POSETS OF FINITE PRINJECTIVE TYPE AND A CLASS OF ORDERS

The main aim of this paper is to give a characterisation of finite pos ets J having only finitely many isomorphism classes of indecomposable socle projective K-linear representations over a given field K, or equ ivalently, finite posets J having only finitely many indecomposable ca nonical forms of partitioned matrices of the shape (2.12) (with coeffi cients in K) with respect to the J-elementary transformations (E1) and (E2) defined in Section 2. The characterisation is given in Theorem 3 .1 in terms of the Tits quadratic form associated to J, in terms of a class of algebraic varieties with an algebraic group action, and by pr esenting a critical list of 114 minimal posets having infinitely many isomorphism classes of indecomposable socle projective representations . An application of posets of finite prinjective type to the study of indecomposable lattices over a class of orders is given.