of powers zwhich are either all powers
of 2 or all powers of 3 or product of all powers of 2 and 3. Then Sforms a semi-
group under functional composition. It is a finitely generated polynomial semigroup
generated by two polynomials z→z2and z→z3. In this case, S=hz2, z3i.
Example 1.2. Let α∈Csuch that Re α≥0. For any k∈N, the function
fk(z) = e−αkzfor all z∈C
is holomorphic in the complex plane Cand so S={fk:k∈N}is a holomorphic
(in particular, entire) semigroup.
The particular examples of holomorphic semigroup given in Example 1.2 are
S={e−kz:k∈N},
a dilation semigroup, and
S={eikz:k∈N},
a rotation semigroup. Note that semigroup of Example 1.2 is a linear action on
Cor certain subsets thereof. That is, each one parameter semigroup (discrete or
continuous) which is a linear action has the form
S={fk:k∈N}
where fk(z) = e−αkzfor some α∈Cand Re α≥0.
A STUDY OF HOLOMORPHIC SEMIGROUPS 3
Example 1.3. If we choose index set ∆, a set of positive rational numbers Q+in
Example 1.2, then each function
fr(z) = e−αrzfor all z∈Cand r∈Q+
for some α∈Csuch that Re α≥0, can be written as a finite composition of
ft(z) = e−αtzwith t∈Q+. Hence S=hftit∈Q+is a holomorphic semigroup
generated by the set {ft:t∈Q+}.
Note that if we choose ∆ = R+∪ {0}, the set of non-negative real numbers, then
the semigroup of Example 1.2 is a (semi) flow on C. Such a semigroup is also an
example of one parameter continuous holomorphic semigroup. That is, in general,
if the index set of holomorphic semigroup is a set of non-negative real numbers,
then the semigroup is known as flow or one parameter continuous semigroup. If
∆ = (−R, R), R > 0 (a real number), then Sis a group which is called one
parameter holomorphic group and each element ftin Sis an biholomorphic mapping
(automorphism) of C∞whose inverse is f−t.
Example 1.4. Define a function ft, t ∈R, by
ft(z) = z+ tan ht
ztan ht + 1.
It is easy to see that for each t∈R, ftis an automorphism, Hence, one parameter
holomorphic group S={ft:t∈R}is a flow of automorphisms.
2. Ideal Theory of holomorphic semigroups
Unless otherwise stated, we may assume onward that Sis a discrete holomorphic
semigroup. There are certain subsets of semigroups with a stronger closure property
rather than that of subsemigroups.
Definition 2.1 (Left ideal, right ideal and two sided ideal).Let Ibe a non-
empty subset of holomorphic semigroup S. We say Iis a left ideal (or right ideal)
of Sif f◦h∈I(or h◦f∈I) for all f∈Sand h∈I, that is, SI ⊂I(or IS ⊂I).
We say Iis two sided ideal (or simply ideal) if it both left and right ideal.
Note that if Sis an abelian semigroup, then the notions of left ideal, right
ideal, and two sided ideal coincide. For any non-empty subset Kof a holomorphic
semigroup S, the sets
SK ={f◦g:f∈S, g ∈K}=[
g∈K
S◦g,
KS ={g◦f:f∈S, g ∈K}=[
g∈K
g◦S,
and
SK S ={f◦g◦h:f , h ∈S, g ∈K}=[
g∈K
S◦g◦S
are respectively left, right and two sided ideals. Likewise, for any g∈S, the sets
S◦g,g◦Sand S◦g◦Sare respectively left, right and two sided ideals. In general,
gmay not be in g◦S( or S◦gor S◦g◦S) for each g∈S. If it happens to be
in g◦S( or S◦gor S◦g◦S), then g=g◦f(or g=f◦gor g=f◦g◦h)
for some f, h ∈S. In this case, g◦S(or S◦gor S◦g◦S) is a smallest right (or
left or two sided) ideal containing g, which is a right (or left or two sided) ideal
4 BISHNU HARI SUBEDI
generated by g. Otherwise, g◦S(or g◦Sor S◦g◦S) is said to be quasi-generated
by g. It is obvious that the union of any non-empty family of left (or right or two
sided) ideals of Sis a left (or right or two sided) ideal of S. On the basis of some
topological structure of the complement of g◦Sin S, we can define the following
types of holomorphic semigroup. Recall that a space Xis compact if every open
covering of Xcontains a finite subcover.
