Ultrafilters and Topologies on Groups (De Gruyter Expositions in Mathematics)

Follow the Author

---

It is clear that. Chapter 5 Topological Groups with Extremal Properties In this chapter, assuming MA, we construct important topological groups with extremal properties. We also show that some of them cannot be constructed in ZFC. Recall that starting from this chapter, all topological groups are assumed to be Hausdorff.

LFor every n Section 5. But then, since G is extremally disconnected, 0 … cl UE nD.

To see that F is dense in itself, let x 2 F. Pick n0 [ An: Finally, to see that F is nowhere dense, let x 2 G and let U be a neighborhood of 0 in T. Pick A 2 p such that max supp. Assume p D c. Then there exists a maximal topological group. Let R denote the root of B 0. Every maximal topological group contains a countable open Boolean subgroup. Let G be a maximal topological group. Then G is extremally disconnected. L Consequently by Theorem 5.

Then every nondiscrete group topology on an Abelian group of character 73 Section 5. Construct inductively a decreasing sequence. Then the set V D Vn is as required. Let G be a nondiscrete metrizable Abelian topological group of prime period and let X be a nowhere dense subset of G such that 0 … X. Let p be the period of G. Since p is prime, the mapping x 7!

Then by Lemma 5. We shall construct a sequence. Fix n 2 N and suppose that we have constructed a sequence. Since an 2 Un ,. Now we are ready to prove Theorem 5. Proof of Theorem 5. Let X be an extremally disconnected space, let D be a strongly discrete subset of X , and let x 2 cl DnD. Then there are disjoint subsets A and B of D with x 2. Suppose that p contains a countable 77 Section 5.

Then there is a mapping f W G! Without loss of generality one may assume that G is Boolean and p converges to 0. Let D be a countable discrete subset of G such that D 2 p. Then by Lemma 3. It follows that 0 … cl Q and so 0 … cl F. It is consistent with ZFC that there is no extremally disconnected topological group containing a countable discrete nonclosed subset.

Our second theorem deals with countable topological Boolean groups containing no nonclosed discrete subset, and even more generally, containing no subset with exactly one accumulation point. A nonempty subset of a topological space is called a P-set if the intersection of any countable family of its neighborhoods is again its neighborhood. Note that every isolated point of a P -set is a P -point.

Then the following statements 1 F is a P-set, 2 for every decreasing sequence. Let T be a nondiscrete Q group topology on! We claim that for every n m. We say that a topological group. Clearly, every maximal topological group is maximally nondiscrete. Since T1 is maximally nondiscrete, it follows that T1 D T. The existence of a maximal topological group implies the existence of a P -point in! Hence by Theorem 5. It is consistent with ZFC that there is no maximal topological group.

As another consequence we obtain from Theorem 5. Now assume on the contrary that. Then by Theorem 5. Sirota [71], the implication. Z2 are due to A. We show that every compact Hausdorff right topological semigroup has an idempotent and, as a consequence, a smallest two sided ideal which is a completely simple semigroup. The structure of a completely simple semigroup is given by the Rees-Suschkewitsch Theorem. If S is a semigroup, the extended operation is associative. For each a 2 S, the left translation la W S 3 x 7!

T be a continuous mapping such that '. If 'jS is a homomorphism, so is '. Let S be a discrete semigroup and let ' W S! T be any homomorphism of S into a compact Hausdorff right topological semigroup T such that '. T of ' is a homomorphism. T is a continuous homomorphism such that '.

Then A is a neighborhood of sq. Assume on the contrary that A … sq. Consequently, S n A 2 sq. Assume on the contrary that A … pq. Consequently, S n A 2 pq. Let S be a cancellative left cancellative, right cancellative semigroup. Suppose that S is left cancellative. Now suppose that S is right cancellative. Assume on the contrary that pq D a 2 S. Then xz D yz, a contradiction. S by putting f. As usual, for every semigroup S, we use S 1 to denote the semigroup with identity obtained from S by adjoining one if necessary.

