Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures(revisited)

Florentin  Smarandache

Authors

Florentin Smarandache University of New MexicoMathematics Department705 Gurley Ave., Gallup, NM 87301, USA

Keywords:

Neutrosophic Triplets, (Axiom, NeutroAxiom, AntiAxiom), (Law, NeutroLaw, AntiLaw), (Associativity, NeutroAssociaticity, AntiAssociativity), (Commutativity, NeutroCommutativity, AntiCommutativity), (WellDefined, NeutroDefined, AntiDefined), (Semigroup, NeutroSemigroup, AntiSemigroup), (Group, NeutroGroup, AntiGroup), (Ring, NeutroRing, AntiRing), (Algebraic Structures, NeutroAlgebraic Structures, AntiAlgebraic Structures), (Structure, NeutroStructure, AntiStructure), (Theory, NeutroTheory, AntiTheory), S-denying an Axiom, S-geometries, Multispace with Multistructure.

Abstract

In all classical algebraic structures, the Laws of Compositionson a given set are well-defined. But this is a restrictive case, because there are many moresituations in science and in any domain of knowledgewhen a law of composition defined on a set maybe only partially-defined (or partially true) and partially-undefined(or partially false), that we call NeutroDefined, or totally undefined (totally false)that we call AntiDefined.Again, in all classical algebraic structures, the Axioms(Associativity, Commutativity, etc.)defined on a set are totally true, but it is again a restrictive case, because similarly there are numerous situations in science and in any domain of knowledgewhen an Axiom defined on a set may be only partially-true (and partially-false), that we call NeutroAxiom, or totally false that we call AntiAxiom. Therefore,we open for the first time in 2019 new fields of research called NeutroStructures andAntiStructures respectively.