Do synapses have special types?
This question seems to be asking "How is knowledge represented in neurons" and then jumps to the assumption "synapses represents relations". As discussed in the linked question, there is a lot of evidence that this method of binding is not biologically plausible, given that it doesn't scale well to the level of human vocabulary.

So, to answer your question, it is hypothesised that there are "special" synapses to do certain operations/relations, however the model you seem to be holding in your mind is incorrect. For more information, please check out "How to Build a Brain" by Chris Eliasmith or the other reference below.

Reference

Eric Crawford, Matthew Gingerich, and Chris Eliasmith. Biologically plausible, human-scale knowledge representation. In 35th Annual Conference of the Cognitive Science Society, 412–417. 2013.