This paper relates to a central problem in synthetic biology, which is that of designing Recombinase-based biological devices by matching a functional specification expressed as a Boolean function. This task is challenging as exploring the space of possibilities is typically unfeasible, and therefore many non-trivial design alternatives remain unexplored. Also, the issue has been so far regarded mainly from a practical perspective and is still lacking of formal foundations on which the definition of algorithms for assisting the biologists in their design tasks can be based. In this work, we present the first formal study of the problem, and give a formal semantics for a family of Recombinase-based biological devices. We then exhibit a set of semantic properties leading to the definition of representative devices, a notion that allows one to express infinitely large classes of design possibilities in a finite way. Building on this, we then provide a terminating algorithm for generating representative devices for n-input Boolean functions. An open online database of 18M design solutions for 4-inputs devices generated with our method has been released at http://recombinator.lirmm.fr.