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This paper studies the concepts of a totally compatible dialgebra and atotally compatible Lie dialgebra, defined to be a vector space with two binaryoperations that satisfy individual and mixed associativity conditions and Liealgebra conditions respectively. We show that totally compatible dialgebras areclosely related to bimodule algebras and semi-homomorphisms. Moresignificantly, Rota-Baxter operators on totally compatible dialgebras provide auniform framework to generalize known results that Rota-Baxter relatedoperators give tridendriform algebras. Free totally compatible dialgebras areconstructed. We also show that a Rota-Baxter operator on a totally compatibleLie dialgebra gives rise to a PostLie algebra, generalizing the fact that aRota-Baxter operator on a Lie algebra gives rise to a PostLie algebra.

Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras