A Classification of Two-dimensional Endo-commutative Algebras over F_2

We introduce a new class of algebras called endo-commutative algebras in which the square mapping preserves multiplication, and provide a complete classification of endo-commutative algebras of dimension 2 over the field F_2 of two elements. We list all multiplication tables of the algebras up to isomorphism. This clarifies the difference between commutativity and endo-commutativity of algebras.