Event and Its Application in Algebraic Structures

Abstract

The notion of eventful element and eventful set is introduced, and its application is considered in BCI and/or BCK algebras. Several properties are investigated, and (positive implicative) ideal in an eventful BCK algebra and closed ideal in an eventful BCI algebra are discussed. Given the subsets A and B of a BCI and/or BCK algebra, the eventful subset E(A, B) which consists of BCI and/or BCK events with a condition is established. Conditions for the eventful subset E(A, B) to be a (positive implicative) ideal of an eventful BCK algebra are provided, and conditions for the eventful subset E(A, B) to be a (closed) ideal of an event BCI algebra are given. An example to show that the set {(0) over tilde} is not a positive implicative ideal in an eventful BCK algebra is provided, and then the conditions for the set {(0) over tilde} to be a positive implicative ideal in an eventful BCK algebra are considered.