This paper deals with a chain of three conjectures in the theory of semigroups. Conjecture 1: if a (pseudo) variety V of finite semigroups is decidable (i.e. one can decide if a given finite semigroup is in the variety), then so is the variety which is the Malcev product of V with the variety of all finite groups. Conjecture 2 (due to Rhodes): the kernel of a finite monoid M (i.e. the intersection of the kernels of all the relational morphisms from M into a finite group) is equal to the smallest submonoid D of M closed under weak conjugation (i.e. \(s,t\in M\) with \(sts=s\) or \(tst=t\), and \(u\in D\) \(\Rightarrow\) sut\(\in D)\). Conjecture 3: a recognizable (by a finite automaton) subset L of a finitely generated free monoid \(A^*\) is closed for the Hall topology (which is the initial topology for all the homomorphisms from \(A^*\) into a finite discrete group) if and only if for any words s,e,t: \(\forall n\geq 1\), \(se^ nt\in L\) \(\Rightarrow\) st\(\in L\). Conjecture 3 \(\Rightarrow\) Conjecture 2 \(\Rightarrow\) Conjecture 1. In a forthcoming paper of the author and the reviewer, is given a conjecture 4 (concerning the Hall topology in a free group) which implies conjecture 3.