Add to ORKGRecently, a special algebra called EQ-algebra has been introduced by Vilem Novák in [31], which aims at becoming the algebra of truth values for fuzzy type theory. It has three binary operations – meet, multiplication and fuzzy equality – and a top element. The mul-tiplication, in EQ-algebra, is assumed to be both commutative and associative. In this paper, we generalize the concept of EQ-algebra by excluding both the commutativity and the associativity of the multi-plication showing that nothing is lost. We call such type of algebra a semicopula-based EQ-algebra. We show that all proved properties of EQ-algebras remain valid and applicable in semicopula-based EQ-algebras and vice versa. Besides these main results, a lot of new and importan...