If S is a cancellative semigroup, so is S 1. To see this, let x 2 S. Multiplying the equality e D ee by x from the left gives us xe D xee. Then cancellating the latter equality by e from the right we obtain x D xe. Similarly x D ex. Now assume on the contrary that S 1 is not cancellative.

Then some translation in 1 S is not injective. One may suppose that this is a left translation. It follows that there exist a; b 2 S such that ab D a. Hence b D 1 2 S , a contradiction. Every compact Hausdorff right topological semigroup has an idempotent. Let S be a compact Hausdorff right topological semigroup. Consider the set P of all closed subsemigroups of S partially ordered by the inclusion.

For every chain C in P , C 2 P. Pick x 2 A. We shall show that xx D x. We start by showing that Ax D A. Let B D Ax.

16 337,67 RUB

Clearly B is nonempty. Thus B 2 P. Since x 2 A D Ax, C is nonempty. Thus C 2 P. Hence x 2 C and so xx D x as required. A left ideal L of a semigroup S is minimal if S has no left ideal strictly contained in L. Then S contains a minimal left ideal. Every minimal left ideal of S is closed and has an idempotent. It follows that every minimal left ideal of S is closed and, by Theorem 6. Thus we need only to show that S contains a minimal left ideal. To this end, consider the set P of all closed left ideals of S , partially ordered by the inclusion. Since L is minimal among closed left ideals and every left ideal contains a closed left ideal, it follows that L is a minimal left ideal.

A semigroup can have many minimal left right ideals. However, it can have at most one minimal two-sided ideal. For every semigroup S, let K.

Post navigation

Let S be a semigroup and assume that there is a minimal left ideal L of S. Then 1 for every a 2 S , La is a minimal left ideal of S , 2 every left ideal of S contains a minimal left ideal, 3 different minimal left ideals of S are disjoint and their union is K. Pick a 2 N. Clearly K is an ideal. It is easy to see that a smallest ideal a minimal left ideal is a simple left simple semigroup. A semigroup S is completely simple if it is simple and there is a minimal left ideal of S which has an idempotent. Every compact Hausdorff right topological semigroup has a smallest ideal which is a completely simple semigroup.

Given a semigroup S, let E. Let S be a left simple semigroup with E. Then 1 e is a right identity of S, and 2 eS is a group. By 1 , e is also a right identity, so e is identity of eS. Now let x 2 eS. Let y D es. A semigroup satisfying the identity xy D x xy D y is called a left right zero semigroup. Then f is an isomorphism. To see that f is a homomorphism, let. To see that f is surjective, let x 2 S. Since S is left simple, there is y 2 S such that yx D e. Then xyxy D xey D xy, so xy 2 E. To see that f is injective, let.

We claim that a D ex and u D xy where y is the inverse of ex 2 eS. Let S be a semigroup and let e 2 E. If Se is a minimal left ideal of S , then eS is a minimal right ideal. We have to show that e 2 eaS. Then e D eu D eaeb 2 eaS. Let S be a semigroup and let R and L be minimal right and left ideals of S. Let G D RL. It is straightforward to check that M. The next theorem tells us that every completely simple semigroup is isomorphic to some Rees matrix semigroup.

Let S be a completely simple semigroup. Pick e 2 E. We call G the structure group of S. The next proposition and lemma are useful in identifying the smallest ideal of subsemigroups and homomorphic images. Let S be a semigroup and let T be a subsemigroup S. Suppose that both S and T have a smallest ideal which is a completely simple semigroup.

For the reverse inclusion, let x 2 K. Let S and T be semigroups and let f W S! T be a surjective homomorphism. If S has a smallest ideal, so does T and K. We conclude this section with discussing standard preorderings on the idempotents of a semigroup. Let S be a semigroup with E. Hence, e D f. Denote g D ef.

