Dendriform algebra is a generic term both for dendriform dialgebras and dendriform trialgebras.
Dendriform dialgebras are algebras over a Koszul quadratic operad which is Koszul dual to the associative dialgebra operad. They have two binary operations $\prec$, $\succ$ (commonly also denoted $\dashv$ and $\vdash$) satisfying 3 quadratic relations. The sum of the two binary operations is an associative operation.
Dendriform dialgebras were introduced by Loday in his study of K-theory,
Jean-Louis Loday, Dialgebras, in: Dialgebras and Related Operads, Springer Lecture Notes in Mathematics 1763 (2001) 766, 1–61
Kurusch Ebrahimi-Fard, Li Guo?, Rota–Baxter algebras and dendriform algebras, J. Pure Appl. Alg. 212:2 (2008) 320-339
Sara Maradiaga, Splitting of operations for alternative and Malcev structures, arXiv:/1312.5710
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