Extended Nonstandard Neutrosophic Logic, Set, and Probability based on Extended Nonstandard Analysis

TL;DR

This paper extends nonstandard analysis by adding new monads and binads, then develops a corresponding nonstandard neutrosophic logic, set, and probability framework with lattice structures and algebraic properties.

Contribution

It introduces a second extension of nonstandard analysis with new structures, enabling the development of a nonstandard neutrosophic logic and set theory with lattice and algebraic properties.

Findings

Extended nonstandard space is closed under key operations.

Established nonstandard neutrosophic lattices of first and second types.

Provided theorems, new terms, and examples of operations.

Abstract

We extend for the second time the Nonstandard Analysis by adding the left monad closed to the right, and right monad closed to the left, while besides the pierced binad (we introduced in 1998) we add now the unpierced binad - all these in order to close the newly extended nonstandard space under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations. Then, we extend the Nonstandard Neutrosophic Logic, Nonstandard Neutrosophic Set, and Nonstandard Probability on this Extended Nonstandard Analysis space - that we prove it is a nonstandard neutrosophic lattice of first type (endowed with a nonstandard neutrosophic partial order) as well as a nonstandard neutrosophic lattice of second type (as algebraic structure, endowed with two binary neutrosophic laws, inf_N and sup_N). Many theorems, new terms introduced, and examples of nonstandard neutrosophic operations are given.