Specker spaces and their absolutes, I

A (Tychonoff) space is Specker if for each nonzero real-valued continuous function there is a nontrivial clopen set on which it is constant and nonzero. The following question is posed: if X is a Specker space, then is the absolute of X a Specker space as well? The answer is no, in general, but there are several interesting classes of spaces for which it is true. The dual notion of a Specker boolean algebra turns out to be equivalent to so-called (ω, 2)-distributivity, a fact which appears crucial in finding an example showing that the answer to the question raised above is indeed no. In the final section the totally ordered Specker spaces are characterized.