Extended implicative groupoids

Abstract

In this paper, we generalize the concept of extended order algebras in order to get a new algebraic structure called “extended implicative groupoid”. First, we define the notions of pre-weak extended, weak extended, right extended and left extended implicative groupoid. Then we introduce the concept of extended implicative groupoid by using these notions. In addition, the special properties of these structures, such as the existence of MacNeille completion and adjoint product are studied. Finally, we prove that the class of symmetrical associative complete distributive extended implicative groupoids, coincides with complete residuated lattices.