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Combined Algebraic Properties of IP* and Central* Sets Near 0
It is known that for an IP* set in and a sequence there exists a sum subsystem of such that . Similar types of results also have been proved for central* sets. In this present work we will extend the results for dense subsemigroups of .
One of the famous Ramsey theoretic results is Hindman's Theorem.
Theorem 1.1. Given a finite coloring there exists a sequence in and such that where for any set is the set of finite nonempty subsets of .
The original proof of this theorem was combinatorial in nature. But later using algebraic structure of a very elegant proof of this theorem was established in [1, Corollary 5.10]. First we give a brief description of algebraic structure of for a discrete semigroup .
We take the points of to be the ultrafilters on , identifying the principal ultrafilters with the points of and thus pretending that . Given , is a basis for the closed sets of . The operation on can be extended to the Stone-Čech compactification of so that is a compact right topological semigroup (meaning that for any , the function defined by is continuous) with contained in its topological center (meaning that for any , the function defined by is continuous). A nonempty subset of a semigroup is called a left ideal of if , a right ideal if , and a two-sided ideal (or simply an ideal) if it is both a left and right ideal. A minimal left ideal is the left ideal that does not contain any proper left ideal. Similarly, we can define minimal right ideal and smallest ideal.
Any compact Hausdorff right topological semigroup has a smallest two-sided ideal:
Given a minimal left ideal and a minimal right ideal , is a group, and in particular contains an idempotent. An idempotent in is a minimal idempotent. If and are idempotents in we write if and only if . An idempotent is minimal with respect to this relation if and only if it is a member of the smallest ideal.
Given , and , if and only if , where . See  for an elementary introduction to the algebra of and for any unfamiliar details.
is called an set if it belongs to every idempotent in . Given a sequence in , we let be the product analogue of Finite Sum. Given a sequence in , we say that is a sum subsystem of provided there is a sequence of nonempty finite subsets of such that and for each .
Theorem 1.2. Let be a sequence in and let be an set in . Then there exists a sum subsystem of such that
Definition 1.3. A subset is called central if and only if there is an idempotent such that .
The algebraic structure of the smallest ideal of has played a significant role in Ramsey Theory. It is known that any central subset of is guaranteed to have substantial additive structure. But Theorem 16.27 of  shows that central sets in need not have any multiplicative structure at all. On the other hand, in  we see that sets which belong to every minimal idempotent of N, called sets, must have significant multiplicative structure. In fact sets in any semigroup are defined to be those sets which meet every central set.
Theorem 1.4. If is a set in then it is central in .
Proof. See [2, Theorem 2.4].
In case of sets a similar result has been proved in  for a restricted class of sequences called minimal sequences, where a sequence in is said to be a minimal sequence if
Theorem 1.5. Let be a minimal sequence and let be a set in . Then there exists a sum subsystem of such that
Proof. See [3, Theorem 2.4].
A strongly negative answer to the partition analogue of Hindman's theorem was presented in . Given a sequence in , let us denote = and and and .
Theorem 1.6. There exists a finite partition of with no one-to-one sequence in such that is contained in one cell of the partition .
Proof. See [4, Theorem 2.11].
A similar result in this direction in the case of dyadic rational numbers has been proved by V. Bergelson et al..
Theorem 1.7. There exists a finite partition such that there do not exist and a sequence with
Proof. See [5, Theorem 5.9].
In , the authors also presented the following conjecture and question.
Conjecture 1.8. There exists a finite partition such that there do not exists and a sequence with
Question 1. Does there exist a finite partition such that there do not exist and a sequence with
Definition 1.9. If is a dense subsemigroup of one defines .
It is proved in , that is a compact right topological subsemigroup of which is disjoint from and hence gives some new information which are not available from . Being compact right topological semigroup contains minimal idempotents of . A subset of is said to be -set near 0 if it belongs to every idempotent of and a subset of is said to be set near 0 if it belongs to every minimal idempotent of . In  the authors applied the algebraic structure of on their investigation of image partition regularity near 0 of finite and infinite matrices. Article  used algebraic structure of to investigate image partition regularity of matrices with real entries from .
2. IP* and Central* Set Near 0
In the following discussion, we will extend Theorem 1.2 for a dense subsemigroup of in the appropriate context.
Definition 2.1. Let be a dense subsemigroup of . A subset of is said to be an IP set near 0 if there exists a sequence such that converges and such that . One calls a subset of an set near 0 if for every subset of which is IP set near 0, is IP set near 0.
