In a paper of 1993, Pin and Thérien establish an algebraic characterization of the closure of a pseudovariety V under bideterministic product making use of an expansion of finite monoids.  

We use a profinite approach to the Pin-Thérien expansion and results about J-reduced factorizations of pseudowords to show that a pseudovariety in the interval [Sl, DS]  closed under bideterministic product admits a basis of pseudoidentites between finite products of regular pseudowords. Conversely, a pseudovariety with a basis of pseudoidentities between finite products of regular pseudowords is closed under bideterministic product.

Let RS be the class of all finite semigroups whose regular elements form a subsemigoup. For pseudovarieties V such that Sl is in V and V is in the intersection DS and RS we deduce that if V is local then the closure of the pseudovariety V is local.

This is a joint work with Alfredo Costa.