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