# Akbar RezaeiPayame Noor University | PNU · Department of Mathematics

Akbar Rezaei

Ph.D.

## About

101

Publications

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661

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Citations since 2017

Introduction

Akbar Rezaei currently works at the Department of Mathematics, Payame Noor University. Akbar does research in Algebra and Logic and Foundations of Mathematics.

Additional affiliations

September 2006 - September 2020

## Publications

Publications (101)

The notion of a ((complete-) normal) vague weak interior ideal on a (regular) Γ-semiring is defined. It is proved that the set of all vague weak interior ideals forms a complete lattice. Also, a characterization theorem for a regular Γ-semiring in terms of vague weak interior ideals is derived. Another interesting consequence of the main result is...

In this paper, we define the notion of Smarandache pseudo-CI algebras and we investigate their properties. We also define and study the notions of Smarandache filters, pseudo-CI Smarandache homomorphisms and modal Smarandache operators on pseudo-CI algebras. The classes of Smarandache fantastic, implicative and positive implicative filters of Smara...

In this study, a neutrosophic N−subalgebra and neutrosophic N −ideal of a Sheffer stroke BCK-algebras are defined. It is shown that the level-set of a neutrosophic N −subalgebra (ideal) of a Sheffer stroke BCK-algebra is a subalgebra (ideal) of this algebra and vice versa. Then we present that the family of allbneutrosophic N −subalgebras of a Shef...

In this paper, we define the notion of hyper CI-algebras as a generalization of CI-algebras and hyper BE-algebras and present some properties. Also, the concepts of a weak hyper filter and a hyper filter over hyper CI-algebras are defined. Moreover, we define the commutative hyper CI-algebra and find the number of commutative hyper CI-algebras of o...

The concept of BE-ringoid is introduced, and some properties of it are investigated. We discuss on a distributive BE-ringoid. Further, transitive BE-ringoid is defined and the relationship between distributive BE-ringoids and transitive BE-ringoids is considered, and we prove that every distributive BE-ringoid is a transitive BE-ringoid, but the co...

In this paper, we introduce the notion of a ringoid, and we obtain left distributive ringoids over a field which are not rings. We introduce several different types of ringoids, and also we discuss on (r, s)-ringoids. Moreover, we discuss geometric observations of the parallelism of vectors in several ringoids.

We present and analyse the concept of vague BE-algebras in this research, as well as some of its properties. In terms of vague-cut subalgebra of BE-algebra, we characterize vague BE-algebra. We also discuss and introduce the idea of normal vague BE-algebras, as well as some of their features.

This paper aims is to introduce states, Bosbach states and state-morphism operators on BI-algebras. We define state ideals on BI-algebras and give a characterization of the least state ideal of a BI-algebra. It is proved that the kernel of a Bosbach state on a BI-algebra X is an ideal of X. Further, by these concepts, we introduce the notions of st...

In this research, we use averages and relative measures of interval grey numbers to introduce grey vertices, grey edges, and grey graphs (graphs are based on interval grey numbers). To do so, we design a grey graph based on a graph (as the underlying graph). Also, we find a relation between grey vertices and grey edges of a grey graph...

The aim of the study is to introduce a neutrosophic N −subalgebra andneutrosophic N −ideal of
a Sheffer stroke BCH-algebras. We prove that the level-set of a neutrosophic N −subalgebra (neutrosophic
N −ideal) of a Sheffer stroke BCH-algebra is its subalgebra (ideal) and vice versa. Then it is shown that the
family of all neutrosophic N −subalgebras...

In this paper, we introduce and study the notion of a [Formula: see text]positive implicative[Formula: see text] left ray in groupoids, and we show that every normal subgroup of a group is a left ray of a group, and in every finite group, left rays are normal subgroups. Further, left absorptive subsets of groupoids are discussed and several example...

The aim of this paper is to derive pseudo-BCK algebras from directoids and vice versa. We generalize some results proved by Ivan Chajda et al. in the case of BCK-algebras. We assign to an arbitrary pseudo-BCK algebra a semilattice-like structure and observe that this is the point where directoids are different from the semilattice-like structures....

We extend the notions of right (left) independency and absorbent from groupoids to Bin(X) as a semigroup of all the groupoids on a set X and study and investigate many of their properties. We show that these new concepts are different by presenting several examples. In general, the concept of right (left) independence is a generalization and altern...

