He doesn’t carry a six shooter or ride horseback around town, but Edgar A. Domenech, 48, was recently appointed by Mayor Michael R. Bloomberg to be New York City’s next sheriff. The position of city sheriff dates back to 1626, and includes arresting and taking to court parents who owe child support, investigating the sale of illegal cigarettes, enforcing domestic protection orders and helping people with mental illness who are under court order to receive treatment. Domenech is currently the special agent in charge of the Washington office of the Bureau of Alcohol, Tobacco, Firearms and Explosives. Some of the lesser-known duties of the sheriff, whose office is part of the Department of Finance, include, being in charge of shipwrecks, and deputizing and arming citizens to help quash rebellions against the government. Domenech will become the city’s 117th sheriff, and will lead a staff of 174 employees, working with an annual budget of $16 million.