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[Merged by Bors] - feat(RingTheory/Idempotents): generalize to Semiring, add Corner and direct product decomposition#20531

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Idempotent_DirectProduct
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[Merged by Bors] - feat(RingTheory/Idempotents): generalize to Semiring, add Corner and direct product decomposition#20531
alreadydone wants to merge 2 commits intomasterfrom
Idempotent_DirectProduct

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@alreadydone alreadydone commented Jan 6, 2025 •
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  • Add CompleteOrthogonalIdempotents.iff_ortho_complete: if a family is complete orthogonal, it consists of idempotents.

  • Add CompleteOrthogonalIdempotents.pair_iff'ₛ: x and y form a complete orthogonal family iff x * y = y * x = 0 and x + y = 1. Golf pair_iff.

  • Given an element e in a semigroup R, define Subsemigroup.corner = eRe. If R is a non-unital semiring and e is an idempotent, then eRe is a semiring. Define IsIdempotentElem.Corner as the Type version of corner, and provide unital (commutative) (semi)ring instances on it.

  • CompleteOrthogonalIdempotents.mulEquivOfIsMulCentral: A complete orthogonal family of central idempotents in a semiring give rise to a direct product decomposition. There exists CompleteOrthogonalIdempotents.bijective_pi but it uses subtraction and quotient so it's not suitable for semirings.

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@alreadydone alreadydone added the t-algebra Algebra (groups, rings, fields, etc) label Jan 6, 2025
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github-actions bot commented Jan 6, 2025 •
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PR summary 56237b65cc

Import changes for modified files

No significant changes to the import graph

Import changes for all files
Files Import difference

Declarations diff

+ CompleteOrthogonalIdempotents.iff_ortho_complete
+ CompleteOrthogonalIdempotents.mulEquivOfComm
+ CompleteOrthogonalIdempotents.mulEquivOfIsMulCentral
+ CompleteOrthogonalIdempotents.pair_iff'ₛ
+ CompleteOrthogonalIdempotents.pair_iffₛ
+ NonUnitalRing.corner
+ NonUnitalSubsemiring.corner
+ _root_.IsIdempotentElem.Corner
+ corner
+ instance [NonUnitalCommRing R] (idem : IsIdempotentElem e) : CommRing idem.Corner
+ instance [NonUnitalCommSemiring R] (idem : IsIdempotentElem e) : CommSemiring idem.Corner
+ instance [NonUnitalRing R] (idem : IsIdempotentElem e) : Ring idem.Corner
+ instance [NonUnitalSemiring R] (idem : IsIdempotentElem e) : Semiring idem.Corner
+ mem_corner_iff
+ mem_corner_iff_mem_range_mul_left
+ mem_corner_iff_mem_range_mul_right
+ mem_corner_iff_mul_left
+ mem_corner_iff_mul_right

You can run this locally as follows 
## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary
  • The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
  • The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

@alreadydone alreadydone mentioned this pull request Jan 6, 2025
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@alreadydone alreadydone force-pushed the Idempotent_DirectProduct branch from e179a79 to 56237b6 Compare January 8, 2025 16:18
@alreadydone alreadydone mentioned this pull request Jan 13, 2025
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jcommelin commented Jan 20, 2025

Thanks 🎉

bors merge

@ghost ghost added the ready-to-merge This PR has been sent to bors. label Jan 20, 2025
mathlib-bors bot pushed a commit that referenced this pull request Jan 20, 2025
@alreadydone
feat(RingTheory/Idempotents): generalize to Semiring, add Corner and …
6313738 
…direct product decomposition (#20531)

+ Add `CompleteOrthogonalIdempotents.iff_ortho_complete`: if a family is complete orthogonal, it consists of idempotents.

+ Add `CompleteOrthogonalIdempotents.pair_iff'ₛ`: x and y form a complete orthogonal family iff `x * y = y * x = 0` and `x + y = 1`. Golf `pair_iff`.

+ Given an element `e` in a semigroup R, define `Subsemigroup.corner` = eRe. If R is a non-unital semiring and `e` is an idempotent, then eRe is a semiring. Define `IsIdempotentElem.Corner` as the Type version of `corner`, and provide unital (commutative) (semi)ring instances on it.

+ `CompleteOrthogonalIdempotents.mulEquivOfIsMulCentral`: A complete orthogonal family of central idempotents in a semiring give rise to a direct product decomposition. There exists `CompleteOrthogonalIdempotents.bijective_pi` but it uses subtraction and quotient so it's not suitable for semirings.
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mathlib-bors bot commented Jan 20, 2025

Pull request successfully merged into master.

Build succeeded:

@mathlib-bors mathlib-bors bot changed the title feat(RingTheory/Idempotents): generalize to Semiring, add Corner and direct product decomposition [Merged by Bors] - feat(RingTheory/Idempotents): generalize to Semiring, add Corner and direct product decomposition Jan 20, 2025
@mathlib-bors mathlib-bors bot closed this Jan 20, 2025
@mathlib-bors mathlib-bors bot deleted the Idempotent_DirectProduct branch January 20, 2025 12:24
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