Consequently g D e. Thus, to say that a semigroup S contains a minimal left ideal which has an idempotent is the same as saying that S has a minimal idempotent. It is clear also that if a semigroup S has a minimal idempotent, then the minimal idempotents of S are precisely the idempotents of K. An idempotent e of a semigroup S is right left maximal if e is maximal in E. Thus, Tp is a closed subsemigroup of S.

Now by Theorem 6. It is immediate from Lemma 6. Furthermore, if S is cancellative, then T is Hausdorff. Then for every U 2 N , 1 2 U and and by Lemma 6. Hence by Theorem 4. Now suppose that S is cancellative. It follows that T is Hausdorff. Let S be a left topological semigroup with identity and let U be an open subset of S such that 1 2 cl U. Pick x1 2 U. Fix n 2 N and suppose we have chosen a sequence. Let F D FP.. The statement is obvious if p 2 S. Let U D intT A. We have that U is an open subset of.

Then apply Lemma 6. Pick by Theorem 6. Then apply Theorem 6. As a special case of Corollary 6. In other words, supp2. Then there is a sum subsystem 1. We proceed by induction on k. Now assume the statement holds for some k. Then there are F1 ; F2 2 Pf. Consequently, there is an idemFP.. Before proving Proposition 6.

To see that it is a subsemigroup, let p; q 2 F and let A 2 F. We have to show that A 2 pq. Hence, A 2 pq. Pick F 2 Pf. T 1 Consequently, by Lemma 6. For the second part, apply Theorem 6. If F D G , we say F -syndetic instead of. Then for every q 2 L, one has p 2 Lq. Now to show that A is G -syndetic, let V 2 G. Then p 2 K. Suppose that p 2 K.

Indeed, let y 2 W. Conversely, suppose that p … K. Pick q 2 K. We claim that B is not F -syndetic. Assume on the contrary that B is F -syndetic. Then B is also. Hence by Lemma 6. But then A 2 rqp, which is a contradiction. Day [15] using a multiplication of the second conjugate of a Banach algebra, in this case l1. Numakura [52] for topological semigroups and by R. Ellis [20] in the general case. Rees [63] in the general case. For more information about compact right topological semigroups, including references, see [67]. An introduction to this topic can be found also in [39]. The exposition of Section 6.

In this chapter we study the relationship between algebraic properties of Ult. We conclude by showing how to construct homomorphisms of Ult. Since U 2 p and Vx 2 q, it follows that U 2 pq, so pq 2 U.

Clearly, one may suppose that U is open. We refer to Ult. To show that F is open, let U 2 F. To show that F is closed, assume the contrary. Then there is U 2 F such that for every V 2 F ,. Then by Proposition 3. Since F is open, F is a right ideal of Ult. To see that F is a left ideal, let p 2 Ult. Hence pq 2 F. Then there are two disjoint open subsets U and V of. But then by Lemma 7. But then 1 2. Hence T is not extremally disconnected. Then by Theorem 3. Then Q is left saturated. Since T is regular, N is closed. Then by Lemma 7. Hence, Q is left saturated.

Then there is a regular left invariant topology T on G such that Ult. Statement i is obvious. For every q 2 L,. Indeed, otherwise xp 2 C for some x 2 V and p 2 Q. Take any q 2 L. It follows from i — ii and Theorem 4. We now show that T is regular. Then there is a neighborhood U of 1 such that for every neighborhood V of 1,. For every open neighborhood V of 1, choose xV 2.

Pick any q 2 L. We have obtained that Q is not left saturated, which is a contradiction. To see that C. Then xqp D rp, and so xp D p. If x 2 G, this equality implies that x D 1 by Corollary 6. Hence x 2 C. Now we deduce from Theorem 7. Then 1 there is a regular left invariant topology T on G such that Ult. Then applying by Theorem 7. To see 2 , let T 0 be any regular left invariant topology on G in which p converges to 1, let Q D Ult.

