Abstract
In this paper, we define kleene algebras with implication and investigate the property of a suitable logic (called a Kleene logic) characterised by those algebras. We show the following;(1) Kleene algebras with implication can be embedded to the simplest such algebra 3; (2) The Lindenbaum algebras of Kleene logics are Kleene algebras with implication;(3) The completeness theorem holds for Kleene logics;(4) Kleene logics are decidable.
