Boolean algebras realized by c.e. equivalence relations

Let $E$ be a computably enumerable (c.e.) equivalence relation on the set of natural numbers $\omega$. We consider countable structures where basic functions are computable and respect $E$. If the corresponding quotient structure is a Boolean algebra $B$, then we say that the c.e. relation $E$ realizes $B$. In this paper we study connections between algorithmic properties of $E$ and algebraic properties of Boolean algebras realized by $E$. Also we compare these connections with the corresponding results for linear orders and groups realized by c.e. equivalence relations.