grey congruences on s-acts: a structural and categorical approach

In this paper, we investigate the notion of grey congruences in the category of s-acts. we begin by extending the classical concept of congruences by incorporating grey set theory to allow degrees of equivalence under partial information. we prove analogs of the first, second, and third isomorphism theorems in the grey context. furthermore, we develop several new structural results on grey congruences, including their behavior under direct products, restrictions to subacts and lattice-theoretic completeness. we show that the collection of grey congruences on an s-act forms a complete lattice. we analyze the behavior of grey congruences under pullbacks in the category of s-acts.