Categories with negation

Homology of systemic modules

“…See [46] for a relatively brief introduction; more details are given in [31], [34], and [45]. Throughout the paper, we let N be the additive monoid of nonnegative integers.…”

“…We round out the structure with a surpassing relation ( [45, Definition 1.31] and also described in [34,Definition 2.11]). Definition 2.11.…”

“…In §6 •-idempotence will play a special role. Following [34], we consider the category of systemic modules (M, T M , (−), ) over a semiring system (A, T A , (−), ).…”

“…In this paper we explore homology theory for tropical mathematics. This has been initiated in a strong paper of Connes and Consani [11], which we utilize, but we work more algebraically in the theory of triples and systems which was espoused in [34,45], as surveyed in [46], (and applied in [3,18,31]) in order to unify classical algebra with the algebraic theories of supertropical algebra, symmetrized semirings, hyperfields, and fuzzy rings.…”

“…The surpassing relation plays a key structural role. 1 In [34] we have considered both "ground systems" which often are semirings, and systemic modules over ground systems. In this paper we investigate the rudiments of homology theory of systemic modules over a given ground system.…”