Some Operators on Quantum B-Algebras

The aim of this paper is to investigate several operators on quantum B-algebras. At first, we introduce closure and interior operators on quantum B-algebras and consider their relations on bounded quantum B-algebras. Furthermore, we discuss very true operators on quantum B-algebras by three cases via the unit element, and present some similar conclusions and different results. Finally, by constructing a very true operator on a quotient very true perfect quantum B-algebra, we establish a homomorphism theorem on very true perfect quantum B-algebras.

An Arithmetical-like Theory of Hereditarily Finite Sets

This paper presents the (second-order) theory of hereditarily finite sets according to the usual pattern adopted in the presentation of the (second-order) theory of natural numbers. To this purpose, we consider three primitive concepts, together with four axioms, which are analogous to the usual Peano axioms. From them, we prove a homomorphism theorem, its converse, categoricity, and a kind of (semantical) completeness.

Ternary Menger Algebras: A Generalization of Ternary Semigroups

Let n be a fixed natural number. Menger algebras of rank n, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups is. In this article, we first introduce the notion of ternary Menger algebras of rank n, which is a canonical generalization of arbitrary ternary semigroups, and discuss their related properties. In the second part, we establish the so-called a diagonal ternary semigroup which its operation is induced by the operation on ternary Menger algebras of rank n and then investigate their interesting properties. Moreover, we introduce the concept of homomorphism and congruences on ternary Menger algebras of rank n. These lead us to study the quotient ternary Menger algebras of rank n and to investigate the homomorphism theorem for ternary Menger algebra of rank n with respect to congruences. Furthermore, the characterization of reduction of ternary Menger algebra into Menger algebra is presented.

Intersection soft ideals and their quotients on KU-algebras

<abstract><p>In this paper, we have discussed quotient structures of KU-algebras by using the concept of intersection soft ideals. In general, the soft sets are parameterized families of sets that are used to dealt with uncertainty. In particular, We have given the fundamental homomorphism theorem of quotient KU-algebras. A characterization of commutative quotient KU-algebras, implicative quotient KU-algebras and positive implicative quotient KU-algebras are also presented.</p></abstract>

On Finite NeutroGroups of Type-NG[1,2,4]

The NeutroGroups as alternatives to the classical groups are of different types with different algebraic prop- erties. In this paper, we are going to study a class of NeutroGroups of type-NG[1,2,4]. In this class of Neu- troGroups, the closure law, the axiom of associativity and existence of inverse are taking to be either partially true or partially false for some elements; while the existence of identity element and axiom of commutativity are taking to be totally true for all the elements. Several examples of NeutroGroups of type-NG[1,2,4] are presented along with their basic properties. It is shown that Lagrange’s theorem holds for some NeutroSub- groups of a NeutroGroup and failed to hold for some NeutroSubgroups of the same NeutroGroup. It is also shown that the union of two NeutroSubgroups of a NeutroGroup can be a NeutroSubgroup even if one is not contained in the other; and that the intersection of two NeutroSubgroups may not be a NeutroSubgroup. The concepts of NeutroQuotientGroups and NeutroGroupHomomorphisms are presented and studied. It is shown that the fundamental homomorphism theorem of the classical groups is holding in the class of NeutroGroups of type-NG[1,2,4].

On Homomorphism Theorem for Perfect Neutrosophic Extended Triplet Groups

Some homomorphism theorems of neutrosophic extended triplet group (NETG) are proved in the paper [Fundamental homomorphism theorems for neutrosophic extended triplet groups, Symmetry 2018, 10(8), 321; doi:10.3390/sym10080321]. These results are revised in this paper. First, several counterexamples are given to show that some results in the above paper are not true. Second, two new notions of normal NT-subgroup and complete normal NT-subgroup in neutrosophic extended triplet groups are introduced, and their properties are investigated. Third, a new concept of perfect neutrosophic extended triplet group is proposed, and the basic homomorphism theorem of perfect neutrosophic extended triplet groups is established.

Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups

In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related.