Homology of systemic modules

Abstract

In this paper, we develop the rudiments of a tropical homology theory. We use the language of “triples” and “systems” to simultaneously treat structures from various approaches to tropical mathematics, including semirings, hyperfields, and super tropical algebra. We enrich the algebraic structures with a negation map where it does not exist naturally. We obtain an analogue to Schanuel’s lemma which allows us to talk about projective dimension of modules in this setting. We define two different versions of homology and exactness, and study their properties. We also prove a weak Snake lemma type result.