Andries P. J. van der Walt

PRIMITIVITY IN MATRIX NEAR-RINGS

Abstract The relationships between modules of a near-ring R and the matrix near-ring IMn(R) are studied, especially as regards primitivity. It is shown that R is 2-primitive iff IMn(R) is.

A shrinking lemma for random forbidding context languages

Random context grammars belong to the class of context-free grammars with regulated rewriting. Their productions depend on context that may be randomly distributed in a sentential form. Context is classified as either permitting or forbidding, where permitting context enables the application of a production and forbidding context inhibits it. We concentrate on non-erasing grammars that use forbidding context only. We show that they are strictly weaker than the non-erasing random context grammars and prove a shrinking lemma for their languages.

A pumping lemma for random permitting context languages

Random context grammars belong to the class of context-free grammars with regulated rewriting. Their productions depend on context that may be randomly distributed in a sentential form. Context is classified as either permitting or forbidding, where permitting context enables the application of a production and forbidding context inhibits it. For random context languages of finite index a generalization of the well-known pumping lemma for context-free languages has been proven. We drop the finite index restriction and concentrate on non-erasing grammars that use permitting context only. We prove a pumping lemma for their languages that generalizes and refines the existing one, and show that these grammars are strictly weaker than the non-erasing random context grammars

Generating pictures using random forbidding context

We use random context picture grammars to generate pictures through successive refinement. At any stage a picture consists of a shape divided into smaller shapes, each containing a variable or terminal. A variable may be rewritten according to a production of the underlying grammar, which entails either dividing the shape containing it into smaller shapes, or substituting a variable or terminal for it. A production may depend on context randomly distributed in the intermediate picture. Context is classified as either permitting or forbidding, the former enabling the application of a production, the latter inhibiting it. For visualization purposes every terminal is associated with a color, and its shape filled with that color. We show examples of pictures generated with random context picture grammars. Then we concentrate on grammars which use permitting context only and present a pumping lemma for the corresponding picture sets.

The J2-radical in structural matrix near-rings

Abstract We show that the two obvious definitions for a structural matrix near-ring somewhat unexpectedly yield the same near-ring. The strictly maximal left ideals of a structural matrix near-ring are characterized and used to describe its J 2 -radical.

On Two-Sided Ideals in Matrix Near-Rings

Publisher Summary This chapter presents an account of this correspondence, which turns out to be more complex than in the ring case. Let (R,+,·) be a right near-ring with identity 1. R n denotes the direct sum of n copies of (R,+) and similarly for subgroups of (R,+). The elements of R n are considered as column vectors and written in transposed form with pointed brackets.

A property of random context picture grammars

We use random context picture grammers to generate pictures through successive refinement. The productions of such a grammar are context free, but their application is regulated by context randomly distributed in the developing picture. Grammars using this relatively weak context often succeed where context-free grammars fail, e.g., in generating the typical iteration sequence of the Sierpinski carpet. On the other hand, it proved possible to develop iteration theorems for three subclasses of these grammars; finding necessary conditions is problematic in the case of most models of context-free picture grammars with context-sensing ability, since they consider a variable and its context as a connected unit.We present a property of all picture sets generated with random context picture grammers, and then construct a picture set that does not belong to this class.

Bag Context Tree Grammars

We introduce bag context, a device for regulated rewriting in tree grammars. Rather than being part of the developing tree, bag context (bc) evolves on its own during a derivation. We show that the class of bc tree languages is the closure of the class of random context tree languages under linear top-down tree transductions. Further, an interchange theorem for subtrees of dense trees in bc tree languages is established. This result implies that the class of bc tree languages is incomparable with the class of branching synchronization tree languages.

Solution of an Open Problem Concerning 2-Primitive Near-Rings

Abstract We construct a 2-primitive non-ring with identity which will clarify an open problem posed in 1971.

Necessary conditions for subclasses of random context languages

Random context grammars belong to the class of context-free grammars with regulated rewriting. Their productions depend on context that may be randomly distributed in a sentential form. Context is classified as either permitting or forbidding, where permitting context enables the application of a production and forbidding context inhibits it. We have proven a pumping lemma for random permitting context languages and a shrinking lemma for random forbidding context languages. We now present new necessary conditions for both these classes of languages and illustrate them with examples. We also present and illustrate a new necessary condition for context-free languages.