Quantum groups (or more precisely, function algebras on quantum groups), i.e. bialgebras with certain additional properties, can be gained by deforming an appropriate function algebra on a group. In a similar way, we show that a polynomial-like algebra on the (function algebra of the) Manin plane leads to a so-called trialgebra (as suggested by Crane and Frenkel), i.e. an algebraic structure possessing a coproduct and two products in a compatible way. We show how to deform this trialgebra to a noncocommutative and totally noncommutative (i.e. in both products) one. Trialgebras are of interest for various reasons, e.g. the search for topological invariants for four manifolds or the duality operation for non-Abelian lattice gauge theories, recently suggested by the authors.