  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.
