## Abstract

In this paper, we introduce the notion of a Г-field as a generalization of field, study them properties of a Г -field and prove that *M* is a Г-field if and only if *M* is an integral, simple and commutative Г-ring.

In this paper, we introduce the notion of a Г-field as a generalization of field, study them properties of a Г -field and prove that *M* is a Г-field if and only if *M* is an integral, simple and commutative Г-ring.

In this paper, we construct the fundamental theorem of UP-homomorphisms in UP-algebras. We also give an application of the theorem to the first, second, third and fourth UP-isomorphism theorems in UP-algebras.

In this paper we define root selections and 2* ^{p}*-th root selections for hyperfields: these are multiplicative subgroups whose existence is equivalent to the existence of a well behaved square root function and 2

Since the reduct of every residuated lattice is a semiring, we can ask under what condition a semiring can be converted into a residuated lattice. It turns out that this is possible if the semiring in question is commutative, idempotent, G-simple and equipped with an antitone involution. Then the resulting residuated lattice even satisfies the double negation law. Moreover, if the mentioned semiring is finite then it can be converted into a residuated lattice or join-semilattice also without asking an antitone involution on it. To a residuated lattice **L** which does not satisfy the double negation law there can be assigned a so-called augmented semiring. This can be used for reconstruction of the so-called core *C*(**L**) of **L**. Conditions under which *C*(**L**) constitutes a subuniverse of **L** are provided.

In this paper, we study the planar and outerplanar indices of some graphs associated to a commutative ring. We give a full characterization of these graphs with respect to their planar and outerplanar indices when *R* is a finite ring.

We investigate the class **BA** of ordered regular semigroups in which each element has a biggest associate *x*
^{†} = max {*y *| *xyx *= *x*}. This class properly contains the class **PO **of principally ordered regular semigroups (in which there exists *x*
^{⋆} = max {*y *| *xyx *⩽*x*}) and is properly contained in the class **BI **of ordered regular semigroups in which each element has a biggest inverse *x*◦. We show that several basic properties of the unary operation *x *↦ *x*
^{⋆} in **PO **extend to corresponding properties of the unary operation *x *↦ *x*
^{†} in **BA**. We consider naturally ordered semigroups in **BA **and prove that those that are orthodox contain a biggest idempotent. We determine the structure of some such semigroups in terms of a principal left ideal and a principal right ideal. We also characterise the completely simple members of **BA**. Finally, we consider the naturally ordered semigroups in **BA **that do not have a biggest idempotent.

One of an important problems in finite groups theory, is characterization of groups by specific property. However, in the way the researchers, proved that some of groups by properties such as, elements order, set of elements with same order, graphs, . . . , are characterizable. One of the other methods, is group characterization by using the order of group and the largest elements order. In this paper, we prove that projective special unitary groups *PSU*
_{3}(3* ^{n}*), where 3

A quasimodel is an algebraic axiomatisation of the hyperspace structure based on a module. We initiated this structure in our paper [2]. It is a generalisation of the module structure in the sense that every module can be embedded into a quasi module and every quasi module contains a module. The structure a quasimodel is a conglomeration of a commutative semigroup with an external ring multiplication and a compatible partial order. In the entire structure partial order has an intrinsic effect and plays a key role in any development of the theory of quasi module. In the present paper we have discussed order-morphism which is a morphism like concept. Also with the help of the quotient structure of a quasi module by means of a suitable compatible congruence, we have proved order-isomorphism theorem.

In a recent paper, Çeven and Öztürk have generalized the notion of derivation on a lattice to *f*-derivation, where *f* is a given function of that lattice into itself. Under some conditions, they have characterized the distributive and modular lattices in terms of their isotone *f*-derivations. In this paper, we investigate the most important properties of isotone *f*-derivations on a lattice, paying particular attention to the lattice (resp. ideal) structures of isotone *f*-derivations and the sets of their *f*-fixed points. As applications, we provide characterizations of distributive lattices and principal ideals of a lattice in terms of principal *f*-derivations.

In this paper, we study the concept of ordered (*m, n*)-Г-hyperideals in an ordered LA-Г-semihypergroup. We show that if (*S,* Г, ◦,⩽) is a unitary ordered LA-Г-semihypergroup with zero 0 and satisfies the hypothesis that it contains no non-zero nilpotent (*m, n*)-Г-hyperideals and if *R*(*L*) is a 0-minimal right (left) Г-hyperideal of *S*, then either (*R*◦ Г ◦*L*] = {0} or (*R*◦ Г ◦ *L*] is a 0-minimal (*m, n*)-Г-hyperideal of *S*. Also, we prove that if (*S,* Г, ◦,⩽) is a unitary ordered LA-Г-semihypergroup; *A* is an (*m, n*)-Г-hyperideal of *S* and *B* is an (*m, n*)-Г-hyperideal of *A* such that *B* is idempotent, then *B* is an (*m, n*)-Г-hyperideal of *S*.