we show that the lattice of all ideals of a ring $r$ can be embedded in the lattice of all its fuzzyideals in uncountably many ways. for this purpose, we introduce the concept of the generalizedcharacteristic function $chi _{s}^{r} (a)$ of a subset $a$ of a ring $r$ forfixed $r , sin [0,1] $ and show that $a$ is an ideal of $r$ if, and only if, its generalizedcharacteristic function $chi _{s}^{...

1 . Introduction. Let S be a set and f(x) a function which makes correspond to every x E S a subset fix) of S so that x(f (x) . Such a function f(x) we shall call a set-mapping defined on S. A subset S' 9 S is called free (or independent) with respect to the set-mapping f (x), if for every x E S' and Y E S', x ~f(y) and y ~ ftx) . Let S = m o and n < m. Assume that f(x) < n for every x E S. RUZ...

let tx be the full transformation semigroups on the set x. for an equivalence e on x, let te(x) = {α ∈ tx : ∀(x, y) ∈ e ⇔ (xα, yα) ∈ e}it is known that te(x) is a subsemigroup of tx. in this paper, we discussthe green&apos;s *-relations, certain *-ideal and certain rees quotient semigroup for te(x).

We show that the lattice of all ideals of a ring $R$ can be embedded in the lattice of all its fuzzyideals in uncountably many ways. For this purpose, we introduce the concept of the generalizedcharacteristic function $chi _{s}^{r} (A)$ of a subset $A$ of a ring $R$ forfixed $r , sin [0,1] $ and show that $A$ is an ideal of $R$ if, and only if, its generalizedcharacteristic function $chi _{s}^{...

In this research, the effects of important parameters, including the molar ratio of acetic acid to sulfur(S) , sonication time, temperature, and hydrogen the molar ratio of peroxide to sulfur on the performance of ultrasound-assisted oxidative desulfurization were studied using the response surface method (RSM). To this end, a model fuel containing n-decane and dibenzothiophene at a concentrati...

In the present paper, measure differential equations involving the distributional Henstock-Kurzweil integral are investigated. Theorems on the existence and structure of the set of solutions are established by using Schauder$^prime s$ fixed point theorem and Vidossich theorem. Two examples of the main results paper are presented. The new results are generalizations of some previous results in t...

This is Section 5.4 in Mehlhorn/Sanders [DMS14, MS08] Quicksort is a divide-and-conquer algorithm. Let S be the set to be sorted. We select a uniformly random element p from S and split S into three parts. The set S< of elements smaller than p, the set S= of elements equal to p and the set S> of elements larger than p. The element p is usually called the pivot. Then we apply the algorithm recur...