Hence by Lemma 7. Hence by Proposition 7. Since a regular extremally disconnected space is zero-dimensional, we obtain from Theorem 7. The proof of Theorem 7. Proof of Theorem 7. Now we show that B possesses the following properties: To this end, pick A 2 p such that U D Ap0. Then Uq0 Uq0 D. It follows from i and ii that there is a left invariant T1 -topology T on G such that Ult.

A space is locally regular if every point has a neighborhood which is a regular subspace. Choose a regular open neighborhood X of 1 2 G in T. Assume on the contrary that xq 2 X for some x 2 X. Since X is regular, U can be chosen to be closed. Put X D q2F Xq. We claim that X is regular. Then there is x 2 X and a neighborhood U of x such that for every neighborhood V of x, clX. Since Q is left saturated, it follows from a that q 2 F. It is clear that xq 2 Xq, and b gives us that xq 2 X. Let T be a compact Hausdorff right topological semigroup and let f W X! T is the continuous extension of f.

For every x 2 X , one has f. Then there is a zero-dimensional Hausdorff left invariant topology T on S such that Ult. Clearly this is also an M -decomposition. Next, we have that y1 2 N. Choose a basic mapping M W S! For every x 2 M. Extend f0 W M. N to a mapping f W. We also extend in some sense this result to the case where G is an arbitrary countable group. To this end, we develop a special technique based on the concepts of a local left group and a local homomorphism.

A basic example of a local left group is an open neighborhood of the identity of a T1 left topological group. Let X be a local left group and let Xd be the partial semigroup X reendowed with the discrete topology. Given a local left group X , Ult. It is straightforward to check that Ult. If X is an open neighborhood of the identity of a left topological group. Let X and Y be local left groups. A mapping f W X! Y is a local homomorphism if f. We say that f W X!

Special offers and product promotions

Y is a local isomorphism if f is a local homomorphism and homeomorphism. Note that if f W X! Y be a continuous local homomorphism. Let X be a local left group and let S be a semigroup.

About the Author

Let T be a compact right topological semigroup and let f W X! T be a local homomorphism such that f. Furthermore, if for every neighborhood U of 1 2 X , f. To check the second statement, let t 2 T. It follows from this that there exists p 2 Ult. We say that an element a 2! Z2 is basic if supp. Each nonzero a 2! We call such L a decomposition canonical.

Let S be a semigroup and let f W! Z2 and let n D max supp. We now come to the main result of this section. Then there is a continuous bijection h W X! Notice that condition 2 in Theorem 8. X is a local homomorphism, by Lemma 8. Since h is continuous, it follows that h also is a local homomorphism. To see that h is open, let x 2 X. Put n D max supp. Since h is continuous, there is a neighborhood V of 1 such that h. Now let U be any neighborhood of 1 contained in V. If w is basic, X. In any case, X.

It follows that X. Now for every x 2 X , there is w 2 W with nonzero last letter such that x D x. Pick w; v 2 W with nonzero last letters such that x D x. Then y 2 X. Let X be a countable nondiscrete regular L local left group. Then there is a continuous bijective local homomorphism f W X! Z2 , and consequently, Ult. It is immediate from Theorem 8. Let X and Y be countable nondiscrete regular local left groups. Then there is a bijection f W X!

Z2 be bijections guaranteed by Theorem 8. Then for every local homomorphism f W X! Q and for every surjective homomorphism g W T! Q, there is a local homomorphism h W X! Q, there is a proper homomorphism W Ult. Z2 be a bijection guaranteed by by Theorem 8. For every basic a 2 L! It is clear that h is a local homomorphism.

We say that a local homomorphism f W X! S is surjective if for every neighborhood U of 1 2 X , f. Then for every local L homomorphism f W X! Q and for every surjective local homomorphism g W! Q, there is a continuous local L homomorphism h W X! Z2 be a bijection guaranteed by Theorem 8. For every n n such that g. Then for every basic a 2! Z2 , L pick a nonzero. If m Section 8.

