[2] **viXra:1503.0115 [pdf]**
*submitted on 2015-03-14 14:58:16*

**Authors:** Florentin Smarandache

**Comments:** 11 Pages.

In this paper we introduce for the first time a new type of structures, called (T, I, F)-Neutrosophic Structures, presented from a neutrosophic logic perspective, and we show 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, that structure is a (T, I, F)-Neutrosophic Structure.
The (T, I, F)-Neutrosophic Structures [based on the components T=truth, I=indeterminacy, F=falsehood] are different from the Neutrosophic Algebraic Structures [based on neutrosophic numbers of the form a+bI, where I=indeterminacy and I^n = I], that we rename as Neutrosophic I-Algebraic Structures (meaning algebraic structures based on indeterminacy “I” only). But we can combine both and obtain the (T, I, F)-Neutrosophic I-Algebraic Structures, i.e. algebraic structures based on neutrosophic numbers of the form a+bI, but also having indeterminacy 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 related to at least one axiom (or law) acting on the structure space. Then we extend them to Refined (T, I, F)-Neutrosophic Refined I-Algebraic Structures.

**Category:** Set Theory and Logic

[1] **viXra:1503.0085 [pdf]**
*replaced on 2018-10-01 22:31:51*

**Authors:** Takahiro Kato

**Comments:** 380 Pages.

Modules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract framework to explore the theory of structures, in particular, the notion of universal property. In this book we generalize and redevelop the basic notions and results of various universal constructions in category theory using this framework of modules.

**Category:** Set Theory and Logic