arXiv subjects
Publications and source records attributed to Jyotirmoy Poddar.
Sets satisfying Central sets theorem and other Ramsey theoretic large sets were studied extensively in literature. Hindman and Strauss proved that product of some of these large sets is again large. In this paper we show that if we take two combinatorially large sets along idempotent filters, then their product is also a filter large set. The techniques used here to prove our results are completely elementary in nature.
The Central Sets Theorem near zero was originally proved by Hindman and Leader. Later a version of Central Sets Theorem was proved by De, Hindman and Strauss known to be the stronger Central Sets Theorem. Subsequently many other versions of Central Sets Theorem came, among which Dev Phulara proved the theorem for a sequence of central sets instead of taking one set. In this paper, we provide various general versions of the theorem along Dev Phulara's way with some Ramsey theoretic applications.
Using the notions of Topological dynamics, H. Furstenberg defined central sets and proved the Central Sets Theorem. Later V. Bergelson and N. Hindman characterized central sets in terms of algebra of the Stone-Čech compactification of discrete semigroup. They found that central sets are the members of the minimal idempotents of \b{eta}S, the Stone-Čech compactification of a semigroup (S, .). Hindman and leader introduced the notion of Central set near zero algebraically. Later dynamical and combinatorial characterization have also been established. For any given filter F in S a set A is said to be a F- central set if it is a member of a minimal idempotent of a closed subsemigroup of \b{eta}S, generated by the filter F. In a recent article Bergelson, Hindman and Strauss introduced strongly central and very strongly central sets in [BHS]. They also dynamically characterized the sets in the same paper. In the present article we will characterize the strongly F- central sets dynamically and combinatorially. Here we introduce the filter version of strongly central sets and very strongly central sets. We also provide dynamical and combinatorial characterization of such sets.
The Central Sets Theorem was introduced by H. Furstenberg and then afterwards several mathematicians have provided various versions and extensions of this theorem. All of these theorems deal with central sets, and its origin from the algebra of Stone-Cech compactification of arbitrary semigroup, say $βS$. It can be proved that every closed subsemigroup of $βS$ is generated by a filter. We will show that, under some restrictions, one can derive the Central Sets Theorem for any closed subsemigroup of $βS$ . We will derive this theorem using the corresponding filter and its algebra. Later we will also deal with how the notions of largeness along filters are preserved under some well behaved homomorphisms and give some consequences.