Beta-thalassemia is an autosomal recessive blood disorder characterized by abnormalities in the synthesis of β globin. Together with α globin, it is a subunit of globin protein, called hemoglobin, located inside our red blood cells to deliver oxygen from the lungs to all of the tissues throughout our body. Thereby, individuals with β-thalassemia will often feel limp due to a lack of oxygen dissolved in their blood. In this paper, a finite state automaton to detect and classify β globin gene mutations using its DNA sequence is constructed. Finite state automata have a close connection to an algebraic structure, that is, a monoid. Together with the theory of the syntactic monoid, we present a methodology to minimize the number of the internal states of an automaton to have minimal state automata. Therefore, a minimal state automaton can be constructed to detect β globin gene mutation causing the β-thalassemia disease. We have developed a MATLAB program to conduct the appropriate simulations.