Monadic pseudo BCI-algebras and corresponding logics

We introduce the notion of monadic pseudo BCI-algebras and study some related properties. Then, we introduce monadic filters and monadic congruences of monadic pseudo BCI-algebras and discuss the relations between them. We proved that there is a one-to-one correspondence between the set of closed m-congruence relations and the set of normal closed m-filters in a monadic pseudo BCI-algebra. Moreover, we introduce a notion of strong residuated mappings and study the relation between monadic operators and strong residuated mappings in pseudo BCI-algebras. Let A be a pseudo BCI-algebra and \(f:A\rightarrow A\) be a mapping, we obtain that \((f, f^+)\) is a monadic operator on A if and only if f is a strong residuated mapping on A where \(f^+\) is the residual of f. Also we exhibit an axiom system of monadic pseudo BCI-logic, which enrich the language of pseudo BCI-logics. Based on the monadic pseudo BCI-algebras, we prove the completeness and soundness of the monadic pseudo BCI-logic propositional system. Finally, using provable formula set, normal subset and monadic subset in the set of all formulas of a monadic pseudo BCI-logic \(\mathcal {L}\), we characterize filters, normal filters and monadic normal filters in a monadic pseudo BCI-algebra, respectively.