In this paper, we introduce and study the notions of T-topogenous orders and T-syntopogenous structures on a set and investigate their properties, where T stands for any continuous triangular norm. Their definitions subsumes that of fuzzy topogenous orders and fuzzy syntopogenous structures due to A. K. Katsaras (Fuzzy Sets and Systems 36 (1990)), as our Min-topogenous orders and Min-syntopogenous structures. In particular, we study T-syntopogenous maps and an I-topological space associated with a T-syntopogenous structure. Also, we deduce the T-topogenous orders as fuzzy relations in (ordinary) power sets.