In the best mathematical terms, a triadic relation is a cartesian product of three sets together with a specified subset of that cartesian product.
Alternatively, one may think of a triadic relation as a set of 3-tuples contained in a specified cartesian product.
It is important to recognize that sets have formal properties that their elements do not. The greatest number of category mistakes that bedevil errant discussions of relations and especially triadic sign relations arise from a failure to grasp this fact.
For example, the irreducibility (or indecomposability) of triadic relations is a property of sets-of-triples, not of individual triples.
See the articles under the following heading for concrete examples and further discussion: