We use the concept of a regular object with respect to another object in an arbitrary category, defined in [3], in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories R−mod and S−mod, when S is an excellent extension of the ring R. Consequently, we obtain a result of [5]: if S is an excellent extension of the ring R, then S is von Neumann regular ring i...

In a typical regular expression (regex) crossword puzzle, you are given two nonempty lists R1, . . . , Rm and C1, . . . , Cn of regular expressions over some alphabet, and your goal is to fill in an m× n grid with letters from that alphabet so that the string formed by the ith row is in L(Ri), and the string formed by the jth column is in L(Cj), for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. Such a grid is a...

Let G be a graph and F ⊆ V (G). Then F is called an induced forest of G if 〈G〉 contains no cycle. The forest number, f(G), of G is defined by f(G) := max{|F | : F is an induced forest of G}. It was proved by the second author in [6] that if G is an r-regular graph of order n, then f(G) ≤ b nr−2 2(r−1)c. It was also proved that the bound is sharp by constructing an r-regular graph H of order n w...

for some real number X, the symbol V denoting the lattice least upper bound. Any ring R is regular [10] if for each xER there is an xaER such that xx°x = x. It is evident that every regular F-ring R contains a maximal bounded sub-F-ring R, the F-ring of all xER satisfying equation (1.1). The relationship between a regular F-ring and its maximal bounded sub-F-ring is analogous to that between th...

For the set of graphs with a given degree sequence, consisting of any number of 2s and 1s, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of m-matchings. We find the expected value of the number of m-matchings of r-regular bipartite graphs on 2n vertices with respect to the two standard measures. We state and discuss the conjectured u...

For a set X , an equivalence relation ρ on X , and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T (X, ρ, R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y) ∈ ρ then (xa, ya) ∈ ρ) and R (if r ∈ R then ra ∈ R). The semigroup T (X, ρ, R) is the centralizer of the idempotent transformation with kernel ρ and imageR. We determine the str...

Call a noncommutative rational function r regular if it has no singularities, i.e., r(X) is defined for all tuples of self-adjoint matrices X. In this talk regular noncommutative rational functions r will be characterized via the properties of their (minimal size) linear systems realizations r = c∗L−1b. Our main result states that r is regular if and only if L = A0 + ∑ j Ajxj is privileged. Rou...

Over ten years ago Wright [4] proved a fundamental theorem in the theory of random graphs. He showed that if M = M(n) is such that almost no labelled graph of order n and size M has two isolated vertices or two vertices of degree n — 1, then the number of labelled graphs of order n and size M divided by the number of unlabelled graphs of order n and size M is asymptotic to n\. The result is bes...

A context-free e-free grammar with regular restrictions is a context-free £-free grammar G over which a context-free rule r or G is applicable on a string x only if x e y(r) where y(r) is a regular set. It is known from [1] that the context-free £-free grammars with regular restrictions are as powerful as Chomsky's grammars of the type 0. It is shown, that the same result holds for the grammars...

In this paper we define a new type of rings ”almost powerhermitian rings” (a generalization of almost hermitian rings) and establish several sufficient conditions over a ring R such that, every regular matrix admits a diagonal power-reduction.