Peirce gives his clearest and most complete definition of signs and sign relations in the context of defining logic. Here’s a link to a couple of variants:
There is more discussion in the following article and section:
The proper unit of analysis is the whole sign relation where and are the object, sign, and interpretant sign domains, respectively. In that sense, one could say that the individual sign is always incomplete until one specifies the sign relational setting in which it is conceived to have significance.