(t, i, f)-Neutrosophic Structures & I-Neutrosophic Structures (Revisited)

• Main Author: Florentin Smarandache
• Format: Article
• Language: English
• Published: University of New Mexico 2015-03-01
• Series: Neutrosophic Sets and Systems
• Subjects: (t; i; f)-neutrosophic structure; truth-indeterminacy-falsehood; neutrosophic axiom; indeterminacy; degree of indeterminacy; neutrosophic algebraic structures; neutrosophic groupoid; neutrosophic semigroup; neutrosophic group; neutrosophic linear algebras; neutrosophic bi-algebraic structures; neutrosophic N-algebraic structures
• Online Access: http://fs.gallup.unm.edu/NSS/(t,%20i,%20f)-Neutrosophic%20Structures.pdf
• ISSN: 2331-6055; 2331-608X

This paper is an improvement of our paper “(t, i, f)-Neutrosophic Structures” [1], where we introduced for the first time a new type of structures, called (t, i, f)Neutrosophic Structures, presented from a neutrosophic logic perspective, and we showed particular cases of such structures in geometry and in algebra. In any field of knowledge, each structure is composed from two parts: a space, and a set of axioms (or laws) acting (governing) on it. If the space, or at least one of its axioms (laws), has some indeterminacy of the form (t, i, f) ≠ (1, 0, 0), that structure is a (t, i, f)-Neutrosophic Structure. The (t, i, f)-Neutrosophic Structures [based on the components t = truth, i = numerical indeterminacy, f = falsehood] are different from the Neutrosophic Algebraic Structures [based on neutrosophic numbers of the form a + bI, where I = literal indeterminacy and In = I], that we rename as I-Neutrosophic Algebraic Structures (meaning algebraic structures based on indeterminacy “I” only). But we can combine both and obtain the (t, i, f)-INeutrosophic Algebraic Structures, i.e. algebraic structures based on neutrosophic numbers of the form a+bI, but also having indeterminacy of the form (t, i, f) ≠ (1, 0, 0) related to the structure space (elements which only partially belong to the space, or elements we know nothing if they belong to the space or not) or indeterminacy of the form (t, i, f) ≠ (1, 0, 0) related to at least one axiom (or law) acting on the structure space. Then we extend them to Refined (t, i, f)- Refined I-Neutrosophic Algebraic Structures.