A grouplike, which has been introduced earlier, is an algebraic structure between semigroups and groups and its axioms are generalization of the four group axioms. We observe that every grouplike is a homogroup (a semigroup containing an ideal subgroup) with a unique central idempotent. On the other hand, decomposer and associative functions on groups, semigroups and even magmas have been introduced in 2007. In this paper, we introduce special type of grouplikes (namely f-grouplike) that is motivated from the both topics. We prove that f-grouplikes is a proper subclass of Class United Grouplikes, and we study some of their properties.

The "forgotten topological index" or "􀜨 – index" has been introduced by Furtula and Gutman in 2015. The 􀜨 – index of a (molecular) graph is defined as the sum of cubes of the vertex degrees of the graph. In this paper, we compute this topological index for some special graphs such as Wheel graph, Barbell graph and friendship graph. Moreover, the effects on the 􀜨 – index are observed when some operations such as edge switching, edge moving and edge separating are applied to the graphs. Finally, we investigate degeneracy of 􀜨 – index for small graphs.

Let $C= {}_SC_R$ be a (faithfully) semidualizing bimodule. This paper begins with the introduction of the concepts of $C$-$fp_n$-injective $R$-modules and $C$-$fp_n$-flat $S$-modules. Subsequently, we investigate various properties associated with classes of modules characterized by $C$-$fp_{n}$-injective and $C$-$fp_{n}$-flat dimensions. For instance, we explore Foxby equivalence and the existence of preenvelopes and covers in relation to these classes of modules. Finally, we analyze the exchange properties of these classes and the connections between preenvelopes (or precovers) and Foxby equivalence, particularly within the context of almost excellent extensions of rings.

Cat f hnur has lng been cmmnly referred t as the cmmn dress given t ministers r emirs by a Muslim ruler, king, r caliph. Cat f hnur is nt nly a categry related t the mnarchy and the curt, but als an expressin that mystics, especially Sanai, Attar and Rumi, have used in their bks. Assuming that mystics refrain frm being explicit in speech and tend t refer t their intended meaning in speech, we set ut t answer the questin f what meanings and allegries can be interpreted frm the interpretatin f Cat f hnur in the bks f Sanai, Attar and Rumi? In this research by analytical-descriptive methd and Based n the evidence, we have shwn that if Gd gives a special Cat f hnur t the mystic, it is gd and pleasing, therwise Cat f hnur will be the veil f the mystic. The frgiveness r achievable f Cat f hnur has dimensins. That is, accrding t sme evidences, the mystic can be given a cat f hnur by Gd, and accrding t sme ther evidences, the mystic must fight with ne’s wn self t btain it. Ennbling f cat f hnur is its mst imprtant aspect. Als, Cat f hnur can mean frgiving and cvering the shrtcmings f mytics.

The essential oil isolated by water and steam distillation from aerial parts of Lomatopodium staurophyllum (Rech.f.) Rech.f. was analyzed by capillary GC annd GC/MS. Among the 15 compounds identified, the major componeents were (E)-b-ocimene (26.8%%), myrcene (26.3%), (z)- b - ocimene (17.7%), b -caaryophyllene (4.6%) and limonene (4.6%).

In this article, we discussed the zero-divisor graph of a commutative ring with identity $\mathbb{F}_p+u\mathbb{F}_p+u^2 \mathbb{F}_p$ where $u^3=0$ and $p$ is an odd prime. We find the clique number, chromatic number, vertex connectivity, edge connectivity, diameter and girth of a zero-divisor graph associated with the ring. We find some of topological indices and the main parameters of the code derived from the incidence matrix of the zero-divisor graph $\Gamma(R).$ Also, we find the eigenvalues, energy and spectral radius  of both adjacency and Laplacian matrices of $\Gamma(R).$

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.

‎Krasner $F^{(m,n)}$-hyperrings were introduced and  investigated by Farshi and Davvaz. In this paper, our purpose is to  define and characterize  three particular classes of $F$-hyperideals in a Krasner $F^{(m,n)}$-hyperring, namely prime $F$-hyperideals, maximal $F$-hyperideals and  primary $F$-hyperideals, which extend similar concepts of ring context. Furthermore, we  examine the relations between these structures. Then a number of major conclusions are given to explain the general framework of these structures.

In this paper, we introduced the concept of f-derivation on equality algebra E by using the notions of inner and outer f-derivations. Then, we investigated some properties of (inner, outer) f-derivation and introduced some suitable conditions that they help us to define an f-derivation on E. We defined kernel and fixed point sets of f-derivation on E and proved that under which condition they are filters of E. Finally, we proved that under some conditions the equivalence relations on (E, →, 1) coincidewith the equivalence relations on E with derivation d_f . Also, by using the properties of equality algebra, we made two outer f-derivations and by introducing an implication operation “\ri ” on the set of all these outer f-derivations, we showed that it make a BE-algebra and dual BCK-algebra.

It is a common experience in processing land seismic reflection data to be confronted with data containing weak reflection signals submerged by coherent, strong linear noise such as ground roll or air waves. Ground roll is a type of Rayleigh wave that has high amplitude, low frequency and low velocity. Ground roll is also dispersive and overwhelm the desired reflection signals unless special steps are taken in data acquisition and processing to get rid of it. Various methods to filter ground roll have been proposed and applied in recent years. Coherent linear noise is recognized by lateral continuity and its predictability from nearby traces. Therefore, to eliminate them, it is requested a kind of prediction filter which can predict coherent energy from trace to trace. Since linear events are coherent in space, f-x prediction filter is used to eliminate these events from seismic data in frequency-space domain. In the frequency-space domain, for a given frequency, coherent noise is predictable. In this paper, complex Wiener filter theory has been used to predict coherent noise. In the frequency-wave number domain signal and ground roll can be separated. But when the difference between the apparent velocities of signals and noise is small, part of signals exist in the ground roll rejoin. To retain this part of signals we can use f-x prediction filter. This method is named f-k and f-x filtering. This method was tested on synthetic and real shot records contaminated with ground rolls. Results in reducing ground rolls are acceptable.