On endo-commutative algebraic structures on two-dimensional vector spaces over an arbitrary field

In this paper, we describe the class of all two-dimensional endo-commutative algebras over any base field. Thereby, we extend recent results of Takahasi, Shirayanagi, and Tsukada on description of the class of two-dimensional endo-commutative algebras to the case of an arbitrary field. The concept of an endo-commutative algebra was first introduced by aforementioned authors; in the same works, the motivations to study this class of algebras also were presented. In this paper, we present the canonical representatives of the isomorphism classes of two-dimensional endo-commutative algebras over an arbitrary field. The authors would like to thank the anonymous reviewer for very attentive reading the manuscript and the valuable suggestions made.

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