I've been looking at generalized definitions of symmetry. Symmetries, the way we usually define them, form a group. However, there have been efforts to generalize this, creating algebras which have weaker properties. I have heard of a non-invertable concept of symmetry which yields a symmetry semigroup. I have also heard of one which leads to a symmetry groupoid.
Has there been any significant attempt to define a generalized concept of symmetry whose algebra is a quasigroup or a loop? I have been biased by answers like this and my own love of the associative property, so I have been having trouble breaking free of those assumptions.
This is not intended to be a proper 'answer', but it's far too long to fit in a comment. It's an attempt to elaborate both on the question by Cort Ammon and on the comment by Noah Schweber.
Imagine we have a semigroup (it's easy to adapt this reasoning for a monoid or a group if one wishes) whose composition law changes with time. More precisely, we start with a set $G$ equipped with a function
$${\cal T}\times G\times G\rightarrow G,\qquad(t,g,g')\mapsto g'\circ_t g.$$
It seems natural to require the composition to behave as usually with respect to a fixed instant of time. This means: for each $t\in\cal T$ and $g,g',g''\in G$, the static composition shall be associative:
$$g''\circ_t(g'\circ_t g)=(g''\circ_t g')\circ_t g.$$
However, with respect to different instants of time, the temporal composition doesn't have necessarily to be associative anymore. This means: for some $t_1,t_2\in\cal T$ and $g,g',g''\in G$, we may have
$$g''\circ_{t_2}(g'\circ_{t_1} g)\ne(g''\circ_{t_2} g')\circ_{t_1} g.$$
Of course, this construction looks somewhat artificial, and indeed I've never heard of anything like that before. But, doesn't it throw some light on a possible hidden relationship between associativity and permanence in time?
Now, if someone wanted to try and think about this 'notion' as the 'symmetry group' of some concrete object, maybe a tensegrity structure could be a nice starting point. For instance, one such structure that happened to admit multiple solutions, possibly related by some abrupt deformations, would probably have a very interesting 'symmetry group' to think about.