Projective systemic modules

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.…”

“…Any surpassing relation induces a partial preorder ≤ on A given by a 0 ≤ a 1 iff a • 0 a • 1 . This restricts to a partial preorder on A • , which ties in with the modulus, as seen in [31,Example 2.16]. Definition 2.13.…”

“…The relation ∇ is reflexive and symmetric, but not transitive. According to experience, ∇ has been more effective in linear algebra [2,3], but serves better in developing representation theory, and features in [31] as well as this paper.…”

“…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.…”

“…In [31] we studied versions of projective modules and Schanuel's Lemma. Here we continue with projective dimension via a version of Schanuel's Lemma which is more technical but which fits in well with the systemic environment, i.e., by means of negation maps and symmetrization as in [3], [45], §2.7.…”