<p>The well-known notion of <em>associativity</em> in algebra leads to structures with interesting properties, such as groups or semigroups. According to a <a href="http://math.berkeley.edu/~rhodes/JohnRhodes.pdf" rel="nofollow noreferrer">paper</a> 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.</p>

<p>It seems that such a suggestion was made <a href="https://cstheory.stackexchange.com/questions/21631/notion-of-associativity-for-ternary-relations">anonymously</a> 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.</p>

CC BY-SA 3.0
2014-04-06T20:32:10.957Z
Whether *others* are interested in that definition, I personally think that the definition by @PilarLoof was interesting. But what leads you to pose the question? Is there any application or similar motivation in mind?
Niel de Beaudrap
CC BY-SA 3.0
2014-04-07T08:07:26.053Z
I don't know how closely it relates to the paper you mention, but one active area of algebra where non-associative structures are studied is the theory of operads (https://en.wikipedia.org/wiki/Operad_theory).
Noam Zeilberger
<algebra><advice-request>
Possible generalizations of associativity?
SiboKikoo
# Possible generalizations of associativity?

(Asked by: username_0 on 2014-04-06)

The well-known notion of *associativity* in algebra leads to structures with interesting properties, such as groups or semigroups. According to a [paper](http://math.berkeley.edu/~rhodes/JohnRhodes.pdf) 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](https://cstheory.stackexchange.com/questions/21631/notion-of-associativity-for-ternary-relations) 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.

**Tags:** algebra, advice-request

> **Comments:**
> **username_1**: Whether *others* are interested in that definition, I personally think that the definition by @PilarLoof was interesting. But what leads you to pose the question? Is there any application or similar motivation in mind?  
> **username_2**: I don't know how closely it relates to the paper you mention, but one active area of algebra where non-associative structures are studied is the theory of operads (https://en.wikipedia.org/wiki/Operad_theory).  

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