Confluence and Koszulity

A lattice-theoretic approach to Koszulity is well known. This paper gives a lattice-theoretic approach to polynomiality, i.e., existence of a P.B.W. basis in Priddyʹs sense. This provides a straightforward proof for Priddyʹs theorem. Polynomiality is interpreted in terms of confluence in noncommutative computational algebra. The basic notion of a reduction operator is investigated in some detail. The reduction algebras are introduced to study the confluence of two reduction operators in terms of representation theory. The meet and the join of two reduction operators are constructed in an algorithmic way. A geometric characterization of the confluence is obtained.