Digital representation of semigroups and groups

A digital representation of a semigroup (S,⋅) is a family 〈F _t 〉 _t∈I , where I is a linearly ordered set, each F _t is a finite non-empty subset of S and every element of S is uniquely representable in the form Π _t∈H   x _t where H is a finite subset of I, each x _t ∈F _t and products are taken in increasing order of indices. (If S has an identity 1, then Π _t∈∅   x _t =1.) A strong digital representation of a group G is a digital representation of G with the additional property that for each t∈I, for some x _t ∈G and some m _t >1 in ℕ where m _t =2 if the order of x _t is infinite, while, if the order of x _t is finite, then m _t is a prime and the order of x _t is a power of m _t . We show that any free semigroup has a digital representation with each | F _t |=1 and that each Abelian group has a strong digital representation. We investigate the problem of whether all groups, or even all finite groups have strong digital representations, obtaining several partial results. Finally, we give applications to the algebra of the Stone-Čech compactification of a discrete group and the weakly almost periodic compactification of a discrete semigroup. Dedicated to Karl Heinrich Hofmann on the occasion of his 75th birthday. Stefano Ferri was partially supported by a research grant of the Faculty of Sciences of Universidad de los Andes. The support is gratefully acknowledged. Neil Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803.