Euclidean transitive relations
The correspondence theory of modal logic is concerned with the study of specific types of relations definable by modal formulae. My one sentence description does not do justice to the numerous motivations behind correspondence theory; consider it an explanation of why it relates to your sentence.

Specific examples arising in correspondence theory are the following.


The modal formula $\Box p \implies p$ is true in a frame if and only if the frame is reflexive.
The formula $\Box p \implies \Box \Box p$ is true in a frame if and only if the frame is transitive. 


So the types of relations you are asking for can be viewed as being defined by a finite set of modal formulae. These formulae can in turn be viewed as the axioms of a logic, so your question is directly related to the study of modal logics. You can look up material on the model theory of modal logic for more details.


Modal Correspondence Theory, van Benthem
Modal Logic, Blackburn, de Rijke, Venema
Model Theory of Modal Logic, Goranko and Otto