From [6, Theorem 3.1] it follows that for a dense subsemigroup of a subset of is an IP set near 0 if and only if there exists some idempotent with . Further it can be easily observed that a subset of is an set near 0 if and only if it belongs to every idempotent of .
Given and , the product is defined in . One has is a member of if and only if is a member of .
Lemma 2.2. Let be a dense subsemigroup of such that is a subsemigroup of . If is an IP set near 0 in then is also an IP set near 0 for every . Further if is a an IP* set near 0 in then is also an set near 0 for every .
Proof. Since is an IP set near 0 then by [6, Theorem 3.1] there exists a sequence in with the property that converges and . This implies that is also convergent and . This proves that is also an set near 0.
For the second let be a an set near 0 and . To prove that is a an set near 0 it is sufficient to show that if is any IP set near 0 then . Since is an IP set near 0, is also an IP set near 0 by the first part of the proof, so that . Choose and such that . Therefore so that .
Given and , : , and : .
Theorem 2.3. Let be a dense subsemigroup of such that is a subsemigroup of . Also let be a sequence in such that converges and let be a set near 0 in . Then there exists a sum subsystem of such that
Proof. Since converges, from [6, Theorem 3.1] it follows that we can find some idempotent for which . In fact and . Again, since is a set near 0 in , by Lemma 2.2 for every , . Let : . Then by [1, Lemma 4.14] . We can choose . Inductively let and , in be chosen with the following properties:(1); (2)if then and .
We observe that : . Let : , let and . Now consider Then . Now choose and such that . Putting shows that the induction can be continued and proves the theorem.
If we turn our attention to sets then the above result holds for a restricted class of sequences which we call minimal sequence near 0.
Definition 2.4. Let be a dense subsemigroup of . A sequence in is said to be a minimal sequence near 0 if
The notion of piecewise syndetic set near 0 was first introduced in .
Definition 2.5. For a dense subsemigroup of , a subset of is piecewise syndetic near 0 if and only if .
The following theorem characterizes minimal sequences near 0 in terms of piecewise syndetic set near 0.
Theorem 2.6. Let be a dense subsemigroup of . Then the following conditions are equivalent: (a) is a minimal sequence near 0. (b) is piecewise syndetic near 0. (c)There is an idempotent in .
Proof. follows from (see [6, Theorem 3.5]).
To prove that (b) implies (a) let us consider that be a piecewise syndetic near 0. Then there exists a minimal left ideal of such that . We choose . By [6, Theorem 3.1], is a subsemigroup of , so it suffices to show that for each , . In fact minimal left ideals being closed, we can conclude that and so is a compact right topological semigroup so that it contains idempotents. To this end, let with . Then . So we must have one of the following: (i), (ii), (iii) for some .
Clearly (ii) does not hold, because in that case becomes a member of while it is a member of minimal left ideal. If (iii) holds then we have for some . Since , we have . But , a contradiction. Hence (i) must hold so that .
Let us recall following lemma for our purpose.
Lemma 2.7. Let be a dense subsemigroup of such that is a subsemigroup of and assume that for each and each , and . If and is a central set near 0, then is also a central set near 0.
Proof. See [6, Lemma 4.8].
Lemma 2.8. Let be a dense subsemigroup of such that is a subsemigroup of and assume that for each and each , and . If is central set near 0 in then is also central set near 0.
Proof. Since and is central set near 0 then by Lemma 2.7, is central set near 0.
Lemma 2.9. Let be a dense subsemigroup of such that is a subsemigroup of and assume that for each and each , and . If is a set near 0 in then is also set near 0.
Proof. Let be a set near 0 and . To prove that is a set near 0 it is sufficient to show that for any central set near 0 , . Since is central set near 0, is also central set near 0 so that . Choose and such that . Therefore so that .
Theorem 2.10. Let be a dense subsemigroup of such that is a subsemigroup of and assume that for each and each , and . Also let be a minimal sequence near 0 and let be a set near 0 in . Then there exists a sum subsystem of such that
The authors would like to acknowledge the referee for her/his constructive report. The first named author is partially supported by DST-PURSE programme. The work of this paper was a part of second named authors Ph.D. dissertation which was supported by a CSIR Research Fellowship.
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Copyright © 2012 Dibyendu De and Ram Krishna Paul. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.