In usual algebraic structures, like groups, rings, monoids, etc, or in algebras coming from logics like Boolean algebras, Heyting algebras and the like the operations are usually of arity 0 (constants), 1 or 2. My question is two-fold:
Provide examples of algebras arising naturally in some field (I'm mainly interested in algebras coming from logics, but I'm open to any field) with operations of arity 3 or bigger.
Is there any reason (more or less profound) for being so few algebras with operations of arity bigger than 2?
Thank you in advance.