TITLE: 
                        
                            Second-Order Formulas in Action
                                
                                
                                    AUTHORS: 
                                            Yuri Movsisyan 
                                                    
                                                        KEYWORDS: 
                        Boolean Algebra, Boolean Function, Distributive Lattice, Monotone Boolean Function, De Morgan Algebra, De Morgan Function, Boole-De Morgan Algebra, Quasi-De Morgan Function, Hyperidentity, Coidentity 
                                                    
                                                    
                                                        JOURNAL NAME: 
                        Applied Mathematics,  
                        Vol.15 No.9, 
                        September
                                                        27,
                        2024
                                                    
                                                    
                                                        ABSTRACT: This survey article illustrates many important current trends and perspectives for the field and their applications, of interest to researchers in modern algebra, mathematical logic and discrete mathematics. It covers a number of directions, including completeness theorem and compactness theorem for hyperidentities; the characterizations of the Boolean algebra of n-ary Boolean functions and the bounded distributive lattice of n-ary monotone Boolean functions; the functional representations of finitely-generated free algebras of various varieties of lattices via generalized Boolean functions, etc.