@article{id:Siminski2026Arithmetic,
title = {Arithmetic with extensional fuzzy numbers},
journal = {Fuzzy Sets and Systems},
volume = {544},
pages = {110067},
year = {2026},
issn = {0165-0114},
doi = {https://doi.org/10.1016/j.fss.2026.110067},
url = {https://www.sciencedirect.com/science/article/pii/S0165011426003039},
author = {Krzysztof Siminski},
keywords = {Fuzzy numbers, Extensional fuzzy numbers, t-norms},
abstract = {Fuzzy numbers are commonly used to represent imprecise data and are typically
modelled using fuzzy sets. However, arithmetic operations on fuzzy numbers are not as 
straightforward as those on crisp values. The classical approach based on Zadeh’s extension 
principle often leads to three major issues: (1) high computational cost due to iterative 
procedures, (2) lack of closure under operations (e.g., multiplication of triangular fuzzy 
numbers does not yield a triangular fuzzy number), and (3) progressive increase of fuzziness 
with successive operations. These limitations restrict the practical applicability of fuzzy 
numbers. In this paper, we address these issues by introducing arithmetic based on 
extensional fuzzy numbers. We define fundamental arithmetic operations and relational 
operators (=,  > ,  ≥ ,  < ,  ≤ ) for this representation. The proposed approach provides 
closed-form operations without the need for discretisation or iterative numerical procedures, 
preserves the structure of fuzzy numbers under repeated computations, and prevents 
uncontrolled growth of fuzziness. As a result, extensional fuzzy numbers can be directly 
used in mathematical models and standard algorithms as substitutes for real-valued quantities, 
without requiring algorithmic modifications. The usefulness of the approach is demonstrated 
through several application examples. A C++ implementation is available in a public GitHub 
repository.}
}