Homology of systemic modules

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