According to Wikipedia, the proper term is “open front toilet seat,” and is required by code in all public restrooms. The code is from a plumbing organization and is not law, but everyone follows it as a standard that avoids legal challenges. (“Gee, Honey, I don’t know where I possibly got that VD infection. Must have been from that closed toilet seat at work.”)
Open front seats are mostly for the benefit of women, but are universal even in men’s rooms. The reason for the split is best quoted directly from Wikipedia: “The purpose for this seat design is to allow women to wipe the perineal area after using the toilet without contacting the seat. It also omits an area of the seat that could be contaminated with urine, and avoids contact between the seat and the user’s genitals.”
I have no idea what goes on behind closed stall doors, but the benefits sound so desirable, maybe I’ll get one for our guest bathroom. Come on over and try it out.