In this paper, by using the notion of prime filter, we show representation theorem of hoops and we prove that every nontrivial ∨-hoop is a subdirect product of hoop-chains. In the following, by using the concept of maximal filter of hoops, we introduce radical of hoops. Then some equivalence definitions of it and some related properties are investigated. Then by using this notion, we introduce the concepts of r-filters and p-filters on hoops and the relation between them and other filters of hoops are investigated. Finally, by using p-filters on hoops, we define new open sets on hoop that could be used to construct a Zarisky topology.

We investigated maximal Prym varieties on  nite  elds by attaining their upper bounds on the number of rational points. This concept gave us a motivation for de ning a generalized de ni-tion of maximal curves i. e., maximal morphisms. By MAGMA, we give some non-trivial examples of maximal morphisms that results in non-trivial examples of maximal Prym varieties.

This paper investigates the properties of prime and maximal filters in hoops, introducing two distinct types of  prime  filters and analyzing their interrelationships. Furthermore, it investigates the existence of maximal filters  with a special  focus on bounded hoops. Moreover, this  paper presents the spectrum of a hoop, and shows that  when it has the  topology of the spectrum, it forms a compact topological space T0. It is also shown that the maximal spectrum, regarded  as a subspace of the spectrum,  constitutes a compact T1-topological space. Finally, using the Boolean center of a hoop,  the paper establishes that the clopen subsets of the hoop algebra coincide with the set of closed subsets generated by  Boolean elements.

In this paper , the concepts of weakly prime filters and super-max filters in $\mathrm{BL}$-algebras are introduced , and the relationships between them are discussed . Also , some properties and relations between these filters and other types of filters in $\mathrm{BL}$-algebras are given . With some examples , it is shown that these filters have differences . After that , the notions of weakly linear $\mathrm{BL}$-algebras and weak top $\mathrm{BL}$-algebras are defined and investigated . Finally , using the notion of a weakly prime filter , a new topology on $\mathrm{BL}$-algebras is defined and studied .

An optimal labeling of a graph with $n$ vertices and $m$ edges is an injective assignment of the first $n$ nonnegative integers to the vertices‎, ‎that induces‎, ‎for each edge‎, ‎a weight given by the sum of the labels of its end-vertices with the property that the set of all induced weights consists of the first $m$ positive integers‎. ‎We explore the connection of this labeling with other well-known functions such as super edge-magic and $\alpha$-labelings‎. ‎A graph with $n$ vertices is maximal when the number of edges is $2n-3$; all the results included in this work are about maximal graphs‎. ‎We determine the number of optimally labeled graphs using the adjacency matrix‎. ‎Several techniques to construct maximal graphs that admit an optimal labeling are introduced as well as a family of outerplanar graphs that can be labeled in this form.

The notion of ω,-filters is introduced in distributive lattices and their properties are studied. A set of equivalent conditions is derived for every maximal filter of a distributive lattice to become an ω,-filter which leads to a characterization of quasi-complemented lattices. Some sufficient conditions are derived for proper D-filters of a distributive lattice to become an ω,-filter. Finally, ω,-filters of a distributive lattice are characterized with the help of minimal prime D-filters.

In this research, we studied isolation of the cellulolitic fungi from the persimmon tree, grapevine, pomegranate tree and walnut soils. Among of them, persimmon soil was selected because of maximum cellulolitic fungi. Seven fungi werw isolated from persimmon soil, which three of them had suitable cellulose activity and they were identificated by 18S rRNA and named: Aspergillus niger MZM 89-a2, Penicillium decumbens ZHE 89-p3, Penicillium decumbens MMH 89-p1. Cellulase activities of these fungi were respectively (U/d): FPA 3.1671, 3.5740, 3.1812 and Avicelase 1.6605, 0.3869, 1.1451 and CMCase 2.950, 0.2644, 0.4604. Response surface activity (RSM) was studied to evaluate the effects of temperature, moisture content and particle size for FPA by Aspergillus niger MZM 89-a2. The optimum FPA was in temperature, moisture content and particle size respectively: 28.49oC, 66.62٪ and 1.5-2 (mm). The maximum predicted FPA was 4.35 (U/g) and obtained FPA under this condition was 4.42 (U/g), which indicates the efficacy of the model for prediction of FPA activity under different conditions of the medium.