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In this paper, we establish a structure theorem of naturally ordered rpp semigroups with max-idempotents. Results on naturally ordered semigroups with max-idempotents in the literature are extended and amplified. Some previous results obtained by Blyth, McFadden, Guo and Xie on regular and abundant ordered semigroups are extended to ordered rpp semigroups. [ABSTRACT FROM AUTHOR]

We introduce the concept of vague ideals in a distributive implication groupoid and investigate their properties. The vague ideals of a distributive implication groupoid are also characterized. [ABSTRACT FROM AUTHOR]

We characterize the semigroups which are semilattices of groups by using the properties of their generalized fuzzy ideals. [ABSTRACT FROM AUTHOR]

The concepts of soft rings and idealistic soft rings are introduced. Three basic isomorphism theorems for soft rings are established, and consequently, some properties of soft rings and idealistic soft rings are given. [ABSTRACT FROM AUTHOR]

Let L2(q) be the projective special linear group, where q is a prime power. In the present paper, we prove that L2(q) is OD-characterizable by using the classification of finite simple groups. A new method is introduced in order to deal with the subtle changes of the prime graph of a group in the discussion of its OD-characterization. This not only generalizes a result of Moghaddamfar, Zokayi and Darafsheh, but also gives a positive answer to a conjecture put forward by Shi. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is called an ℋ-subgroup of G if NG(H) ∩ Hg ≤ H for all g ∈ G. The set of all ℋ-subgroups of a finite group G is denoted by ℋ(G). In this paper, a sufficient condition about p-nilpotency is given and some new results for a finite group G to be p-nilpotent or supersolvable are obtained based on the assumption that some subgroups belong to ℋ(G). [ABSTRACT FROM AUTHOR]

Basic properties of branches of weak BCC-algebras and connections between various types of ideals are described. Next, a new type of ideals with the property that the quotient algebras induced by these ideals are uniquely determined by some groups is introduced and characterized. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of n-isoclinism to the class of all pairs of groups (G,M), where M is a normal subgroup of a group G. The details of that notion are studied and some equivalent conditions are given for two pairs of groups to be n-isoclinic. In addition, the concept of subgroup- and quotient-irreducibility for a pair of groups is defined and we derive how to see whether (G,M) has a pair of subgroups or a pair of factor groups n-isoclinic with (G,M). [ABSTRACT FROM AUTHOR]

We take in this paper an arbitrary class $\mathcal{K}$ of groups as a base, and define a radical property 풫 for which every group in $\mathcal{K}$ is 풫-semisimple. This is called the upper radical property determined by the class $\mathcal{K}$. At the same time, we define a radical property 풫 for which every group in $\mathcal{K}$ is a 풫-radical group. This is called the first lower radical property determined by the class $\mathcal{K}$. Also, we give another construction leading to the second lower radical property which is proved to be identical with the first one. [ABSTRACT FROM AUTHOR]

The degree pattern of a finite group G was introduced in [10]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and same degree pattern as G. When a group G is 1-fold OD-characterizable, we simply call it OD-characterizable. In recent years, a number of authors attempt to characterize finite groups by their order and degree pattern. In this article, we first show that for the primes p=53, 61, 67, 73, 79, 83, 89, 97, the alternating groups Ap+3 are OD-characterizable, while the symmetric groups Sp+3 are 3-fold OD-characterizable. Next, we show that the automorphism groups Aut(O7(3)) and Aut(S6(3)) are 6-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected. [ABSTRACT FROM AUTHOR]

Let G be any of the (binary) icosahedral, generalized octahedral (tetrahedral) groups or their quotients by the center. We calculate the automorphism group Aut(G). [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a group G. Then, we call H weakly s-supplemented in G if G has a subgroup T such that HT = G and H ∩ T ≤ HsG, where HsG is the largest s-permutable subgroup of G contained in H. In this paper, we use the weakly s-supplemented subgroups to characterize the structure of groups. A series of known results in the literature are unified and generalized. [ABSTRACT FROM PUBLISHER]

It has been well known that the band of idempotents of a naturally ordered orthodox semigroup satisfying the "strong Dubreil-Jacotin condition" forms a normal band. In the literature, the naturally ordered orthodox semigroups satisfying the strong Dubreil-Jacotin condition were first considered by Blyth and Almeida Santos in 1992. Based on the name "epigroup" in the paper of Blyth and Almeida Santos and also the name "epigroups" proposed by Shevrin in 1955; we now call the naturally ordered orthodox semigroups satisfying the Dubreil-Jacotin condition the epiorthodox semigroups. Because the structure of this kind of orthodox semigroups has not yet been described, we therefore give a structure theorem for the epi-orthodox semigroups. [ABSTRACT FROM AUTHOR]

Abundant semigroups expressed as a perfect rectangular band of adequate semigroups will be investigated. A known theorem of Pastijn–Petrich in 1984 on perfect rectangular bands of inverse semigroups will be generalized from the class of regular semigroups to the class of abundant semigroups. A Rees matrix representation of such abundant semigroups will be established and some characterization theorems will be consequently given. [ABSTRACT FROM AUTHOR]