Definition 2.2 (F-semigroup and C-semigroup).Let Sis a holomorphic semi-
group and g∈S. Then we say
(1) Sis a F-(right) semigroup if S−g◦Sis finite;
(2) Sis a C-(right) semigroup if S−g◦Sis relatively compact (that is, S−g◦S
is compact in S).
Analogously, we can define F-(left) semigroup and C-(left) semigroup of any
holomorphic semigroup S. We say only F-semigroup and C-semigroup onward for
such holomrphic semigroup Son the assumption that left/right is clear from the
context. For example, holomorphic semigroup of Example 1.2 is both F-semigroup
and C-semigroup and that of Example 1.3 is a C-semigroup. The holomorphic
trivial semigroup is also both F-semigroup and C-semigroup. Every (semi) flow,
that is, one parameter continuous semigroup is a C-semigroup.
There are certain type of left (or right) ideals which are connected to two sided
ideals. That is, on the basis of such ideals of holomorphic semigroups, we can
construct two sided ideals. This ideal structure is defined as follows.
Definition 2.3 (Minimal left (or right) ideal).A left (or right) ideal Mof
holomorphic semigroup Sis minimal if for every left (or right) ideal Iof Ssuch
that I⊆M, we have M=I.
Note that minimal left (or right) ideal of Smay be empty. If it is non-empty for a
certain holomorphic semigroup S, then for every f∈M, there must be M◦f=M
(or f◦M=M) and S◦f=M(or f◦S=M). Also, a minimal left (or right)
ideal is always contained in every two sided ideal of S. We can also make two sided
ideal by the help of minimal left (or right) ideals. For any holomorphic semigroup
S, let us define
K(S) = [{M:Mis a minimal left (or right) ideal of S}
Since K(S) is non-empty if and only if Shas at least one minimal left (or right)
ideal and in such a case it is itself a minimal left (or right) ideal. So, as stated
above, it is contained in every two sided ideal of S. Since for all f∈S
K(S)◦f=[{M◦f:Mis a minimal left (or right) ideal of S} ⊆ K(S)
So, K(S) is also left (or right) ideal as well. From this discussion, we can conclude
the following result.
Proposition 2.1 (Example of minimal two sided ideal).For any holomorphic
semigroup S, K(S)is a minimal two sided ideal of Sif it is non-empty.
Proof. See for instance in [2, Theorem 2.9].
We can define a special type of holomorphic semigroup Swhere K(S) is non-
empty. This type semigroup has some special features such as every left (or right)
A STUDY OF HOLOMORPHIC SEMIGROUPS 5
ideal of Sinclude minimal one and every left (or right) ideal of Scontains a spe-
cial element which is called an idempotent. Recall that an element e∈Sis an
idempotent if e◦e=e.
Definition 2.4 (Abundant semigroup).A holomorphic semigroup Sis abun-
dant if every left (or right) ideal of Sincludes a minimal one and every minimal
left (or right) ideal contains an idempotent element.
It is obvious that K(S)6=∅if Sis abundant. There is a topologically significant
nice examples of abundant semigroups which can be defined as follows.
Definition 2.5 (Compact right topological holomorphic semigroup).Let
Sbe a holomorphic semigroup and g∈S.
(1) We define a right translation map Fg:S→Swith respect to gby Fg(h) =
h◦gfor all h∈S.
(2) We define a compact holomorphic right topological semigroup by the pair
(S, τ ), where τis a topology on Ssuch that the space (S, τ ) is compact and
Hausdorff and right translation map Fgfor every g∈Sis continuous with
respect to τ.
Note that a left translation map and a compact holomorphic left topological
semigroup are defined similarly. Also note that in a compact right topological
semigroup, we do not require that left translation maps are continuous. We say
only translation map and compact holomorphic topological semigroup if left or right
is clear from the context.
Example 2.1. Let S={ft:t∈Q+∪ {0}} be a holomrphic semigroup, where ft
is a function of Example 1.3 for all t∈Q+∪ {0}. The collection of all subsets of
Sforms a topology τon Sand hence it is also open cover of S. There are some
finite number of elements in τthat can cover S. Hence space (X, τ ) is compact and
Hausdorff as well. Therefore, this semigroup is compact holomrphic topological
semigroup and hence abundant.
