Sheaf Representation of an Information System

Ever since Pawlak introduced the concepts of rough sets, it has attracted many researchers and scientists from various fields of science and technology. Particularly for algebraists as it presented a gold mine to explore the algebraic and topological connections with rough set theory. The present article deals with the connections between rough sets and sheaves. The authors studied sheaf representation of an information system in rough set framework and illustrated how it helps information retrieval.

Truncated Vector Lattices

In analysis, truncation is the operation of replacing a nonnegative real-valued function a (x) by its pointwise meet a (x) ∧ 1 with the constant $1$ function. A vector lattice A is said to be closed under truncation if a ∧ 1 ∈ A for all a ∈ A+. Note that A need notcontain 1 itself.Truncation is fundamental to analysis. To give only one example, Lebesgue integration generalizes beautifully to any vector lattice of real-valued functions on a set X, provided the vector lattice is closed under truncation. But vector lattices lacking this property may have integrals which cannot be represented by any measure on X. Nevertheless, when the integral is formulated in a context broader than RX, for example in pointfree analysis, the question oftruncation inevitably arises.What is truncation, or more properly, what are its essential properties? In this paper we answer this question by providing the appropriate axiomatization, and then go on to present several representation theorems. The first is adirect generalization of the classical Yosida representation of an archimedean vector lattice with order unit. The second is a direct generalization of Madden's pointfree representation of archimedean vector lattices. If time permits, we briefly discuss a third sheaf representation which has no direct antecedent in the literature.However, in all three representations the lack of a unit forces a crucial distinction from the corresponding unital representation theorem. The universal object in each case is some sort of family of continuous real-valued functions. The difference is that these functions must vanish at a specified point of the underlying space or locale or sheaf space. With that adjustment, the generalization from units to truncations goes remarkably smoothly.

Rings in Which the Annihilator of an Ideal Is Pure

A ring R is a left AIP-ring if the left annihilator of any ideal of R is pure as a left ideal. Equivalently, R is a left AIP-ring if R modulo the left annihilator of any ideal is flat. This class of rings includes both right PP-rings and right p.q.-Baer rings (and hence the biregular rings) and is closed under direct products and forming upper triangular matrix rings. It is shown that, unlike the Baer or right PP conditions, the AIP property is inherited by polynomial extensions and has the advantage that it is a Morita invariant property. We also give a complete characterization of a class of AIP-rings which have a sheaf representation. Connections to related classes of rings are investigated and several examples and counterexamples are included to illustrate and delimit the theory.

Fully idempotent near-rings and sheaf representations

Fully idempotent near-rings are defined and characterized which yields information on the lattice of ideals of fully idempotent rings and near-rings. The space of prime ideals is topologized and a sheaf representation is given for a class of fully idempotent near-rings which includes strongly regular near-rings.