A Polynomial Ring Construction for the Classification of Data

Abstract

Drensky and Lakatos (Lecture Notes in Computer Science, 357 (Springer, Berlin, 1989), pp. 181-188) have established a convenient property of certain ideals in polynomial quotient rings, which can now be used to determine error-correcting capabilities of combined multiple classifiers following a standard approach explained in the well-known monograph by Witten and Frank (Data Mining: Practical Machine Learning Tools and Techniques (Elsevier, Amsterdam, 2005)). We strengthen and generalise the result of Drensky and Lakatos by demonstrating that the corresponding nice property remains valid in a much larger variety of constructions and applies to more general types of ideals. Examples show that our theorems do not extend to larger classes of ring constructions and cannot be simplified or generalised.2000 Mathematics subject classification: primary 16S36; secondary 20M25, 94B60.

Ideal basis in constructions defined by directed graphs

“…[33, §3.15], [39], [49] and [59]). If R has an identity element 1, then to simplify notation we identify every edge e in E with the element 1e in I D (R).…”

“…The following definition was introduced in [29] by analogy to a similar concept of ring theory considered, for example, in [49].…”

“…Visible ideal bases were introduced in [7] by analogy with different constructions (cf. [29,49]). Our main theorem establishes that, for each balanced digraph D and every idempotent semiring R with identity element, the incidence semiring I D (R) always has a convenient visible ideal basis B D (R), and elements of B D (R) can be used to generate two-sided ideals with the largest possible weight among the weights of all two-sided ideals in the incidence semiring (see Theorem 4.1).…”

Ideals of Largest Weight in Constructions Based on Directed Graphs

“…This result belongs to the large area studying the weights of ideals in various constructions. For instance, the ideals of the largest possible weight have been described for certain classes of the polynomial quotient rings in [26], structural matrix semirings in [17], Munn semirings in [9], and incidence semirings in [8,10]. The investigation of ideals with largest weight has been motivated by applications of ideals in data mining (cf.…”

On Hyperideals in Left Almost Semihypergroups

“…Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics and its applications such 2 ISRN Algebra as in informatics, physics, and others. As an example of applications of the concept of an ideal in informatics, let us mention that ideals of algebraic structures have been used recently to design efficient classification systems, see [8][9][10][11][12] .…”