I know only about identities and quasi-identities as axioms of  structures called "algebras". Are there other kinds of such formulas?

I perceive identities and quasi-identities as specific for algebra, because they define varieties and quasi-varieties, which can be defined also in terms of homomorphisms, subalgebras, products and subdirect products - notions specifically algebric. Are there other algebraic notions same strongly correlated with fundamental formulas describing them? 

This question was motivated by my another [question](http://mathoverflow.net/questions/177630/adjunction-algebra-is-there-anything-similar-to-this-in-abstract-algebra), where I named "adjunction *algebra*" an algebra with a binary operation and axioms among which one is not an identity, and I don't know whether or not it is equivalent with one or several quasi-identities.