  Up-directed rough sets are introduced and studied by the present author in
earlier papers. This is extended by her in two different granular directions in
this research, with a surprising algebraic semantics. The granules are based on
ideas of generalized closure under up-directedness that may be read as a form
of weak consequence. This yields approximation operators that satisfy cautious
monotony, while pi-groupoidal approximations (that additionally involve
strategic choice and algebraic operators) have nicer properties. The study is
primarily motivated by possible structure of concepts in distributed cognition
perspectives, real or virtual classroom learning contexts, and student-centric
teaching. Rough clustering techniques for datasets that involve up-directed
relations (as in the study of Sentinel project image data) are additionally
proposed. This research is expected to see significant theoretical and
practical applications in related domains.