The condition also implies that min supp. To see this, suppose that max supp. Pick any such b. Then by the inductive assumption, min supp. Now, applying the condition, we obtain that min supp. Also note that G.

Download e-book for iPad: Ultrafilters and Topologies on Groups (De Gruyter by Yevhen Zelenyuk

Indeed, if xu 2 Q, then xQ D x. That this mapping is injective follows from Lemma 6. To see that this is a homomorphism, let x; y 2 G. Assume on the contrary that Q is nontrivial while G. Let u be the identity of Q. Furthermore, C is left saturated. Consequently, x 2 C [ G. Since Q is a minimal left ideal of C , it follows that C has only one minimal right ideal. Consequently, T is extremally disconnected, by Proposition 7. Being regular extremally disconnected, T is zero-dimensional. Note that we showed zero-dimensionality of T not using the fact that G is countable.

We claim that f is a local homomorphism. Then y 2 Ux;q for some q 2 Q. Q be a surjective homomorphism. Let G be a countable torsion free group. Since G is torsion free, it follows that G. Then by Theorem 8. As an immediate consequence of Corollary 8. Let X be a countable regular local left group.

We say that a family F of subsets of X is invariant if for every Y 2 F , f. Let X be a space with a distinguished point 1 2 X and let f W X! Then exactly one of the following two possibilities holds: We say that f is spectrally irreducible if for every neighborhood U of 1, spec. Also, a neighborhood U of a point x 2 X is spectrally minimal if for every neighborhood V of x contained in U , spec. Then there is an open invariant neighborhood U of 1 such that f jU is spectrally irreducible.

A local automorphism of a local left group X is a local isomorphism of X onto itself. In other words, a mapping f W X! X is a local automorphism if f is both a homeomorphism with f. Let X be a Hausdorff local left group and let f W X! For each x 2 O. Let n be the order of f.

Choose a neighborhood W of 1 such that f i. Then the subsets U. Then xn D x. Now, for every x 2 X , there is w 2 W with nonzero last letter such that x D Lx. We prove that p1 D q1 and k D s. Without loss of generality one may suppose that f is spectrally irreducible. For every x 2 Y , consider the sequence of coordinates of h. Then A 2 u. For every x 2 A, put n. We now show that k D s. For every x 2 X , consider the sequence of coordinates of h. Then for every u; v 2 Ult. The isomorphism is given by F 3 x 7!

Indeed, that this mapping is injective follows from Lemma 6. To see that this is a homomorphism, let x; y 2 F. Then xyu D xyuu D xuyu. Then u commutes with each element of F and Q D F u. Indeed, since a 2 G. Consider the conjugation f W x 7! We claim that f is a homeomorphism. To see this, let p 2 C. Consequently, f is continuous. We now show that Q D F u. Then for every x 2 F , xq D qx. Let H be the subgroup of G generated by the subset A [ F. Consider the group R D g. We claim that the subgroup L.

To see this, let x 2 L. Pick y 2 H with g. Consequently y 2 F , and so x D 1 2 L. References The results of Sections 8. Its proof is based on Theorem 8. The latter and Theorem 8. The exposition of this chapter is based on the treatment in [92]. We conclude by showing that if is not Ulam-measurable, then K. It is easy to see that the set U.

For every p 2 U. A2p The next theorem is the main result of this section. If is a regular cardinal, then I. Before proving Theorem 9. Clearly, Ip is a closed subset of U. Let D be a decomposition of a compact Hausdorff space X into closed subsets and let Y be the corresponding quotient space of X. Then Y is Hausdorff if and only if D is upper semicontinuous.

Y denote the natural quotient mapping. It then follows that f. Conversely, suppose that D is upper semicontinuous and let A1 ; A2 be distinct members of D. Let J be a decomposition of U. Since V is a neighborhood of p 2 U. This shows that I is a decomposition. It follows from this also that for every Section 9. Thus, I is a decomposition of U. Now we consider decompositions of U.