Recently, research on uncertainty modeling is progressing rapidly and many essential and breakthrough studies have already been done. There are various ways such as fuzzy, intuitionistic and neutrosophic sets to handle these uncertainties. Although these concepts can handle incomplete information in various real-world issues, they cannot address al...

In this study, a neutrosophic N −subalgebra, a (implicative) neutrosophic N − filter, level sets of these neutrosophic N −structures and
their properties are introduced on a Sheffer stroke BE-algebras (briefly, SBE-algebras). It is proved that the level set of neutrosophic N − subalgebras ((implicative) neutrosophic N −filter) of this algebra is th...

In this paper, we introduce and study the concept of Neutrosophic Γ-semiring and study various properties. Also, we prove that there is a one-to-one correspondence between Neutrosophic Γ-semirings and sub Γ-semirings of a Γ-semiring. Further, we prove that the set of all neutrosophic Γ-semirings is a De-Morgan algebra. Moreover, we establish that t...

In this paper, we extend the notion of Hv-semigroups to neutro-Hv-semigroups and anti-Hv-semigroups and investigate many of their properties. We show that these new concepts are different from the classical concept of Hv-semigroups by presenting several examples. In general, the neutro-algebras and anti-algebras are generalizations and alternatives...

The purpose of this paper is to propose a new Pythagorean fuzzy entropy for Pythagorean fuzzy sets, which is a continuation of the Pythagorean fuzzy entropy of intuitionistic sets. The Pythagorean fuzzy set continues the intuitionistic fuzzy set with the additional advantage that it is well equipped to overcome its imperfections. Its entropy determ...

AbstractAs generalizations and alternatives of classical algebraic structures there have been introduced in 2019 the NeutroAlgebraic structures (or NeutroAlgebras) and AntiAlgebraic structures (or AntiAlgebras). Unlike the classical algebraic structures, where all operations are well defined and all axioms are totally true, in NeutroAlgebras and An...

A new algebraic structure was introduced, called an eGE-algebra, which is a generalisation of a GE-algebra and investigated its properties. We explore the definition of filters and the quotient algebra associated with such filters.

As generalizations and alternatives of classical algebraic structures there have been introduced in 2019 the NeutroAlgebraic Structures (or
NeutroAlgebras) and AntiAlgebraic structures (or AntiAlgebras). Unlike
the classical algebraic structures, where all operations are well-defined and all
axioms are totally true, in NeutroAlgebras and AntiAlgebr...

Decision making by the business managerial on framing strategies to foster customer acquisition is a challenging task. The aim of this paper is to introduce a new method of Multi-Strategy Decision-Making (MSDM) integrated with neutrosophic soft relational maps to determine the significant and feasible strategies of customer acquisition and their in...

As generalizations and alternatives of classical algebraic structures there have been introduced in 2019 the Neutro Algebraic Structures (or Neutro Algebras) and Anti Algebraic structures (or Anti Algebras). Unlike the classical algebraic structures, where all operations are well-defined and all axioms are totally true, in Neutro Algebras and Anti...

In this paper, we introduce the notion of a wRM/eRM-algebra as a generalization of a RM-algebra. These structures are studied in details. Also, we derived an eRM-algebra from a RM-algebra and vice versa. The concept of a positive implicative eRM-algebra is defined and we discussed on the medial filters.

In this study, we analyze a neutrosophic N −subalgebra, a (ultra) neutrosophic N −filter, level sets of these neutrosophic N −structures and their properties on a Sheffer stroke BL-algebra. By defining a quasi-subalgebra of a Sheffer stroke BL-algebra, it is proved that the level set of neutrosophic N −subalgebras on the algebraic structure is its...

In this chapter, we introduced the notion of neutrosophic filters in RM-algebras. Moreover, implicative neutrosophic filters on RM-algebras is defined and the relation between implicative neutrosophic filters and neutrosophic filters in investigated. Further, some necessary and sufficient conditions for a neutrosophic filter to be implicative neutr...

In this study, a neutrosophic N-subalgebra, a (implicative) neutrosophic N-filter, level sets of these neutrosophic N-structures and their properties are introduced on a Sheffer stroke BE-algebras (briefly, SBE-algebras). It is proved that the level set of neutrosophic N-subalgebras ((implicative) neutrosophic N-filter) of this algebra is the SBE-s...

