Why the notion of "duality" (when we get two theorems for the price of one) are ubiquitous in mathematics (order and lattice theory, category theory, group theory), but "triality" (three theorems for the price of one) is much less common?
Is there a fundamental reason for this?
Remark: In my research I met a case when there are three different order isomorphic sets, but apart of being order isomorphic, their structures are very different, so that I could not call this "triality".