A2p As in the proof of Lemma 9. If is a regular cardinal, then jI 0. To see that I is nowhere dense in U. Otherwise I D Ip for some p 2 U. The proof of Theorem 9. Then there is a surjective function f W G! Then Ax 2 p and by condition b , f. Then E D f. It then follows from i - iv that J is as required. We conclude this section by the following consequence of Theorem 9. Without loss of generality one may suppose that p 2 U. Let J be the member of I containing p. Pick any K 2 I different from J and any q 2 K.

Then pq 2 K and cl. For every continuous homomorphism h W G! K from G into a compact group K there is a continuous homomorphism hb W bG! Then bG is the dual group of GO d. The mapping e W G! In order to prove Theorem 9. Proof of Theorem 9. Zp1 denotes the quasicyclic p-group. It follows that G admits a homomorphism onto Zp1 if D! Clearly supp2M p D.

By the previous paragraph, for Section 9. Now, using Lemma 9. The dual groups of continuous homomorphic images of bG are the subgroups of GO d and the dual groups of homomorphic images of G are the Q closed subgroups Q of GOL see [34, Theorems Then G L admits a homomorphism onto one of Q the following 1 1 groups: Their dual groups are T ,!

The L second group is algebraically isomorphic to 2! The others contain L torsion-free subgroups of cardinality 2! Then L L onto one of the following groups: The third lemma deals with products of topological spaces. Then the structure group of K. Recall that if a semigroup S has a smallest ideal which is a completely simple semigroup, then for every p 2 E. Then the elements p C q C p 2 K.

Partition A into subsets Ai , i D 1;: Hence, the elements pqi p, i D 1;: Since f is surjective, f. Consequently, by Theorem 9. Pick p 2 E. It follows that in order to prove Theorem 9. There are elements in cl K. Indeed, let q 2. Then clearly, q 2 cl K. To see that q … K.

Then q 2 K. Consequently, q D qq 2. To see that q … E. To prove Theorem 9. Let X be a nondiscrete regular extremally disconnected space and assume that jX j is not T Ulam-measurable. In the case D! Indeed, inTthis case the sequence. Protasov [61] in the case where is a regular cardinal and by M. Salmi [24] for all. Our proof of Corollary 9. An introduction to the Pontryagin duality can be found in [55] and [34]. See also [50] and [17]. A topological or left topological group is said to be almost maximal if the underlying space is almost maximal.

Z2 such that Ult. We show that every countable almost maximal topological group contains an open Boolean subgroup and its existence cannot be established in ZFC. We then describe projectives in F. These are certain chains of rectangular bands. An object S in some category is an absolute coretract if for every surjective morphism f W T! S there exists a morphism g W S! Let C denote the category of compact Hausdorff right topological semigroups.

The next lemma gives us some simple examples of absolute coretracts in C. Finite left zero semigroups, right zero semigroups and chains of idempotents are absolute coretracts in C. S be a continuous surjective homomorphism. Pick a minimal left ideal L of T. Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support?

This book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters. The contents of the book fall naturally into three parts.

The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous.

In the second part, Chapters 6 through 9, the Stone-Cech compactification ssG of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then ssG contains no nontrivial finite groups. Also the ideal structure of ssG is investigated.

In particular, one shows that for every infinite Abelian group G, ssG contains 22 G minimal right ideals. In the third part, using the semigroup ssG, almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized.

Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely -resolvable, and consequently, can be partitioned into subsets such that every coset modulo infinite subgroup meets each subset of the partition. The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed.

But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas.

• How To Pick Up Girls In Nightclubs & Bars
• Gestaltung von Point of Sale-Maßnahmen (German Edition)
• MAKE MONEY WITH VIDEO & AUDIO STREAMING!
• Corporate Management Strategy - Keeping It Very Simple
• Good Friends Great Tastes
• Pourquoi les thérapies énergétiques fonctionnent (French Edition)