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IP* Sets in Product Spaces
Author(s) -
BERGELSON VITALY,
HINDMAN NEIL
Publication year - 1996
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1996.tb49157.x
Subject(s) - cartesian product , compactification (mathematics) , commutative property , mathematics , substructure , semigroup , finite set , combinatorics , product (mathematics) , discrete mathematics , order (exchange) , pure mathematics , mathematical analysis , geometry , structural engineering , finance , engineering , economics
An IP* set in a semigroup ( S ,·) is a set which meets every set of the form FP (〈 x n 〉 ∞ n =1 ) = {∏ nεF x n : F is a finite nonempty subset of ℕ}, where the products are taken in increasing order of indices. We show here, using the Stone‐Cech compactification of the product space S 1 × S 2 ×…× S ℓ , that if each S i is commutative, then whenever C is an IP* set in S 1 × S 2 ×…× S ℓ , and for each i ε {1, 2, …, ℓ}, 〈 x i,n 〉 ∞ n =1 is a sequence in S i , C contains Cartesian products of arbitrarily large finite substructures of FP (〈 x 1, n ℓ n =1 〉) × FP (〈 x 2, n ℓ n =1 〉) ×…× FP (〈 x ℓ,n ℓ n =1 〉). (The notion of “substructure” is made precise in Definition 2.4.) We show further that C need not contain any product of infinite substructures and that the commutativity hypothesis may not be omitted. Similar results apply to arbitrary finite products of semigroups. By way of contrast, we show in Theorem 2.3 that a much stronger conclusion holds for some cell of any finite partition of S 1 × S 2 ×…× S ℓ , without even any commutativity assumptions.