Note that in the holomorphic semigroup Sof Example 2.1, there is an element
f0(z) = e−α0zsuch that f0◦f0=f0which an idempotent by definition. Therefore,
unless holomrphic semigroup in general, abundant semigroup and in particular,
compact holomorphic topological semigroup do have idempotents.
Theorem 2.1 (Idempotents exist for holomorphic semigroup).Let Sbe a
compact holomorphic topological semigroup. Then there is an element e∈Ssuch
that e◦e=e.
This theorem can be proved as a standard application of Zorn’s lemma from set
theory. It states that if every chain Cin a partially ordered set (S, ≤)has upper
bound in S, then (S, ≤)has a maximal element. Note that partial ordered set is
a system consisting of non-empty set Sand a relation denoted by ≤satisfying the
properties of anti-symmetry, reflexivity and transitivity. A chain Cin a partial
ordered set (S, ≤) is subset of Ssuch that for every x, y ∈C, either x≤yor x≥y.
An element m∈Sis a maximal element of (S, ≤) if m≤xfor x∈Simplies m=x.
Sketch of the Proof of Theorem 2.1. The proof of this theorem follows via the fol-
lowing two facts:
Fact 1 :Shas a minimal close subsemigroup.
6 BISHNU HARI SUBEDI
Let τbe a family of all closed subsemigroups of semigroup S. Then τ6=∅and it
is a topology of closed sets partially ordered by the reverse inclusion. Let C⊂τ
be a chain in (τ , ⊇). Since Sis compact and Chas the finite intersection property.
So, ∩T∈CTis non-empty and serves as a least upper bound of C. Then by Zorn’s
lemma, τhas a maximal element M(say) where M⊆ ∩T∈CT. In reality of this
context, Mis minimal closed subsemigroup of S.
Fact 2 : If e∈M, then M={e}and eis an idempotent.
We can consider two cases of the proof of this fact 2.
Case 1 : We prove M=M◦e={e}. Let e∈M. Then M◦eis a subsemigroup
of M. The map Fe:M→M◦e, Fe(h)→h◦eis a right translation map of
topological holomorphic semigroup Srestricted to M. Then it is continuous. Since
Mis compact so M◦eis also compact as a image of compact set under contin-
uous map Fe. Since by fact 1, Mis minimal and M◦eis non-empty, we must
M=M◦e={e}. This proves that eis an idempotent.
Case 2 : We prove N={f∈M:f◦e=e}=M. By the construction of set N,
it is subset of Mand closed under functional composition. This shows that Nis
a subsemigroup of M. By the case 1, e∈M=M◦ecan be written as e=f◦e
for some f∈M. This shows that e∈N6=∅. Finally, Ncan be written as the
intersection of closed subsets of Mand F−1
e{{e}}. This proved Nis a non-empty
closed subsemigroup of M. Then as in the case 1, N={e}.
Since every left (or right) ideal is a subsemigroup of S. So, from the proof
of this theorem 2.1, we can say the following important facts for every compact
holomorphic topological semigroup S.
(1) Every left (or right) ideal of the form S◦f(or f◦S) is the image of
continuous right translation map Ff(g) = g◦ffor all g∈S. So S◦f(or
f◦S) is closed in S.
(2) For every left (or right) ideal Iof S, we have S◦f⊂I(or f◦S⊂I). Thus
every left (or right) ideal contains closed ideal.
(3) For every minimal left (or right) ideal Mof S, we have M=S◦f(or
M=f◦Sfor all f∈M. This shows that every minimal left (or right)
ideal of Sis closed.
(4) Every left (or right) ideal contains an idempotent element.
(5) Let I⊂J, where Jis an arbitrary left (or right) ideal and Iis a closed
left (or right) ideal of S. By Zorn’s lemma, there is a minimal element L
of (I,⊆), where Iis the family of closed left (or right) ideals contained in
I. This shows Lis a minimal left (or right) ideal of S. That is, every left
(or right) ideal Scontains minimal left (or right) ideal.
From all of above facts, we may conclude the following result as well.
Theorem 2.2 (An example of abundant semigroup).Every compact holo-
morphic topological semigroup is abundant.
