Bongartz Completion via<i>c</i>-Vectors

Abstract In the present paper, we first give a characterization for Bongartz completion in $\tau $-tilting theory via $c$-vectors. Motivated by this characterization, definition of cluster algebras using Then prove existence and uniqueness algebras. We also that admits certain commutativity. two applications As application, full subquiver exchange quiver (or known as oriented graph) algebra $\mathcal A$ whose vertices consist seeds containing particular variables is isomorphic to another algebra. second $Y$-pattern over universal semifield, each $Y$-seed (up equivalence) uniquely determined negative $y$-variables $Y$-seed.