In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, (...) and the NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. (shrink)

For the classical Geometry, in a geometrical space, all items (concepts, axioms, theorems, etc.) are totally (100%) true. But, in the real world, many items are not totally true. The NeutroGeometry is a geometrical space that has some items that are only partially true (and partially indeterminate, and partially false), and no item that is totally false. The AntiGeometry is a geometrical space that has some item that are totally (100%) false. While the Non-Euclidean Geometries [hyperbolic and elliptic 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 in general: theorem, concept, idea etc.] and even of more axioms [theorem, concept, idea, etc.] and in general from any geometric axiomatic system (Euclid’s five postulates, Hilbert’s 20 axioms, etc.), and the NeutroAxiom results from the partial negation of any axiom (or concept, theorem, idea, etc.). Clearly, the AntiGeometry is a generalization of Non-Euclidean Geometries. (shrink)

In this paper we extend Neutro-Algebra and Anti-Algebra to geometric spaces, founding Neutro/Geometry and AntiGeometry. While Non-Euclidean Geometries resulted from the total negation of a specific axiom (Euclid's Fifth Postulate), AntiGeometry results from the total negation of any axiom or even more axioms of any geometric axiomatic system (Euclidean, Hilbert, etc. ) and of any type of geometry such as Geometry (Euclidean, Projective, Finite, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.), and Neutro-Geometry results from the partial negation (...) of one or more axioms [and without total negation of any axiom] of any geometric axiomatic system and of any type of geometry. Generally, instead of a classical geometric Axiom, one can take any classical geometric Theorem of any axiomatic system and of any type of geometry, and transform it by Neutrosophication or Antisofication into a Neutro-Theorem or Anti-Theorem respectively to construct a Neutro-Geometry or Anti-Geometry. Therefore, Neutro-Geometry and Anti-Geometry are respectively alternatives and generalizations of Non-Euclidean Geometries. In the second part, the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra and Anti-Algebra, then to Neutro-Geometry and Anti-Geometry, and in general to Neutro-Structure and Anti-Structure that arise naturally in any field of knowledge is recalled. At the end, applications of many Neutro-Structures in our real world are presented. (shrink)

A special issue of the International Journal in Information Science and Engineering “Neutrosophic Sets and Systems” (vol. 71/2024) is dedicated to the Conference on NeutroGeometry, NeutroAlgebra, and Their Applications, organized by the Latin American Association of Neutrosophic Sciences. This event, which took place on August 12-14, 2024, in Havana, Cuba, was made possible by the valuable collaboration of the University of Havana, the University of Physical Culture and Sports Sciences "Manuel Fajardo," the José Antonio Echeverría University of Technology, University of (...) Informatics Sciences and the Cuban Academy of Sciences among other institutions. In 2019 Smarandache 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 on 2020 he continued to develop them. The NeutroAlgebras & AntiAlgebras are a new field of research, which is inspired from our real world. In classical algebraic structures, all operations are 100% well-defined, and all axioms are 100% true, but in real life, in many cases these restrictions are too harsh, since in our world we have things that only partially verify some operations or some laws. Similarly, a classical Geometry structure has all axioms totally (100%) true. A NeutroGeometry structure has some axioms that are only partially true, and no axiom is totally (100%) false. Whereas an AntiGeometry structure has at least one axiom that is totally (100%) false. And in general, in any field of knowledge one has: Structure, NeutroStructure, and AntiStructure which were inspired from our real world where the laws (axioms) do not equally apply to all people and in the same degree. This special issue aims to highlight the most recent advances and applications in the fields of NeutroGeometry and NeutroAlgebra, two areas that are at the forefront of contemporary mathematical and scientific thought. During the conference, the mathematical foundations and practical applications of these disciplines were explored, as well as their relevance in the MultiAlism system and other interdisciplinary areas. The content of this special issue has been carefully selected to reflect the diversity and depth of the topics discussed at the conference. This event and the subsequent publication of these works underline the growing importance of neutrosophic theories in the current scientific landscape. We are confident that the ideas and discoveries shared in these pages will be of great value to researchers, academics, and professionals interested in these innovative areas of knowledge. (shrink)

In this tenth book of scilogs – called via neutrosophica (the neutrosophic way) –, one may find new and old questions and solutions, referring mostly to topics on NEUTROSOPHY, but also MULTISPACE, with miscellaneous addition of topics on Physics, Mathematics, or Sociology – email messages to research colleagues, or replies, notes about authors, articles, or books, spontaneous ideas, and so on. -/- Exchanging ideas with A. Elhassouny, Junhui Kim, Jeong Gon Lee, Kul Hur, Hojjatollah Farahani, W. B. Vasantha Kandasamy, Said (...) Broumi, Mumtaz Ali, Mohamed Abdel-Basset, Ozen Ozer, Madad Khan, Gheorghe Săvoiu, John Mordeson, Adesina Agboola, Waldyr Rodrigues, Ajay Sharma, Stephen Crothers, Vlad, Dmitri Rabounski, Victor Christianto, Trung Duyên, Mirela Teodorescu, Ioan Aurel Pop.‬‬‬. (shrink)