Growth of Generating Sets of Direct Powers

For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A) = (d(A); d(A2); d(A3);  . . .) (direct powers of A). Thus, for example, for a cyclic group of order n we have d(Cn) = (1; 2; 3; 4; . . . ). Wiegold, in a sequence of papers stretching from 1974 to 1989, and involving several co-authors, investigated the d-sequence for finite groups, infinite groups and finite semigroups. As a very rough summary, they prove that the d-sequence of a (non-trivial) finite group grows either linearly or logarithmically; for infinite groups constant sequences are also possible, while for finite semigroups we can have exponential growth. In my talk I am going to report on joint work in progress with Martyn Quick and several other co-authors. In particular: (a) `Wiegold-type' classifications of growth rates for ?classical? structures: rings, algebras, modules and Lie algebras; (b) connections with Universal Algebra via congruence permutability and functional completeness; (c) some curious examples of growth for infinite semigroups; (d) some initial observations for lattices, tournaments and Steiner triple systems.