In the context of Kahn process networks - where processes are vertices, and edges between vertices are FIFO channels - a fair merge process combines sequences of values arriving on two input channels into a single sequence, in such a way that the output sequence produced in any complete computation is always a fair shuffle.
A fair merge process is not to be confused with an angelic merge or an infinitary-fair merge. In the words of [Panangaden and Stark 1998]:
The angelic merge and infinity-fair merge processes perform a function similar to that of the fair merge, but do not necessarily transmit all values that arrive on both inputs. Instead, they both satisfy the basic requirement that the output sequence should be a fair shuffle of prefixes (possibly all) of the input sequences. In addition to this basic requirement, an angelic merge process guarantees that if the sequence arriving on one input is finite, then it will transmit the entire sequence of values that arrives on the other input. An angelic merge process therefore never gets "stuck" waiting for input that might never arrive. An infinity-fair merge process supplements the basic requirement with the guarantee that if the sequence arriving on one input is infinite, then it will transmit the entire sequence of values that arrive on the other input.
Another description, due to [Panangaden and Shanbhogue 1988]:
A fair merge primitive, as well as an angelic merge primitive, has a pair of input channels and a single output channel. Tokens from different channels appear interleaved in the output sequence in an unpredictable fashion, but the relative order of tokens from a single input channel are preserved. The two primitives differ in the sense that, for a fair merge primitive, every token that appears at an input channel appears on the output channel eventually, but for an angelic merge primitive, all that is guaranteed is that the output sequence is infinite if at least one of the input sequences are infinite. Note that the angelic merge primitive is fair, if both its input sequences are finite.
Panangaden, Prakash, and Eugene W. Stark. 1988. ‘Computations, Residuals, and the Power of Indeterminacy’. In Automata, Languages and Programming, edited by Timo Lepistö and Arto Salomaa, 317:439–54. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/3-540-19488-6_133.
Panangaden, Prakash, and Vasant Shanbhogue. 1988. ‘McCarthy’s Amb Cannot Implement Fair Merge’. In Foundations of Software Technology and Theoretical Computer Science, edited by Kesav V. Nori and Sanjeev Kumar, 338:348–63. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/3-540-50517-2_90.