There is a close connection between minimal left ideals and certain idempotents
(minimal idempotents). If we suppose a set of idempotents of arbitrary holomorphic
semigroup by E(S), then it may be empty but it is non-empty for abundant holo-
morphic semigroup. Minimal idempotent can be defined in a partial order relation
on E(S).
Definition 2.6 (Dominated element of set of idempotents).Let E(S) be a
set of idempotents of (abundant) holomorphic semigroup S. We say eis domonated
A STUDY OF HOLOMORPHIC SEMIGROUPS 7
by fon E(S) and write
e≤fif and only if e◦f=f◦e=efor all e, f ∈E(S).
This is clearly partial order relation on E(S). That is, for all e, f, g ∈E(S), we
have
(1) e≤e.
(2) e≤fand f≤e=⇒e=f.
(3) e≤fand f≤g=⇒e≤g.
An element e∈E(S) is minimal if there is no element of E(S) strictly less than
e. That is, if h≤e, then h=efor all h∈E(S). In a abundant holomorphic
semigroup, minimal idempotents are tightly connected to minimal left (or right)
ideals.
Theorem 2.3 (A connection between minimal ideal and minimal idem-
potant).Let Sbe a holomorphic abundant semigroup and e∈E(S). Then
(1) If L⊆S◦e(or L⊆e◦S) is a minimal left (or right) ideal, then there is
some idempotent f∈Lsuch that f≤e.
(2) eis a minimal idempotent if and only if e∈Lfor some minimal left (or
right) ideal L, that is, if and only if e∈K(S).
(3) eis minimal idempotent if and only if the left (or right) ideal L=S◦e(or
L=e◦S) is minimal.
(4) There is some minimal idempotent fsuch that f≤e.
Proof. We prove this theorem for left ideal Lof semigroup S. By symmetry, the
same is true for right ideal.
(1). Since Lis minimal left (or right) ideal, so there an idempotent in L. Since
Sis abundant, so there is an i∈L∩E(S). Since i∈L⊆S◦e, so i=s◦efor
some s∈S. Now
i◦e=s◦e◦e=s◦e=i.
Let e◦i=f. We will show that fis our required element. First of all f∈L
because Lis a left ideal. Secondly, fis an idempotent because
f◦f=e◦i◦e◦i=e◦i◦i=e◦i=f.
Finally, fis an minimal idempotent because
e◦f=e◦e◦i=e◦i=f
and
f◦e=e◦i◦e=e◦i=f.
(2). Let us suppose that eis a minimal idempotent of S. Let us make a set
L={L⊆S:Lis a minimal left ideal of S}
Since Sis abundant, so there is L∈ L such that L⊆S◦e. By (1), there is
f∈L∩E(S) such that f≤e. Since eis a minimal idempotent, we must have
f=e. Therefore, e∈L. Conversely assume that e∈Lfor some minimal left ideal
L∈ L. We have to show that eis an minimal idempotent. Let h∈E(S) with
h≤e. Then S◦h⊆S◦e. By minimality of L, we can write
L=S◦h=S◦e.
So e=g◦hfor some g∈S. By h≤e, we can write h=e◦hand so
h=e◦h=g◦h◦h=g◦h=e.
8 BISHNU HARI SUBEDI
This proves eis a minimal idempotent.
(3). This follows from the observation of converse part of (b) that if e∈Land
Lis minimal left ideal, then L=S◦e.
(4). By abundancy of Sand (a), there is a minimal left ideal such that L⊆S◦e
and f∈L∩E(S) such that f≤e. Then fis minimal by (b).
From the proof of this theorem, it is not hard to conclude that minimal idempo-
tents of abundant holomorphic semigroup Sare just those in K(S), that is, they
are from E(S)∩K(S). This Theorem 2.3 is also an important source of alternative
definition of minimal idempotent. That is, on the basis of this Theorem 2.3 (2),
minimal idempotent can also be defined as follows. An idempotent eon semigroup
Sis minimal if it belongs to the minimal left (or right) ideal. Also, from the same
Theorem 2.3 (3), we can say that minimal left (or right) ideal always is of the form
S◦e(or e◦S), where eis a minimal idempotent in S.
