The new main notion studied here is the notion of a strong onto homomorphic relation R  between structures A and B denoted by A  B. We show that this notion generalizes many fundamental notions of model theory such as substructure and elementary substructure to the situation where an equality symbol is not assumed to be present in the first order language L. We generalize many results of model theory which usually assume that L has an equality symbol to this situation    
                        
                        
                        
                        
                            
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