On the algebraizability of the implicational fragment of Abelian logic

In this paper we consider the implicational fragment of Abelian logic (Formula presented.). We show that although the Abelian groups provide an semantics for the set of theorems of (Formula presented.) they do not for the associated consequence relation. We then show that the consequence relation is not algebraizable in the sense of Blok and Pigozzi (Mem Am Math Soc 77, 1989). In the second part of the paper, we investigate an extension of (Formula presented.) in the same language and having the same set of theorems and show that this new consequence relation is algebraizable with the Abelian groups as its equivalent algebraic semantics. Finally, we show that although (Formula presented.) is not algebraizable, it is order-algebraizable in the sense of Raftery (Ann Pure Appl Log 164:251-283, 2013).