3. On filters and ultrafilters of holomorphic semigroups
Now, we are interested to investigate more rigorous examples of compact topo-
logical holomorphic semigroups (that is, abundant semigroups). First, we define
the following notion of filters and ultrafilters:
Definition 3.1 (Filter and ultrafilter of a holomorphic semigroup).Let S
be a holomrphic semigroup. A filter Fon Sis a family of non-empty subsets of S
with the following properties:
(1) ∅/∈F.
(2) A∈Fand A⊂B=⇒B∈F.
(3) A, B ∈F=⇒A∩B∈F.
A filter Fis an ultrafilter (or maximal filter, or prime filter ) if there is n∈Nsuch
that
S=A1∪A2∪...∪An
for some iin 1 ≤i≤nand Ai∈F. The semigroup Stogether with the (ultra)
filter Fis called a semigroup (ultra) filtered by For an (ultra)filtered semigroup.
The condition stated for ultrafilter implies that there is no filter on Swhich is
strictly finer than F. In other words, Fis an ultrafilter if and only if for every
two disjoint subsets A, B ∈Ssuch that A∪B∈F, then either A∈For B∈F.
Equivalently, if Fultrafilter and E⊂S, then either E∈For S−E∈F.
From the conditions (1) and (3), we can say that an (ultra) filter Fsatisfies the
finite intersection property, that is, intersection of any finite number of sets of the
family is non-empty. From this facts, we can say that every filter is extendible to
an ultrafilter. Not only filters but any family of subsets of Scan be extendibe to
ultrafilters if and only if this family has finite intersection proporty. Also, every
non-empty subsets of Sis contained in an ultrafilter. The space of all ultrafilters on
a semigroup Sis denoted by βS which is a Stone-Cech compactification of S. The
space βS is quite large in the sense that its cardinality is equal to the cardinality of
P(P(S)) where P(S) represets power set of Sand indeed, it is extremely large to
be metrizable of βS. Filter and ultrafilter are topological terms and most powerful
tools in topology and set theory, but we have adopted these terms from [1] where
it is defined as an extension set of the additive semigroup N.
A STUDY OF HOLOMORPHIC SEMIGROUPS 9
Note that each ultrafilter F∈βS can be identified with finitely additive {0,1}-
valued probability measure µFon the power set P(S) by
µF(A) = 1
if and only if A∈F. It is obvious that
µF(∅) = 0 and µF(S) = 1.
With this notion, every f∈Sis naturally identified with an ultrafilter by defining
µf(A) = (1 if f∈A,
0 if f /∈A
This is nothing other than the set {A⊂S:f∈A}. Such type ultrafilters µf, f ∈S
are called fixed or principal. That is, semigroup Sitself can be identified as a subset
of all principal untrafilters from βS. Ultrafilters from the subset of βS −Sare
called free or non-principal. Note that, by assumption, Sis an infinite semigroup,
there are ultrafilters on Swhich don’t have the form µf, f ∈S. In other words,
the canonical embedding f→µf, f ∈Sis not a bijection. That is, for such a
semigroup, an ultrafilter Fis free if and only if every set in Fis infinite. Note
that a filter Fon Sis free if the intersection of all sets in Fis empty, otherwise,
Fis fixed by any element of the intersection of sets of F. We can define cluster
point of a filter Fand on the basis of this term, we can get equivalent principal
and free ultrafilter. If point lis in all sets of a filter F, then it is known as cluster
point. If cluster point is in ultrafilter, then it is set of all set containing that point,
which is nothing other than principal ultrafilter. Clearly, an ultrafilter can have at
most one cluster point. Clearly, an ultrafilter wih no cluster point is free. For every
f, g ∈S, we have
µf◦µg=µf◦g.
This shows that principal ultrafilters forms a semigroup which is isomorphic to S.
Example 3.1. Let E6=∅be a subset of a holomorphic semigroup S. The set
F={F⊆S:E⊆F}=hEi
is a principal filter on Sgenerated by E. It is indeed, the least filter on Scontaining
E. If E={f}for some f∈S, then Fis an ultrafilter on S.
Example 3.2. Let Sbe a holomorphic semigroup. Define
F={E⊂S:S−Eis finite}.
It is easy to verify that Fis generated by the family
{S− {f}:f∈S}
forms a filter, the Frechet filter on S. It is not an ultrafilter.