In this paper, an (implicative) ideal and a fuzzy ideal of Sheffer stroke BG-algebra are defined and some properties are presented. Then a fuzzy implicative and a sub-implicative ideals of a Sheffer stroke BG-algebra are described. Morever, an implicative Sheffer stroke BG-algebra and a medial Sheffer stroke BG-algebra are defined and it is express...

In this paper, we extend the notion of semi-hypergroups (resp. hypergroups) to neutro-semihypergroups (resp. neutro-hypergroups). We investigate the property of anti-semihypergroups (resp. anti-hypergroups). We also give a new alternative of neutro-hyperoperations (resp. anti-hyperoperations), neutro-hyperoperation-sophications (resp. anti-hypersop...

The relationship between a transitive GE-algebra and a belligerent GE-algebra (also, between an antisymmetric GE-algebra and a left exchangeable GE-algebra) is displayed. A condition for the trivial GE-filter to be a belligerent GE-filter is provided. The least GE-filter containing a given GE-filter and one element is formed. Conditions under which...

The aim of the study is to examine a neutrosophic N −subalgebra, a neutrosophic N −filter, level sets
of these neutrosophic N −structures and their properties on a strong Sheffer stroke non-associative MV-algebra.
We show that the level set of neutrosophic N −subalgebras on this algebra is its strong Sheffer stroke nonassociative MV-subalgebra and...

Abstract. In this research, we use averages and relative measures of interval grey numbers to introduce grey vertices, grey edges, and grey graphs (graphs are based on interval grey numbers). To do so, we design a grey graph based on a graph (as the underlying graph). Also, we find a relation between grey vertices and grey edges of a grey graph. Th...

In this research, we use averages and relative measures of interval grey numbers to introduce grey vertices, grey edges, and grey graphs (graphs are based on interval grey numbers). To do so, we design a grey graph based on a graph (as the underlying graph). Also, we find a relation between grey vertices and grey edges of a grey graph. The primary...

In this research, we use averages and relative measures of interval grey numbers to introduce grey vertices, grey edges, and grey graphs (graphs are based on interval grey numbers). To do so, we design a grey graph based on a graph (as the underlying graph). Also, we find a relation between grey vertices and grey edges of a grey graph. The primary...

In this paper, we introduce the notion of commutator of two elements in a specific NeutroGroup. Then we define the notion of a NeutroNilpotentGroup and we study some of their properties. Moreover, we show that the intersection of two NeutroNilpotentGroups is a NeutroNilpotentGroup. Also, we show that the quotient of a NeutroNilpotentGroup is a Neut...

According to Boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions (it can also be described as an OR of AND’s). For each table an arbitrary T.B.T is given (total binary truth table) Boolean expression can be written as a disjunctive normal form. This paper conside...

In this paper, the concepts of a Neutro-algebra and Anti-algebra are introduced, and some related properties are investigated. We show that the class of Neutro-algebra is an alternative of the class of-algebras.

: In this paper, the concepts of a Neutro-𝐵𝐼-algebra and Anti-𝐵𝐼-algebra are introduced,
and some related properties are investigated. We show that the class of Neutro-𝐵𝐼-algebra is an
alternative of the class of 𝐵𝐼-algebras.

In this paper, we define the notion of a pseudo-eBE-algebra as an extension of a pseudo-BE-algebra, and it is studied in detail. The construction of an eBE-algebra from a pseudo-eBE-algebra is given. Further, the notions of filters and ideals are considered. The classes of distributive and commutative pseudo-eBE-algebras are introduced and investig...

This paper introduces the hypergraphs as complex hyper networks based on codes.
Indeed hypergraphs are securated.

In this paper, the notion of a medial filter in a BE-algebra is defined, and the theory of filters in BE-algebras is developed. These filters are very important for the study of congruence relations in BE-algebras. Moreover, the relationships between implicative filters, medial filters and normal filters are investigated.

In this paper, the notion of fuzzy medial filters of a pseudo BE-algebra is defined, and some of the properties are investigated. We show that the set of all fuzzy medial filters of a pseudo BE-algebra is a complete lattice. Moreover, we state that in commutative pseudo BE-algebras fuzzy filters and fuzzy medial filters coincide. Finally, the notio...

In this paper, as an extension of CI-algebras, we discuss the new notions of Neutro-CI-algebras and Anti-CI-algebras. First, some examples are given to show that these definitions are different. We prove that any proper CI-algebra is a Neutro-BE-algebra or Anti-BE-algebra. Also, we show that any NeutroSelf-distributive and AntiCommutative CI-algebr...

