We present a comprehensive study of algebras satisfying the identity \((xy)z=y(zx),\) named as shift associative algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to anti-Poisson-Jordan algebras and algebras of associative type \(\sigma \) . We study algebras of associative type \(\sigma \) to be Koszul and self-dual. A basis of the free shift associative algebra generated by a countable set X is constructed. An analog of Wedderburn-Malcev’s theorem is established. The algebraic and geometric classifications of complex 4-dimensional shift associative algebras are given. In particular, we prove that the first non-associative shift associative algebra appears only in dimension 5.
