Final answer:
The Institutional Revolutionary Party (PRI) established a monopoly on Mexican politics starting in 1929, controlling it until the twenty-first century, by functioning as a one-party state.
Step-by-step explanation:
The political party that established a monopoly on the political system of Mexico in 1929 was the Institutional Revolutionary Party (PRI). After the Mexican Revolution had led to a period of instability and conflict, the PRI was formed to provide a stable and enduring political structure. The party, despite undergoing various name changes over the years, maintained a strong grip on Mexican politics well into the twenty-first century. Although there were regular elections, the PRI's control was so absolute that it effectively ran a one-party state, with all presidents and the vast majority of legislators being party members. This endeavor to create stability turned into a mechanism that allowed the PRI to have a monopoly over the political landscape, stifling true democratic participation. This monopoly on power limited the rise of opposition and the development of a genuinely competitive political environment in Mexico for much of the twentieth century. The PRI's dominance was exemplified by the actions of President Lázaro Cárdenas during his tenure beginning in 1938, as he pushed for socialist policies, supported labor unions, nationalized the oil industry, and promoted land reform to address the needs of ordinary workers.