Semigroups with the Erdös-Turán Property
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Estadístiques de LA Referencia / Recolecta
Inclou dades d'ús des de 2022
Cita com:
hdl:2117/405
Tipus de documentArticle
Data publicació2006-01
Condicions d'accésAccés obert
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Reconeixement-NoComercial-SenseObraDerivada 2.5 Espanya
Abstract
A set X in a semigroup G has the Erdös-Turán property ET if,
for any basis A of X, the representation function rA is ubounded,
where rA(x) counts the number of representations of x as a product
two elements in A. We show that, under some conditions, operations
on binary vectors whose value at each coordinate depends only on
neighbouring coordinates of the factors give rise to semigroups with
the ET{property. In particular countable powers of semigroups with
no mutually inverse elements have the ET{property. As a consequence,
for each k there is N(k) such that, for every ¯nite subset X of a group
G with X \ X¡1 = f1g, the representation function of every basis of
XN ½ GN, N ¸ N(k), is not bounded by k. This is in contrast with
the known fact that each p{elementary group admits a basis of the
whole group whose representation function is bounded by an absolute
constant
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