Is there a standard name for this relation property : " aRb --> there is no c different from b such that aRc "? Maybe this property could be called "exclusivity" ? Does it have a standard name? It recalls the definition of a function as a " single-valued relation" (Enderton). But here, it is not required that any _a_ ( in a given set) be related to some _b_.

Such relations are (in my experience) called " **functional** ", in analogy with functions. Indeed, such a relation _is_ a partial function (and actually I've heard "$R$ is a partial function" more frequently than I've heard "$R$ is functional").

Similarly, relations such that for every $a$ there is at least one $b$ with $aRb$ are called " **total** " (in analogy with partial vs. total functions), or " **serial** " (although I've heard that one much more rarely). And relations such that for each $b$ there is at most one $a$ with $aRb$ are called " **injective** " (or "one-to-one") relations.