In this paper, as an extension of CI-algebras, we discuss the new notions of Neutro-CI-algebras and Anti-CI-algebras. First, some examples are given to show that these definitions are different. We prove that any proper CI-algebra is a Neutro-BE-algebra or Anti-BE-algebra. Also, we show that any NeutroSelf-distributive and AntiCommutative CI-algebr...

Decision making by the business managerial on framing strategies to foster customer acquisition is a challenging task. The aim of this paper is to introduce a new method of Multi-Strategy Decision-Making (MSDM) integrated with neutrosophic soft relational maps to determine the significant and feasible strategies of customer acquisition and their in...

The theory of fuzzy filters in pseudo-BE algebras is developed. Various characterizations of fuzzy filters are given. It is proved that the set of all fuzzy filters of a pseudo-BE algebra is a complete lattice. Some characterizations of Noetherian pseudo-BE algebras by fuzzy filters are obtained. Finally, fuzzy commutative filters are defined and s...

In this paper, we introduce the notion of pseudo-CI algebras and investigate some of their properties. It is a generalization of the notion of pseudo-BE algebras, pseudo-BCK algebras and pseudo-MV algebras. We give and provide some conditions for a pseudo-CI algebra to be a pseudo-BE algebra. Also, we define the class of singular pseudo-CI algebras...

Abstract. In this paper, the notions of orthogonal, dense, regular, zero-divisor, strong and complemented elements in a pseudo BL-algebra are introduced and relation between the orthogonal and zero-divisor elements for perfect (good) pseudo BL-algebras is investigated. In particular, we get some results when a pseudo BL- algebra is good or perfect....

In this paper, the concepts of branchwise commutative pseudo-CI
algebras and pointed pseudo-CI algebras are introduced and some
of their properties are investigated.

Hilbert algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true” and as a generalization of this was defined the notion of g-Hilbert algebra. In this pa...

In this article we introduce the notion of e-group as a new generalization of a group.
The condition for a group to be an e-group is given. The characterization of some properties is established and some results follow.

In this paper we generalize the notion of an implication groupoid and introduce ei-groupoid, and investigate related properties. We study the notion of filters in this structure and we give the construction of implication algebra from ei-groupoid. Finally, we prove that for distributive ei-groupoids filters coincide with ideals.

The notions of a dual pseudo-Q algebra and a dual pseudo-QC algebra are introduced. The properties and characterizations of them are investigated. Conditions for a dual pseudo-Q algebra to be a dual pseudo-QC algebra are given. Commutative dual pseudo-QC algebras are considered. The interrelationships between dual pseudo-Q/QC algebras and other pse...

In this paper, we develop fuzzy filter theory on a BE-algebra and apply the concept of interval-valued hesitant fuzzy filter to BE-algebras. Moreover, some types of interval-valued hesitant fuzzy filters such as interval-valued hesitant fuzzy implicative filter and interval-valued hesitant fuzzy fantastic filter are introduced and some of propertie...

In this paper, we introduce a new algebra, called a BI-algebra, which is a generalization of a (dual) implication algebra and we discuss the basic properties of BI-algebras, and investigate ideals and congruence relations.

In the present paper, we study Bosbach and Riečan states on pseudo BE-algebras. Additionally, we establish new properties of pseudo BE-algebras and we define the pseudo BE(A)-algebras. We introduce the notion of a normal Bosbach state proving that any Bosbach state on a pseudo BE(A)-algebra is normal. We prove that the quotient pseudo BE-algebra vi...

In this paper, we introduce the notion of sBCI/sBCK/eBCI/eBCK-algebras as a generalization of the notion of BCI/BCK-algebras. This structure is studied in detail. Also we introduce a way to make an eBCK-algebra from a BCK-algebra and vice versa.

In this paper, we introduce the notion of hesitant fuzzy (implicative) ﬁlters and get some results on BE- algebras and show that every hesitant fuzzy implicative ﬁlter is a hesitant fuzzy ﬁlter but not the converse. Finally, we state and prove the relationship between hesitant fuzzy (implicative) ﬁlters and γ-inclusive sets.

Inthispaper, weintroducethenotionof(implicative)neutrosophicfilters in BE-algebras. The relation between implicative neutrosophic filters and neutrosophic filters is investigated and we show that in self distributive BE-algebras these notions are equivalent.

