Locally countable pseudovarieties

The purpose of this paper is to contribute the theory profinite semigroups by considering special class consisting those all whose finitely generated closed subsemigroups are countable, which said be locally countable. We also call countable a pseudovariety V (of finite semigroups) for pro-V investigate operations preserving local countability pseudovarieties and show that, in contrast with finiteness, several natural do not preserve it. relationship semigroup being every element expressable terms generators using multiplication idempotent (omega) power. two properties turn out equivalent if there only countably many group elements, gathered regular J-classes. that ordered monoids satisfying inequality $1\le x^n$ $n=1$.