Next, we make βS a topological space which is, in fact, compact and Hausdorff
as shown in the following theorem.
Theorem 3.1 (βS as a compact Housdorff space).Let Sbe a holomorphic semi-
group. The space βS of all ultrafilters of Sis a topological space. In fact, it is a
compact Hausdorff space.
10 BISHNU HARI SUBEDI
Proof. For given a set A⊂S, define
A={F∈βS :A∈F}and B={A:A⊆S}.
Then, for any A, B ⊂S, the following follows easily.
A∪B=A∪B A ∩B=A∩Band S−A=βS −A.
Since S=SAfor all A⊂S. So
S=[
A⊂S
A=[
A∈B
A=βS.
This proves that Bis a basis for the topology on βS where topology τis defined
by the collection of all unions of elements of the basis B. Let T∈τ. Then, there
are some A∈Bsuch that T=SAwhere Ais an open set. We can consider τan
open cover of βS. Each element Tof the topology τis covered by finite open sets
and so βS ∈τis also cover by finite open sets A∈B. This proves βS is a compact
space.
To make βS a Hausdorff space, let us suppose a pair of distinct ultrafilters
E,F∈βS. Since by above construction, every element of basis Bis both open and
closed in βS . So, there exist open sets T1and T2containing Eand Frespectively
where T1=SAifor some i≤nand T2=SAjfor some j≤nwith i6=jsuch
that T1∩T2=∅.
In the proof of Theorem 3.1, we considered the set B={A:A⊂S}where
A={F∈βS :A∈F}as a basis of the topology τon βS. In this case,
we say that Ais the Stone set corresponding to A⊆S,Bis the Stone base or
simply, (ultra)filter base of βS and τis the Stone topology and the space βS is the
Stone-Cech compactification of semigroup S. Two (ultra) filter bases are said to be
equivalent if they generate the same filter. Any family of sets satisfying the finite
intersection property is a subbasis of for a (ultra) filter Fsince the family together
with the finite intersection of its members is a (ultra) filter base.
The space βS with basis Bis a bonafide example of compact Hausdorff space
which is isomorphic to S={F∈βS :S∈F}. So, it seems natural to extend the
operation of functional composition from semigroup Sto compact Hausdorff space
βS as follows. In this sense, βS can be a good candidate for compact holomorphic
topological semigroup. According to Definition 2.3, to make βS a compact topo-
logical holomorphic semigroup, we need to define binary operation and extension
map as a translation map on βS.
Definition 3.2 (Operation on βS).Let βS be the space of ultrafilters of holomr-
phic semigroup S. For any A⊂Sand f∈S. Define a set A◦f={g∈S:g◦f∈A}.
For any E,F∈βS, define convolution by
E⋆F={A⊆S:{f∈S:A◦f∈E} ∈ F}
The operation ⋆on βS is well defined and associative but not commutative. In
fact, we prove the following result.
Theorem 3.2 (βS is a semigroup).For any E,F∈βS,E⋆Fis an ultrafilter
and ⋆on βS is associative.
A STUDY OF HOLOMORPHIC SEMIGROUPS 11
Proof. From Definition 3.2, it is clear that ∅/∈E⋆F. Let A, B ∈E⋆F. Then
again by the definition 3.2, we have
{f∈S:A◦f∈E} ∈ Fand {f∈S:B◦f∈E} ∈ F
To show A∩B∈E⋆F, we need to show
{f∈S: (A∩B)◦f∈E} ∈ F.
But this happens easily as
{f∈S: (A∩B)◦f∈E}={f∈S:A◦f∈E} ∩ {f∈S:B◦f∈E} ∈ F.
To show E⋆F6=∅, assume A⊂Sbut A /∈E⋆F. Then {f∈S:B◦f∈E}/∈F.
This is equivalent with {f∈S:A◦f∈E}c∈F. This is true precisely when
{f∈S:Ac◦f∈E} ∈ F. This shows that Ac∈E⋆F.