In this paper, we introduce the notion of hesitant fuzzy (implicative) filters and get some results on BE-algebras and show that every hesitant fuzzy implicative filter is a hesitant fuzzy filter but not the converse.

In this paper, we are going to introduce a fundamental relation "δ" on (hyper BE-algebra) dual hyper K-algebra and investigate some properties. We show that quotient of any dual hyper K-algebra via a regular relation is a hyper BE-algebra and this quotient, via any strongly regular relation is a BE-algebra. Furthermore, it shows that "δ" under some...

In this paper, we introduce the notion of distributive pseudo BE-algebra and show that the related relation defined on this structure is transitive and prove that every pseudo upper set is a pseudo filter. Also, the pseudo filter generated by a set is define and show that the set of all pseudo filters is distributive complete lattice but it is not...

In this paper, we investigate the relationship between dual (Weak) Subtraction algebras, Heyting algebras and BE-algebras. In fact, the purpose of this paper is to show that BE-algebra is a generalization of Heyting algebra and dual (Weak) Subtraction algebras. Also, we show that a bounded commutative self distributive BE-algebra is equivalent to t...

In this paper, we introduce the notions of N-subalgebras and N-filters based on Smarandache CI-algebra and give a number of their properties. The relationship between N(Q, f)-subalgebras(filters) and N-subalgebras(filters) are also investigated .

In this paper, we introduce the notions of Nsubalgebras and N-filters based on Smarandache CI-algebra and
give a number of their properties. The relationship between
N(Q; f)-subalgebras(filters) and N-subalgebras(filters) are also investigated.

In this paper, we introduce modal BE-algebra and study some structural properties of modal BE-algebra. The notions of modal upper set, modal BE-filter, BE -tautology filter, dual modal BE-algebra and quotient modal BE-algebraare introduced and their basic properties are investigated. We will prove that every self- distributive BE-algebra, induce a...

In this paper, we introduce the notion of anti fuzzy filter and study the
$\mathcal{N }$
N
-subalgebra(filter) of
$CI$
C
I
-algebra
$X$
X
. We prove that every anti fuzzy filter of
$BE$
B
E
-algebra is an anti fuzzy subalgebra, but it is not valid in
$CI$
C
I
-algebra. Finally, the notion of fuzzy translations and fuzzy extensio...

In this paper, we introduce the notions of N- subalgebras and N-filters based on Smarandache CI-algebra and give a number of their properties. The relationship between N(Q; f)-subalgebras(filters) and N-subalgebras(filters) are also in- vestigated

In this paper, we introduce the notion of hyper BE–algebra and investigate some properties. Also, some types of hyper filters in hyper BE–algebras are studied and the relationship between them are stated. We try to show that these notions are independent by some examples. Furthermore, it shows that under special condition hyper BE–algebras are equi...

In this paper, we consider the notion of congruence relation on pseudo BE-algebras and construct quotient pseudo BE-algebra via this congruence relation. Also, we use the notion of normal pseudo filters and get a congruence relation.

This paper is devoted to the study of some structural properties of bounded and involutory BE–algebras and investigate the relationship between them. We construct a commutative monoid by definition of proper operation in an involutory BE–algebra. Some rules of calculus for BE–algebras with a semi-lattice structure are provided. Many results related...

p>In this paper, we introduce the notions of em>N/em>-subalgebras and em>N/em>-filters based on Smarandacheem> CI/em>-algebra andbr /> give a number of their properties./p

In this paper, we introduce some types of filters in BE-algebras and state some theorems which determine the relationship between these filters and other filters of a BE-algebra and by some examples we show that these notions are different.

In this paper, we introduce the concept of ε-generalized fuzzy filters (ideals) and δ- multiplication of fuzzy set μ, of BE-algebras and their basic properties are investigated.

In this paper, the notions of CI-algebras, Smarandache CI-algebra, Q-Smarandache filters and Q-Smarandache ideals are introduced. We show that a nonempty subset F of a CI-algebra X is a Q-Smarandache filter if and only if
${A(x,y) \subseteq F}$
, which A(x, y) is a Q-Smarandache upper set. Finally, we introduced the concepts of Smarandache BE-alg...

Hilbert algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true” and as a generalization of this was defined the notion of g-Hilbert algebra. In this pa...

In this paper, we introduce the notion of pseudo BE-algebra which is a generalization of BE-algebra. We define the concepts of pseudo subalgebras and pseudo filters and prove that, under some conditions, pseudo subalgebra can be a pseudo filter. We prove that every homomorphic image and pre-image of a pseudo filter is also a pseudo filter. Furtherm...