We consider a generalized superabundant semigroup within the class of semiabundant semigroups, called a supercryptogroup since it is an analogy of a cryptogroup in the class of regular semigroups. We prove that a semigroup S is an $\widetilde{\cal H}$-regular supercryptogroup if and only if S can be expressed as a refined semilattice of completely $\widetilde{\cal J}$-simple semigroups. Some results on regular cryptogroups are extended to $\widetilde{\cal H}$-regular supercryptogroups. Some results on superabundant semigroups are also generalized. [ABSTRACT FROM AUTHOR]

This paper concerns a relationship between fuzzy sets and algebraic hyperstructures. It is a continuation of ideas presented by Davvaz(Fuzy Sets Syst. 101: 191.195 1999) and Bhakat and Das(Fuzzy Sets Syst. 80: 359.368 1996). The concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set, which is a generalization of quasi-coincidence of a fuzzy point with a fuzzy set, is introduced. Using this new idea, the notion of interval valued (α, β)-fuzzy sub-hyperquasigroups in a hyperquasigroup, which is a generalization of a fuzzy sub-quasigroup, is defined, and related properties are investigated. In particular, the study of interval valued (∈, ∈, ∨q)-fuzzy sub-hyperquasigroups in a hyperquasigroup is dealt with. Finally, we consider the concept of implication-based interval valued fuzzy sub-hyperquasigroups in a hyperquasigroup. [ABSTRACT FROM AUTHOR]

We give a short proof for the associativity of the binary operation defined on a refined system of semigroups indexed by a semilattice. The main result given by Zhang, Shum and Zhang in 2001 on the refined semilattice of semigroups is substantially strengthened. [ABSTRACT FROM AUTHOR]

The class $$ \mathcal{P}\mathcal{C} $$ of projectively condensed semigroups is a quasivariety of unary semigroups, the class of projective orthomonoids is a subquasivariety of $$ \mathcal{P}\mathcal{C} $$ . Some well-known classes of generalized completely regular semigroups will be regarded as subquasivarieties of $$ \mathcal{P}\mathcal{C} $$ . We give the structure semilattice composition and the standard representation of projective orthomonoids, and then obtain the structure theorems of various generalized orthogroups. [ABSTRACT FROM AUTHOR]

It is well known that a strong semilattice of semigroups is still a semigroup. However, it is not known whether the analogous result carries or not for a strong semilattice of implicative semigroups. The main difficulty is that the implicative operation ∗ is hard to define on the semilattice, especially we need the operation ∗ of the implicative semigroup to be compatible with the negatively ordered relation and the semigroup multiplication on the semilattice of implicative semigroups. In this paper, we provide a practical method of constructing such a strong semilattice of mplicative semigroups. Our method is particularly useful in constructing implicative semigroups of large size. A constructed example is given. [ABSTRACT FROM AUTHOR]

The concept of wreath product of semigroups was first introduced by Neumann in 1960, and later on, this concept was used by Preston to investigate the structure of some inverse semigroups. In this paper, we modify the wreath product given by Neumann and Preston to study the structure of some generalized Clifford semigroups. In particular, we prove that a semigroup is a left C-rpp semigroup if and only if it is the wreath product of a left regular band and a C-rpp semigroup. Our result provides a new insight to the structure of left C-rpp semigroups. [ABSTRACT FROM AUTHOR]

Regular semigroups with multiplicative inverse transversals were first studied by Blyth and McFadden in 1982. In this paper, we extend the concept of inverse transversals to some kind of quasi-ideal adequate transversals of abundant semigroups. Some characterization theorems of abundant semigroups with different kinds of quasi-ideal adequate transversals are obtained. In particular, the structure theory of abundant semigroups having different kinds of quasi-ideal adequate transversals is established. Some results obtained by Blyth, McAlister and McFadden on inverse transversals in regular semigroups are hence extended to abundant semigroups. [ABSTRACT FROM AUTHOR]

In this paper, Kaplansky-type theorems are given to characterize GCD-monoids and valuation monoids. Also, (unique) r-factorable monoids are defined and it is shown that S is a Krull monoid if and only if S is a unique t-factorable (resp., w-factorable) monoid if and only if S is a t-factorable (resp., w-factorable) t-Prüfer monoid. [ABSTRACT FROM AUTHOR]

In this paper we will present the results of Artin-Markov on braid groups by using the Gröbner-Shirshov basis. As a consequence we can reobtain the normal form of Artin-Markov-Ivanovsky as an easy corollary. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of a superabundant semigroup S whose set of idempotents E(S) forms a subsemigroup. We call such a semigroup a cyber-group because it is a generalization of orthogroups in the class of completely regular semigroups studied by Petrich and Reilly. We show that a cyber-group can be expressed as a semilattice of rectangular monoids. Thus, our result generalizes the well-known result obtained by Petrich in 1987 for orthogroups. Some properties of cyber-groups are given and some special superabundant semigroups are discussed. [ABSTRACT FROM AUTHOR]

