Antimonotone quadratic forms and partially ordered sets

Authors:
L. A. Nazarova, A. V. Roiter and M. N. Smirnova

Translated by:
the authors

Original publication:
Algebra i Analiz, tom **17** (2005), nomer 6.

Journal:
St. Petersburg Math. J. **17** (2006), 1015-1030

MSC (2000):
Primary 06-99

DOI:
https://doi.org/10.1090/S1061-0022-06-00938-1

Published electronically:
September 20, 2006

MathSciNet review:
2202449

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Representations of partially ordered sets (posets) and quivers are an important part of the theory of matrix problems and algebra representations. Along with chains (linearly ordered sets), a special role is played by certain special posets; in this paper it is shown that they are in one-to-one correspondence with the rational numbers that are greater than or equal to .

A wattle is a union of nonintersecting chains such that the minimal element of is smaller than the maximal element of (and these are the only possible comparisons). The known lists of critical (i.e., minimal) infinitely representable and wild posets consist of cardinal chains, with the exception of one poset in the first list (namely, ) and one in the second (namely, . At the same time, the authors have assigned a rational number to each poset in such a way that if and only if is finitely representable and if and only if is tame. A poset is said to be -*faithful* if whenever .

From the work of Zeldich, Sapelkin, and the authors it follows that the -faithful posets are cardinal sums of -sets, i.e., they are wattles of a special type (chains can be regarded as a partial case of -sets).

In the present paper, the notion of an antimonotone poset is introduced, and a criterion for a poset to be antimonotone is presented under the assumption that the quadratic form is positive semidefinite. At the same time, we manage to substantially simplify the proof of the criterion for a poset to be -faithful, avoiding an item-by-item examination of several dozens of various cases. Also, simple explicit formulas for calculation of are obtained, which lead in an elementary way to the lists of critical posets (originally, they arose as a result of a cumbersome and complex argument).

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Additional Information

**L. A. Nazarova**

Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkovska 3, Kiev 01601, Ukraine

**A. V. Roiter**

Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkovska 3, Kiev 01601, Ukraine

Email:
roiter@imath.kiev.ua

**M. N. Smirnova**

Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkovska 3, Kiev 01601, Ukraine

DOI:
https://doi.org/10.1090/S1061-0022-06-00938-1

Keywords:
Quiver,
graph,
wattle,
faithful poset,
antimonotone poset

Received by editor(s):
February 14, 2005

Published electronically:
September 20, 2006

Article copyright:
© Copyright 2006
American Mathematical Society