Abstract
It is known that for an IP* setAinℕand a sequence〈xn〉n=1∞there exists a sum subsystem〈yn〉n=1∞of〈xn〉n=1∞such thatFS(〈yn〉n=1∞) ∪ FP(〈yn〉n=1∞)⊆A. Similar types of results also have been proved for central* sets. In this present work we will extend the results for dense subsemigroups of((0,∞),+).
Highlights
One of the famous Ramsey theoretic results is Hindman’s Theorem.Theorem 1.1
The original proof of this theorem was combinatorial in nature
First we give a brief description of algebraic structure of βSd for a discrete semigroup S, ·
Summary
One of the famous Ramsey theoretic results is Hindman’s Theorem.Theorem 1.1. Given a finite coloring N {1, 2, . . . , r} such that r i exists a sequence xn ∞ n1 in N and i ∈ xn∞ n1 xn : F ∈ Pf N ⊆ Ai, 1.1 n∈F where for any set X, Pf X is the set of finite nonempty subsets of X.The original proof of this theorem was combinatorial in nature. In 2 we see that sets which belong to every minimal idempotent of N, called central∗ sets, must have significant multiplicative structure. In case of central∗ sets a similar result has been proved in 3 for a restricted class of sequences called minimal sequences, where a sequence xn
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