Algebra: Hyperstructures, semirings, and blueprints

Core idea

This section organizes generalized additive structures used in tropical geometry, matroid theory, and blueprint geometry. Relationships between different kinds of coefficient objects are represented below by individual definitions, constructions, and theorems.

Semirings and tropical algebra

• Semiring
• Commutative semiring
• Semiring homomorphism
• Semifield
• Parasemifield
• Idempotent semiring
• Natural order of an idempotent semiring
• Idempotent semifield
• Lattice-ordered abelian group
• Idempotent-semifield and lattice-ordered-group equivalence
• Boolean semifield
• Tropical semifield
• Ordered semiring

Valuations

• Valuation on a field
• Value group of a valuation
• Valuation ring of a valued field
• Non-Archimedean absolute value
• Valuations as tropical-hyperfield morphisms

Hyperoperations, hyperrings, and hyperfields

• Hyperoperation
• Canonical hypergroup
• Hyperring
• Hyperfield
• Integral hyperring
• Hyperring homomorphism
• Quotient hyperring
• Hyperfield of a field quotient
• Krasner hyperfield
• Sign hyperfield
• Phase hyperfield
• Tropical hyperfield
• Valuative hyperfield
• Stringent hyperfield
• Doubly distributive hyperfield
• Doubly distributive hyperfields are stringent
• Classification of stringent hyperfields
• Field embeddings into semifields and hyperfields
• Tropical tied-sum distinction

Blueprints and ordered blueprints

• Commutative monoid
• Pre-addition on a monoid
• Blueprint
• Semiring completion of a blueprint
• Semiring as a blueprint
• Commutative monoid with zero as a blueprint
• Semiring and monoid blueprint embeddings
• Ordered blueprint
• Blueprint as an ordered blueprint
• Hyperrings as ordered blueprints
• Ordered blueprint with unique weak inverses
• Unique-weak-inverse reflection

Bands, tracts, and pastures

• Band
• Null set of a band
• Fusion rule for bands
• Fusion band
• Partial field
• Partial hyperfield
• Tract
• Tract associated with a partial field
• Tract associated with a hyperfield
• Tract associated with a partial hyperfield
• Ordered blueprint associated with a tract
• Band as an ordered blueprint
• Ordered blue field
• Idyll
• Idyll as an ordered blue field
• Pasture

Scheme-theoretic layer

• Localization of a blueprint
• $k$-ideal of a blueprint
• Spectrum of a blueprint
• Locally blueprinted space
• Affine blue scheme
• Blue scheme
• Ordered blueprinted space
• Ordered blue scheme