### Sincere posets of finite prinjective type with three maximal elements and their sincere prinjective representations

Justyna Kosakowska
2002 Colloquium Mathematicum
Assume that K is an arbitrary field. Let (I, ) be a poset of finite prinjective type and let KI be the incidence K-algebra of I. A classification of all sincere posets of finite prinjective type with three maximal elements is given in Theorem 2.1. A complete list of such posets consisting of 90 diagrams is presented in Tables 2.2. Moreover, given any sincere poset I of finite prinjective type with three maximal elements, a complete set of pairwise non-isomorphic sincere indecomposable
more » ... omposable prinjective modules over KI is presented in Tables 8.1. The list consists of 723 modules. 0. Introduction. Throughout this paper I = (I, ) is a finite poset (i.e. partially ordered set) with partial order . We write i ≺ j if i j and i = j. For simplicity we write I instead of (I, ). The poset I is said to be connected if I is not the union of two proper subposets I 1 , I 2 such that I 1 ∩ I 2 = ∅. Throughout the paper all posets are assumed to be connected. Following [21] we denote by max I the set of all maximal elements of I (called peaks of I). The poset I is called an r-peak poset if |max I| = r, where |X| denotes the cardinality of the set X. A subposet J of I is said to be a peak subposet if J ∩ max I = max J. Throughout the paper, K is a field. We denote by KI the incidence K-algebra of the poset I, that is, KI is the K-subalgebra of the full I × I matrix algebra M I (K) consisting of all matrices [λ ij ] in M I (K) such that λ ij = 0 if i j in (I, ) (see [20] , [21] ). Given j ∈ I we denote by e j ∈ KI the standard primitive idempotent corresponding to j. A right KI-module X is identified with a system