Words for string algebras #
A string word is a path in the symmetrified displayed quiver. Reduction is expressed by excluding a contiguous arrow--inverse pair. The monomial relations are excluded in both orientations: every contiguous positive path in the word and in its reverse must survive the quotient.
This is the word convention used by Butler--Ringel, phrased without choosing sign functions at the vertices. The sign functions are useful for ordering strings, but are not part of the underlying string or its module.
A path in the quiver obtained by adjoining a formal inverse to every displayed arrow.
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One arrow in the symmetrified displayed quiver.
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Two decompositions of a quiver path at the same length have the same intermediate vertex, prefix, and suffix.
If two paths are prefixes of the same path, the shorter one is a prefix of the longer one.
A path occurs contiguously inside another path when the latter factors as a prefix, followed by that path, followed by a suffix.
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Every path is a contiguous subpath of itself.
Contiguous-subpath containment is transitive.
A contiguous subpath cannot be longer than the ambient path.
If a contiguous subpath of left ++ e ++ right does not contain the
distinguished boundary arrow e, then it lies wholly on one side of that
boundary.
A contiguous subpath short enough not to span a nonempty overlap lies in one of the two overlapping paths.
Reversal carries a contiguous subpath to the reversed contiguous subpath, and conversely.
Reversing a path preserves its length.
The positive signed copy of a displayed arrow, with the symmetrified quiver instance pinned explicitly.
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The negative signed copy of a displayed arrow, traversed backwards.
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The positive copy of an ordinary displayed-quiver path in the symmetrified quiver.
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The signs of a signed path, listed from its final letter backwards.
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A negative signed arrow cannot occur inside a positive ordinary path.
A signed path is reduced when it contains no adjacent formal inverse pair.
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Reduction is invariant under reversing a signed path.
Every signed path of length at most one is reduced.
A two-letter signed path is reduced when its second letter is not the formal inverse of its first, allowing the endpoints to be dependent.
A zero-length path between two noncancelling signed letters does not affect reducedness.
Every contiguous subpath of a reduced signed path is reduced.
Reducedness glues across a nonempty overlap: an inverse pair is too short to span both ends of that overlap.
Every positive ordinary-quiver path occurring in the signed word survives the relation quotient.
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In an admissible quotient, every signed word of length at most one avoids relations: all of its positive subpaths have length below two.
Avoiding the monomial relations is inherited by contiguous subpaths.
A negative signed boundary blocks every positive ordinary subpath, so avoidance of relations glues across it.
The Butler--Ringel string condition: the signed path is reduced and no positive subpath of it or of its inverse belongs to the monomial relation ideal.
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Casting the displayed endpoints of a signed path does not change whether it is a string.
Casting the endpoint of a prefix and casting the source of the appended arrow cancel when the two pieces are composed.
Casting only the target of a quiver path is injective.
Endpoint casts preserve path length.
Casting the target of a composite is the same as casting the target of its second factor.
Reversing an endpoint-cast path casts the reversed path at the swapped endpoints.
A string extension may be transported across endpoint equalities by casting the old path and the new arrow in opposite directions.
Once reducedness is known, two one-ended strings glue across a negative left boundary and a positive right boundary. The signs prevent relations from crossing either seam.
Two strings with a nonempty common middle glue when the new outer boundaries have the hook signs: negative on the left and positive on the right.
Every signed path of length at most one is a string for an admissible bound-quiver presentation.
A string remains a string after reversing every letter and the order of the word.
Every contiguous subpath of a string is a string.
A bundled string word, including its displayed endpoints.
- source : Q
- target : Q
- path : SignedPath self.source self.target
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Two consecutive positive letters cannot cancel.
Two consecutive negative letters cannot cancel.
A negative letter followed by a positive letter is reduced when their underlying arrows are distinct in the common-source star.
A positive letter followed by a negative letter is reduced when their underlying arrows are distinct in the common-target costar.
Bundled words are equal when their endpoints and dependent paths are equal; the string proof is proposition-valued.
The length-zero string at a displayed vertex.
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A displayed arrow as a positive string of length one.
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Reverse a bundled string word.
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A displayed arrow traversed in the inverse direction.
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Length of a string word.
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Append one signed arrow to a string word, provided the extended path is again a string.