Compatible congruences defined via triangular norms and triangular conorms

The main aim of this paper is to investigate relationship between ideals or filters of a bounded lattice and T-compatible or S-compatible congruences which are obtained by using the definition of compatible aggregation functions. Additionally, we introduce some congruences and characterize the conditions under which they belong to the sets of T-compatible or S-compatible congruences, denoted as $ Con_TL $ ConTL and $ Con_SL $ ConSL, respectively. We define an equivalence class denoted by $ (a)<^>A_I $ (a)IA (A is an aggregation function and I is a nonempty set) and elaborate on the properties of this class when A is t-norm or t-conorm.