Is abstract algebra (mostly?) restricted to $2$-ary operators? This may be due to my own pure ignorance but it's my experience that all abstract algebra I've been introduced to, both in actual courses and in self-studies only exclusively deals with algebraic objects consisting of a set together with one or more binary operators defined on that set, perhaps over some other algebraic structure. Not counting categories here, just "low-level"-ish stuff.
I'm an undergraduate so my knowledge is of course very limited but I can't help to wonder why I never stumble upon algebraic structures with $n$-ary operators? Is there a good reason for this, something along the lines of $n$-ary operators behave "badly" when $n>2$ or is it just my ignorance because I'm a beginner in the field?

The reason is simply that many interesting mathematics can be done (or modelled by) operations of arities $\leq 2$. The purpose of abstract algebra is not to study algebraic structures defined by random operations and random rules between them, but rather to give a general framework for algebraic structures which appear from other contexts (which are often more geometric).
On the other hand, there are algebraic structures which are studied and which have $\geq 3$-ary operations which are not reducible to $\leq 2$-ary operations.


*

*ternary fields

*heaps

*ternary algebras
See also the introduction of arXiv:1403.7099 for some background and references on ternary algebras.
