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We introduce a notion of ternary $F$-manifold algebras which is ageneralization of $F$-manifold algebras. We study representation theory ofternary $F$-manifold algebras. In particular, we introduce a notion of dualrepresentation which requires additional conditions similar to the binary case.We then establish a notion of a coherence ternary $F$-manifold algebra.Moreover, we investigate the construction of ternary $F$-manifold algebrasusing $F$-manifold algebras. Furthermore, we introduce and investigate a notionof a relative Rota-Baxter operator with respect to a representation and use itto construct ternary pre-$F$-manifold algebras.

On Ternary $F$-manifold Algebras and their Representations