A circuit in a simple undirected graphG = (V,E) is a sequence of vertices {v1, v2, . . . , vk+1} such that v1 = vk+1 and {vi, vi+1} ∈ E for i = 1, . . . , k. A circuit C is said to be edgesimple if no edge of G is used twice in C. In this article we study the following problem: which is the largest integer k such that, given any subset of k ordered vertices of a graph G, there exists an edge-si...

A method of representing transformations of a finite set pictorially is described. These pictures of a function are used to count certain idempotent transformations, and interesting formulae for the Fibonacci numbers are obtained. No part of any string extends beyond the region ° : : ; z ::; 1, and if two strings intersect then they join together, from the point of intersection, to meet the sam...

We show that for every infinite cardinal number m there exist two totally ordered abelian groups whose free product in any nontrivial variety of lattice-ordered groups has a disjoint set of cardinality m. This answers problem 10.7 of [13] and extends the results in [12]. We further prove that for the variety of abelian lattice-ordered groups or the variety of all lattice-ordered groups, the fre...

Lower subsets of an ordered semigroup form in a natural way an ordered semigroup. This lower set operator gives an analogue of the power operator already studied in semigroup theory. We present a complete description of the lower set operator applied to varieties of ordered semigroups. We also obtain large families of fixed points for this operator applied to pseudovarieties of ordered semigrou...

As a further extension of the fuzzy set and intuitive set, interval-valued (IIFS) is more effective tool to deal with uncertain problems. However, classical rough based on equivalence relation, which do not apply IIFS. In this paper, we combine IIFS ordered information system obtain (IIFOIS). On basis, three types multiple granulation models dominance relation are established effectively overco...

We address the verification problem of ordered multi-pushdown automata: A multi-stack extension of pushdown automata that comes with a constraint on stack transitions such that a pop can only be performed on the first non-empty stack. First, we show that the emptiness problem for ordered multi-pushdown automata is in 2ETIME. Then, we prove that, for an ordered multi-pushdown automata, the set o...