In this paper we define the Smarandache Hoop-algebras and Q-Smarandache filters, we obtain some related results. After that, by considering the notions of these filters we determine relationships between filters in Hoop-algebras and Q-Smarandache filters in Hoop-algebras. Finally, we introduce the concept of Smarandache 2-structure and Smarandache 2-filter on Hoop-algebras.

In this paper, we construct a topology using the concept of Hoop algebras and investigate its topological properties. These include examining specific types of topological spaces, such as Hausdorff, $T_0$-spaces, and $T_1$-spaces, as well as exploring the concept of connectedness. Additionally, we analyse the relationship between closed and compact sets within this topology. Finally, by incorporating the binary operation $\ri$ and the defined topology on Hoop algebras, we introduce the notion of semi-topological algebra and demonstrate that every Hoop algebra is a right semi-topological algebra.

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, first we introduce the concept of fuzzy n-fold positive implicative filters on Hoop algebras and study some of properties. Next we define and study fuzzy n-fold fantastic filters on Hoop algebras. Also, the relationship between fuzzy n-fold positive implicative filter and some other fuzzy filters likeness fuzzy n-fold implicative filter and fuzzy n-fold fantastic filters are investigated for example every fuzzy n-fold implicative filter is a fuzzy n-fold positive implicative filter on Hoop algebras. Then we obtain some condition equivalent with fuzzy n-fold fantastic filters and every fuzzy filter on chain Hoop algebra is a fuzzy n-fold fantastic filter. Finally we show that the quotient of fuzzy n-fold fantastic filters is a Boolean algebra.

In this paper, we introduce the concepts of n-fold obstinate pseudo-Hoop and n-fold obstinate  lter in pseudo-Hoops. Then we investigated these notions and proved some properties of them. Also, we discussed the relationship between n-fold obstinate pseudo-Hoop and n-fold obstinate  lter and other types of n-fold pseudo-Hoops and n-fold  lters such as n-fold(positive) implicative  lter and n-fold fantastic  lter in pseudo-Hoops. For example, we proved that any n-fold obstinate  lter is a maximal  lter. Finally, we obtain a characterization of n-fold obstinate  lters in terms of congruences and we show that any n-fold obstinate pseudo-Hoop is an n-fold fantastic, n-fold positive implicative, n-fold implicative pseudo-Hoop and simple pseudo-Hoop.

This article investigates derivation theory within the framework of Hoop algebras. We introduce and examine f-derivations and (f, g)-derivations, exploring their properties and providing illustrative examples. We demonstrate that, under specific  conditions, the sets of these derivations form bounded distributive lattices and bounded ∨-semilattices. Furthermore, we  investigate s-derivations, where s is a square root on the Hoop algebra. Notably, we prove that under certain conditions,  the set of fixed points of the s-derivation coincides with the set of idempotents of the Hoop algebra. This work  contributes to the understanding of derivation structures in Hoop algebras, particularly focusing on the interplay between derivations, square roots, and idempotent elements.

In this article, the anti-plane deformation of an orthotropic sector with multiple defects is studied analytically. The solution of a Volterra-type screw dislocation problem in a sector is first obtained by means of a finite Fourier cosine transform. The closed form solution is then derived for displacement and stress fields over the sector domain. Next, the distributed dislocation method is employed to obtain integral equations of the sector with cracks and cavities under anti-plane traction. These equations are of Cauchy singular kind, which are solved numerically by generalizing a numerical method available in the literature by means of expanding the continuous integrands of integral equations with different weight functions in terms of Chebyshoff and Jacobi polynomials. A set of examples are presented to demonstrate the applicability of the proposed solution procedure. The geometric and force singularities of stress fields in the sector are also studied and compared to the earlier reports in the literature.

In cylindrical continua, Hoop stresses are induced due to the circumferential failure. This mainly happens when the cylinder is subjected to mechanical loads that vary in the circumferential directions. On the other hand, radial stress is stress in the direction of or opposite to the central axis of a cylindrical body. In the present study, the influence of curvilinear anisotropy on the radial and tangential stresses of the polar-orthotropic hollow cylinder is presented. The governing equations were derived to evaluate the radial and Hoop stresses inside the material. A semi-analytical method through differential transform method (DTM) for the polarorthotropic hollow cylinder is implemented in the solution. The findings, based on polar-orthotropy, of the effect of the radial and circumferential loads on the radial and Hoop stresses of the growing cylinders, show elastic responses that assist in identifying some of the outstanding properties of the curvilinear anisotropic continuums. It is also revealed that the characteristic response of various wall thicknesses of the cylindrical segment is influenced by the fibre orientation, radial and tangential stresses. This work has shown that the curvilinear anisotropy momentously affects the radial and Hoop stresses on the polar-orthotropic hollow cylinder.

In this paper, the influence of confinement on tensile forces in concrete, using fibers in the form of closed fibers (Hoops), is theoretically investigated. In addition to the resistance of tension, the Hoops confine concrete. Physical analysis is based on the elongation of concrete under tension. However, Hoops prevent this elongation and consequently compress the concrete. Assumptions are made to simplify the complex analysis of the Hoop within the concrete structure. The theoretical results indicate that tensile force and stress within the Hoop are not uniform and shear force direction is not constant. Final formulations depict distribution of tensile force, normal stress and shear stress in circumference of Hoop with respect to the axis of tensile force imposed on the concrete. These formulations are capable of including strength and ductility of concrete and Hoop materials. Theoretical behavior of Hoop and fiber within a continuum reveals that fiber is under shear and bearing stresses while a Hoop is under normal tensile stress. However, due to confinement, the concrete structure reinforced with Hoop possess higher strength and ductility than the concrete structure reinforced with fiber.