An informal overview of triples and
                    systems

Rowen, Louis

doi:10.1090/conm/727/14644

Cited by 12 publications

(12 citation statements)

References 41 publications

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“…See [35] for a relatively brief introduction of systems; more details are given in [25], [27], and [34]. Throughout the paper, we let N be the additive monoid of nonnegative integers.…”

Section: Basic Notionsmentioning

confidence: 99%

$\mathcal{T}$-semiring pairs

Jun¹,

Atanasov²,

Rowen³

2022

Preprint

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We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context. 4.1. Fractions 10 4.2. Integral extensions 10 4.3. Hilbert Nullstellensatz 12 5. Growth in semialgebras 13 5.1. Growth in a pair 13 References 14

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“…See [35] for a relatively brief introduction of systems; more details are given in [25], [27], and [34]. Throughout the paper, we let N be the additive monoid of nonnegative integers.…”

Section: Basic Notionsmentioning

confidence: 99%

$\mathcal{T}$-semiring pairs

Jun¹,

Atanasov²,

Rowen³

2022

Preprint

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“…We review the basic definitions and properties of T -modules, triples, and systems from [40]; more details are given in [26] and [39].…”

Section: Basic Notionsmentioning

confidence: 99%

Projective systemic modules

Jun

Mincheva

Rowen

2020

Journal of Pure and Applied Algebra

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We develop the basic theory of projective modules and splitting in the more general setting of systems. Systems provide a common language for most tropical algebraic approaches including supertropical algebra, hyperrings (specifically hyperfields), and fuzzy rings. This enables us to prove analogues of classical theorems for tropical and hyperring theory in a unified way. In this context we prove a Dual Basis Lemma and versions of Schanuel's Lemma.

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“…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.…”

Section: Basic Notionsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Homology of systemic modules

Jun

Mincheva

Rowen

2020

Preprint

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We develop the rudiments of a tropical homology theory, based on "triples" and "systems." Results include a version of Schanuel's lemma, projective dimension, the homology semi-module, and a weak Snake lemma. Contents 15 3.3. Chain complexes and exact sequences of systemic modules 19 4. (h, )-Projective dimension 20 4.1. Resolutions 20 5. Homological theory of systems 22 5.1. Homology of precongruences 22 5.2. Chains of systemic modules 25 6. Categorical aspects in the perspective of homological categories 29 6.1. Categories with negation 29 6.2. Towards semiexact categories and homological categories 31 References 34

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An informal overview of triples and systems

Abstract: We describe triples and systems, expounded as an axiomatic algebraic umbrella theory for classical algebra, tropical algebra, hyperfields, and fuzzy rings.

Cited by 12 publications

References 41 publications

$\mathcal{T}$-semiring pairs

$\mathcal{T}$-semiring pairs

Projective systemic modules

Homology of systemic modules

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