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 [Formula: see text] and [Formula: see text] of a BCI and/or BCK algebra, the eventful subset [Formula: see text] which consists of BCI and/or BCK events with a condition is established. Conditions for the eventful subset [Formula: see text] to be a (positive implicative) ideal of an eventful BCK algebra are provided, and conditions for the eventful subset [Formula: see text] to be a (closed) ideal of an event BCI algebra are given. An example to show that the set [Formula: see text] is not a positive implicative ideal in an eventful BCK algebra is provided, and then the conditions for the set [Formula: see text] to be a positive implicative ideal in an eventful BCK algebra are considered.
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