Hammocks have been considered by Brenner [1],
who gave a numerical criterion
for a finite translation quiver to be the Auslander–Reiten quiver
of some
representation-finite algebra. Ringel and Vossieck [11]
gave a combinatorial definition
of left hammocks which generalised the concept of hammocks in the sense
of Brenner,
as a translation quiver H and an additive function h
on H (called the hammock
function) satisfying some conditions. They showed that a thin left hammock
with
finitely many projective vertices is just the preprojective component of
the
Auslander–Reiten quiver of the category of [Sscr ]-spaces, where [Sscr ]
is a finite partially ordered set (abbreviated as ‘poset’).
An important
role in the representation theory
of posets is played by two differentiation algorithms. One of the algorithms
was
developed by Nazarova and Roiter [8], and it
reduces a poset [Sscr ] with a maximal
element a to a new poset [Sscr ]′=a∂[Sscr ].
The second algorithm was developed by
Zavadskij [13], and it reduces a poset [Sscr ]
with a suitable pair (a, b) of elements a, b
to
a new poset [Sscr ]′=∂(a,b)[Sscr ].
The main purpose of this paper is to construct new left
hammocks from a given one, and to show the relationship between these new
left
hammocks and the Nazarova–Roiter algorithm. In a later paper
[5], we discuss the
relationship between hammocks and the Zavadskij algorithm.