roni said:
(1) Which word is correct?
Either "property" or "attribute" can be used. These words are used in their usual sense from the dictionary, not in a technical sense. The second occurrence of "property", though, is probably used in a technical sense.
From the mathematical standpoint, a property, or predicate, on a set $A$ is any subset of $A$. Alternatively, a predicate on $A$ can be defined as a function from $A$ to the set {true, false}. The the subsets consists of those elements of $A$ that are mapped to true.
A binary relation on a set $A$ is any subset of $A\times A$, a Cartesian product of $A$ and $A$, which is a set of all ordered pairs of elements of $A$. Thus, a binary relation is a property of ordered pairs, or a predicate on $A\times A$. Binary relations can be transitive or not. Thus, "transitivity" is a property of a binary relation, i.e., something that can be true or not about this relation.
roni said:
(2) Why Transitivity is not standalone by itself?
Any adjective, like "transitive", requires a noun in mathematics just like in everyday language.
roni said:
(3) Are there relations of other kind, that no standalone by themselves?
An adjective "continuous" is a property of functions in calculus, so to talk about continuity you must apply it to some functions.