In this paper we study properties of commutative BE-algebras and we give the construction of quotient (X/I; *, I) of a commutative BE-algebra X via an obstinate ideal I of X. We construct upper semilattice and prove that is a nearlattice. Finally we define and study commutative ideals in BE-algebras.

In this paper, we introduce the notion of fuzzy filters(ideals) in CI-algebras. We study fuzzy congruence relation in detail and construct quotient of (X/(theta)over-bar; o, 1/(theta)over-bar) via a fuzzy congruence relation on CI-algebra X.

In this paper, we introduce the notion of intuitionistic (T,S)(T,S)-fuzzy subalgebras in CICI-algebras and study their fundamental properties. We get a fuzzy subalgebra from an intuitionistic (T,S)(T,S)-fuzzy subalgebra. Also the notion of intuitionistic (T,S)(T,S)-fuzzy (closed) filters of CICI-algebras is introduced. We investigate the relationsh...

In this paper we use the regular congruence relation ∼ I to construct a quotient B-algebra X∖I from a self distributive BE-algebra X. Then we study the notion of homomorphisms on BE-algebras and the properties are studied. Finally we stated and prove the first, second and third isomorphism theorems in self distributive BE-algebras.

In this paper we introduced notion of fuzzy filters and fuzzy ideals on CI-algebras. We give the construction of quotient (Xthetao,1theta) via a fuzzy congruence relation of CI-algebra X.

## Projects

Projects (4)

- A classical Geometry has only totally true Axioms.
- While a NeutroGeometry is a geometry that has at least one NeutroAxiom (partiallt true, partially indeterminate, and partially false) and no AntiAxiom.
- Also, an AntiGeometry is a geometry that has at least
one AntiAxiom (i.e. an axiom that is totally (100%) false).
While the Non-Euclidean Geometries resulted from the total negation of only one specific axiom (Euclid’s Fifth Postulate),
the AntiGeometry results from the total negation of any axiom and even of more axioms from any geometric axiomatic system (Euclid’s five postulates, Hilbert’s 20 axioms, etc.),
and the NeutroAxiom results from the partial negation of one or more axioms [and no total negation of
no axiom] from any geometric axiomatic system.
Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries.
In our real world, the spaces are not homogeneous, but mixed, complex, even ambiguous. And the elements that populate them and the rules that act upon them are not perfect, uniform, or complete -
but fragmentary and disparate, with unclear and conflicting information, and they do not apply in the same degree to each element. That's why we need the NeutroGeometries and AntiGeometries.
The NeutroGeometry and AntiGeometry are the NeutroAlgebra and AntiAlgebra transposed opon a geometric space.

The project goal is to receive papers on NeutroAlgebra and AntiAlgebra for a collective book. The papers sshould be submitted by email to the editors: Prof. Dr. F. Smarandache (Email: fsmarandache@gmail.com) and Prof. Dr. M. Şahin (Email: docdrmemetsahin@gmail.com).
Description.
In 2019 and 2020 Smarandache [1, 2, 3] generalized the classical Algebraic Structures to NeutroAlgebraic Structures (or NeutroAlgebras) {whose operations and axioms are partially true, partially indeterminate, and partially false} as extensions of Partial Algebra, and to AntiAlgebraic Structures (or AntiAlgebras) {whose operations and axioms are totally false}. And in general, he extended any classical Structure, in no matter what field of knowledge, to a NeutroStructure and an AntiStructure.
References and links:
[1] Florentin Smarandache, NeutroAlgebra is a Generalization of Partial Algebra, International Journal of Neutrosophic Science (IJNS), Vol. 2, No. 1, PP. 08-17, 2020, http://fs.unm.edu/NeutroAlgebra.pdf.
[2] F. Smarandache, Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures, in Advances of Standard and Nonstandard Neutrosophic Theories, Pons Publishing House Brussels, Belgium, Ch. 6, pp. 240-265, 2019; http://fs.unm.edu/AdvancesOfStandardAndNonstandard.pdf
[3] F. Smarandache, Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (revisited), Neutrosophic Sets and Systems, vol. 31, pp. 1-16, 2020. DOI: 10.5281/zenodo.3638232, http://fs.unm.edu/NSS/NeutroAlgebraic-AntiAlgebraic-Structures.pdf