| BCI-algcbras and BCK-algcbras arc two kinds of logical algebras, which are algebraic semantics of BCI-system and BCK-system, respectively. Pseudo-BCI/BCk algebras are the non-commutative extension of BCI/BCK-algebras. Monadic operators are the algebraic counterpart of one-variable fragment exis-tential or universal quantifiers in classical predicate logic. Modal operators are a kind of special cases of closure operators. They are the algebraic counterpart of modal terms in modal logic. In the present study, we concentrate on the fun-damental theory of operators on pscudo BCI/BCK-algebras. The main content we have done is as follows:First of all, we introduce the notion of monadic operators on pseudo-BCI algebras and study some related properties. For example, idempotent property, isotonicity and the property of the set of fixed points. Next, we introduce the notion of monadic filters. We give a characterization of monadic filters. As the counterpart of monadic filters, we introduce the notion of monadic congruences. We prove that there is an one-to-one correspondence relation between the set of normal closed m-congruences and the set of normal closed m-filters in monadic pscudo-BCI algebras. Moreover, we introduce the notion of strong rcsiduated mappings and study the relation between monadic operators and strong rcsidu-ated mappings in pseudo-BCI algebras. A strong rcsiduated mapping on pseudo-BCI algebra with its residual arc equivalent to monadic operator.Secondly, we introduce the notion of modal operators on pscudo-BCK al-gebras and we also study their properties. With the help of these properties, we can get that the modal operators of pscudo-BCK(pP) algebras satisfy sep-aration property about (?). Furthermore, we give the characterization of modal operators of pseudo-BCK (pP) algebras. Next we introduce the dual operators of modal operators. On this base, we can obtain modal filters and modal con-grucnces, and then discuss the relations between them. Furthermore, we study the quotient structure of modal pseudo-BCK algebras. The quotient algebras of modal pseudo-BCK(pP) algebras arc still modal pscudo-BCK(pP) algebras. Finally, we study the relation between monadic operators and modal operators. A monadic filter must be a modal filter, but the converse is not true. Therefore, we give a necessary and sufficient condition about a modal operator and its dual to be a monadic operator. |
