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 producttwo elements in A. We show that, under some conditions, operationson binary vectors whose value at each coordinate depends only onneighbouring coordinates of the factors give rise to semigroups withthe ET{property. In particular countable powers of semigroups withno 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 groupG with X \ X¡1 = f1g, the representation function of every basis ofXN ½ GN, N ¸ N(k), is not bounded by k. This is in contrast withthe known fact that each p{elementary group admits a basis of thewhole group whose representation function is bounded by an absoluteconstant