The concept of wreath product of semigroups was initiated by Neumann in 1960, and later on, his concept was used by Preston to investigate the structure of some inverse semigroups. Recently, we start to investigate the structure of left C-rpp semigroups by using wreath products. In this paper, we modify the wreath product to “dual wreath product” so that we can study the structure of right C-rpp semigroups. We prove that a semigroup is a right C-rpp semigroup if and only if it is the dual wreath product of a right regular band and a C-rpp semigroup. Our theorem provides new insight to the structure of right C-rpp semigroups. In particular, a recent result given by Ren and Shum for right C-rpp semigroups is strengthened and enriched. [ABSTRACT FROM AUTHOR]

Hypersubstitutions were introduced in [3] as a way of making precise the concepts of hyperidentity and M-hyperidentity. The monoid of hypersubstitutions has been widely studied by many authors. Knowledge of the monoid of hypersubstitutions can be applied to the concept of M-hyperidentities. In this paper, we show that the order of hypersubstitutions of type τ = (3) is 1, 2, 3 or infinite. [ABSTRACT FROM AUTHOR]

Let P(X) be the semigroup of all partial transformations of a set X. A subsemigroup S of P(X) is factorisable if S = GE = EH, where G, H are subgroups of S and E is the set of idempotents in S. In 2001, Jampachon, Saichalee and Sullivan proved a simple result that generalized most of the previous work on factorisable subsemigroups of P(X). They also determined when the semigroup T(V) of all linear transformations of a vector space V is factorisable. In this paper, we extend that work to partial linear transformations of V and consider the notion of locally factorisable for such semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we consider a finite group in which some subgroups are complemented in a special subgroup of a given group. Some new conditions are obtained for a finite group in a formation ${\cal F}$ containing the class ${\cal U}$ of finite supersolvable groups. Our results generalize the recent results on the complemented subgroups of finite groups. [ABSTRACT FROM AUTHOR]

The investigation of regular F -abundant semigroups is initiated. In fact, F -abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F -abundant semigroups is obtained. In addition, we also prove that a regular F -abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F -inverse semigroups. [ABSTRACT FROM AUTHOR]

Let R be a ring. A right R-module M is called p-injective if every homomorphism from a principal right ideal of R to M can be given by a left multiplication. A ring R is called a right pV-ring if every simple R-module is p-injective. In this paper, Kasch modules are considered. It is proved that if a Kasch module M is finitely generated and quasi-p-injective, then there is a bijective correspondence between the class of maximal submodules of M and the class of all minimal left ideals of its endomorphism ring. Also, it is proved that if M is a pV-module which is a finitely generated projective self-generator, then its endomorphism ring is a right pV-ring. Finally, it is proved that being a right or left pV-ring is a Morita invariant. [ABSTRACT FROM AUTHOR]

The concept of ρG-strong semilattice of semigroups is introduced. By using this concept, we study Green's relation ℋ on a completely regular semigroup S. Necessary and sufficient conditions for S/ℋ to be a regular band or a right quasi-normal band are obtained. Important results of Petrich and Reilly on regular cryptic semigroups are generalized and enriched. In particular, characterization theorems of regular cryptogroups and normal cryptogroups are obtained. [ABSTRACT FROM AUTHOR]

We introduce the concept of a quasi-C-Ehresmann semigroup. Some special quasi-C-Ehresmann semigroups will be considered. We will show that the quasi-C-Ehresmann semigroups are special U-semiabundant semigroups and orthodox U-liberal semigroups, which have been recently studied by several authors. Some characterization theorems of this new class of semigroups are obtained. Many results of quasi-C-semigroups in the literature can be extended to quasi-C-Ehresmann Semigroups. [ABSTRACT FROM AUTHOR]

It is known that a C-rpp semigroup can be described as a strong semilattice of left cancellative monoids. In this paper, we introduce the class of left C-wrpp semigroups which includes the class of left C-rpp semigroups as a subclass. We shall particularly show that the semi-spined product of a left regular band and a C-wrpp semigroup forms a curler which is a left C-wrpp semigroup and vice versa. Results obtained by Fountain and Tang on C-rpp semigroups are extended and strengthened. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ coreH = HG. In this paper, we investigate the structure of finite groups if some special subgroups are c-supplemented in given finite groups. We prove that a finite group G is solvable if there exists a maximal subgroup M of G such that every cyclic subgroup of M with prime power order is c-supplemented in G. As an application of this result, we show that a finite group G is supersolvable if there exist maximal subgroups M1 and M2 such that M1 and M2 are not conjugate in G and every cyclic subgroup of M1 with prime power order (i = 1, 2) is c-supplemented in G. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study some special lpp-semigroups, namely, the left GC-lpp semigroups. After obtaining some properties and characterizations of such semigroups, we establish some structure theorems of this class of semigroups. In addition, we also consider some special cases. As an application, we describe the structure theorems of IC quasi-adequate semigroups whose idempotent band is a regular band. [ABSTRACT FROM AUTHOR]