Hammocks and the Nazarova-Roiter Algorithm

Hammocks have been considered by Brenner 1], who gave a numericalcriterion for a finite translation quiver to be the Auslander–Reitenquiver of some representation-finite algebra. Ringel and Vossieck11] gave a combinatorial definition of left hammocks whichgeneralised the concept of hammocks in the sense of Brenner,as a translation quiver H and an additive function h on H (calledthe hammock function) satisfying some conditions. They showedthat a thin left hammock with finitely many projective verticesis just the preprojective component of the Auslander–Reitenquiver of the category of S-spaces, where S is a finite partiallyordered set (abbreviated as ‘poset’). An importantrole in the representation theory of posets is played by twodifferentiation algorithms. One of the algorithms was developedby Nazarova and Roiter 8], and it reduces a poset S with amaximal element a to a new poset S'=_aS. The second algorithmwas developed by Zavadskij 13], and it reduces a poset S witha suitable pair (a, b) of elements a, b to a new poset S'=_(a,b)S.The main purpose of this paper is to construct new left hammocksfrom a given one, and to show the relationship between thesenew left hammocks and the Nazarova–Roiter algorithm. Ina later paper 5], we discuss the relationship between hammocksand the Zavadskij algorithm.