The well-known notion of associativity in algebra leads to structures with interesting properties, such as groups or semigroups. According to a paper by John Rhodes, some researchers in algebra and computational complexity are interested in extending this notion to obtain more general structures that retain certain properties of semigroups.
It seems that such a suggestion was made anonymously on this site, but I couldn't ascertain whether the definitition is well-founded and interesting. Therefore, I would like to know what other extensions of associativity are known, and which ones were successfully used in TCS, if any. I've heard about things like Moufang loops but couldn't wrap my mind around the defn, so any etra 'intuition' is welcome.