Paul Ricca, clever guy, had the foresight to implement "bearding" in Chicago. When Paul was alive (before he deteriorated), Accardo and Mooney were "bearding" for him, meaning Paul was the real boss but they "wore the boss's badge" and everyone thought they were boss and underboss. After Paul died, Mooney was legitimately the boss, but there was friction between him and Accardo. After Mooney was off the grid, Teets and Phil were legitimate consecutive bosses for a very short period of time, while Accardo enjoyed retirement. When they went to prison, Accardo returned and, drawing on Paul's wisdom, was bearded by Aiuppa and Cerone (and for a short time by Carlisi and DiFronzo) for as long as he was alive. With Accardo, Aiuppa, Cerone, and Carlisi all dead or in prison by the early nineties, DiFronzo became the de facto boss, and pulled his guys back. Whoever the actual boss is at a given time is the guy who gives orders and approves/denies requests.