Axiomatic analysis of state operators in Sheffer stroke BCK-algebras associated with algorithmic approaches

In this paper, we have presented a novel exploration of the construction of Riečan, Bosbach, internal, and general states within the framework of Sheffer stroke BCK-algebra $ \mathcal{B} $. We highlighted the originality of our work by examining key characteristics and the independence of the axiomatic systems associated with these states. Notably, we demonstrated that a Riečan state can correspond to a Bosbach state and vice versa, revealing significant interconnections between these concepts. Additionally, we introduced the innovative concepts of faithful and fixed sets generated by internal states on $ \mathcal{B} $, proving that each Sheffer stroke BCK-algebra retains its structure under an internal state. Our investigation also included internal state-(filters, compatible filters, and prime filters) on $ \mathcal{B} $ and their related results, as well as the relationship between internal state congruence and filters. Furthermore, we explore whether general states imply Riečan and Bosbach states, enhancing our understanding of these relationships. Finally, we introduced the concept of general state-morphism and discuss its implications for $ \mathcal{B} $. To support our findings, we provided compelling examples and fundamental algorithms, underscoring the practical significance of our study across various fields including artificial intelligence, computer science, and quantum logic.