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

Abstract

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: 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)-I-Neutrosophic 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 space, or elements we know nothing if they belong to 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 Algebra-ic Structures.