Finally, we check the associativity of the operation ∗on βS. Let A⊂Sand
E,F,G∈βS. Then
A∈E⋆(F⋆G)⇐⇒ {f∈S:A◦f∈E} ∈ F◦G
⇐⇒ {g∈S: ({f∈S:A◦f∈E}◦g)∈F} ∈ G
⇐⇒ {g∈S:{f∈S:A◦g◦f∈E} ∈ F} ∈ G
⇐⇒ {g∈S:A◦g∈E⋆F} ∈ G
⇐⇒ A∈(E⋆F)⋆G
As we stated before, if ultrafilters µffor f∈Sare principal, then the convolution
operation ∗on βS correspond to the operation ◦of holomorphic semigroup S.
Note that the convolution defined in Definition 3.2 is the unique extension of the
operation ◦in S. This leads to the translation map of Definition 2.3 if we define it
as FF(E) = E⋆F. Indeed, this map is continuous as shown below.
Theorem 3.3 (The convolution operation as a continuous self map).For
a fixed E∈βS, the function FF(E) = E⋆Fis continuous self map of βS for any
F∈βS.
Proof. For F∈βS, let Ube a neighborhood of FF(E). The result can be proved
if there exists a neighborhood Vof Esuch that FF(G)∈ U for any G∈ V where
V={H∈βS :V ∈ H}. Let A⊂Ssuch that FF(E) = E⋆F∈A⊂ U. Then
A∈E⋆F. Define a set V={f∈S:A◦f∈E}. By Definition 3.2 of E⋆F,
V ∈ For F∈V. If G∈ V, then V={f∈S:A◦f∈F} ∈ G. This implies that
A∈F⋆G=FF(G) or FF(G)∈A∈ U.
Now, we are able to get very rigorous example of compact holomorphic topolog-
ical semigroup. This is a space of ultrafilters βS with operation of convolution of
Definition 3.2. Indeed, this space βS is a compact holomorphic topological semi-
group on the basis of Theorems 3.1 and 3.2. It is considered a nice characteristic
property of the space βS. As we mentioned in Theorem 2.1, such semigroups are
known to have idempotents. An idempotent (in this case, it is called idempotent
ultrafilter) in βS is an element Isuch that I⋆I=I. In such a case, we have
A∈I⇐⇒ A∈I⋆I⇐⇒ {f∈S:A◦f∈I} ∈ I. Also if Iis an idempo-
tent ultrafilter, then f◦S∈Ifor all f∈S. Note that an idempotent ultrafilter
12 BISHNU HARI SUBEDI
can not be principal ultrafilter. There is an investigation concerning the number of
idempotent ultrafilters contained in βS . In fact, βS has 2cidempotants, where cis
the power of continuum. This follows from the fact that βS has 2cdisjoint closed
subsemigroups (see for instance [3]).
From the proof of Theorem 2.1 and little bit discussion of above paragraph, we
noticed that a compact holomorphic topological semigroup Shas minimal close sub-
semigroup M. Now, we investigate something more on minimal close subsemigroup
of βS.
Theorem 3.4 (Mnimal closed subsemigroups are left(or right) ideals).
The minimal close subsemigroups of βS are precisely minimal left (or right) ideals.
Proof. Suppose that Iis a minimal close subsemigroup of βS . Then by fact 2 in
the proof of the theorem 2.1, we can write I={I}, where Iis an idempotent
ultrafilter of βS . Which is a minimal (minimum as well) left (or right) ideal of
βS.
This Theorem 3.4 is also a converse of the fact that every left (or right) ideal is a
subsemigroup of the semigroup S. Note that in general such type of converse may
not hold but according to Theorem 3.4, if subsemigroups are closed and minimal,
they are precisely minimal left (or right) ideals. Also, if semigroup is abundant,
such type of close minimal subsemigroups exist and they are minimal left or right
ideals.
References
[1] V. Bergelson, Ergodic theory-an update, Ergodic Theory of Zdaction, London Math. Soc.
Lecture Note Ser. 228 (1996), 1-61.
[2] V. Bergelson, Minimal idempotents and ergodic Ramsey theory, Ergodic Theory of Zdaction,
Topics in dynamics and ergodic theory: London Math. Soc. Lecture Note Ser. 310 (2003),
8-39.
[3] E. K. Douwen, The Cech-Stone compactification of discrete groupoid, Topology and it Ap-
plications, 39 (1991), 43-60.
Central Department of Mathematics, Institute of Science and Technology, Tribhu-
van University, Kirtipur, Kathmandu, Nepal
E-mail address:subedi.abs@gmail.com