A Theoretical Exploration of Rough Approximations in Hilbert Algebras

In this paper, we introduce the concept of roughness in the context of Hilbert algebras, a class of algebraic structures fundamental to studying non-classical logic. By integrating rough set theory with Hilbert algebras, we investigate the lower and upper approximations of subalgebras and ideals. We show that the lower and upper approximations of a subalgebra (or ideal) in a Hilbert algebra also make up a subalgebra (or ideal). This implies that algebraic systems can employ rough set concepts. Our results demonstrate that the approximation spaces induced by ideals in Hilbert algebras provide a robust framework for analyzing algebraic structures under incomplete or uncertain information. Furthermore, we present illustrative examples to validate our theoretical findings and highlight the practical implications of this approach. This study not only enriches the theoretical foundations of rough set theory but also opens new avenues for its application in algebraic logic and related fields.