Generalized Arrow Removal Algebras

• Generalized Arrow Removal Algebras are defined by selectively deleting or neutralizing arrows in quiver, median, and diagrammatic settings to maintain crucial homological invariants.
• They extend classical arrow removal methods by enforcing conditions like pre-removability and strict Gröbner-basis control to preserve properties such as finitistic dimensions and Gorensteinness.
• These techniques facilitate canonical reductions and inverse operations, allowing structural simplification while sustaining invariants across various algebraic frameworks including KLR theory and Arrow calculus.

Generalized Arrow Removal Algebras are not a single standardized class. In the literature, the expression is used for several constructions in which arrows are deleted, quotiented out, neutralized, or made structurally inert. For bound quiver algebras, a generalized arrow removal algebra of $\Lambda$ is any algebra isomorphic to $\Lambda/K_A$ with $A$ removable; for median algebras, trees are exactly the codomains that force all median-preserving aggregators to be essentially unary; for welded-link Arrow calculus, one forms a quotient by Expansion, Arrow moves, w-tree moves, and a removal ideal; for arrow algebras inducing triposes, nuclei provide a restriction mechanism explicitly compared with “removal”; and for KLR theory, the balanced algebra $S_{\Phi_k(\alpha)}$ is the ordered/truncated corner attached to subdivision and runner removal (Giatagantzidis, 17 Jul 2025, Couceiro et al., 2015, Meilhan et al., 2017, Tarantino, 2024, Qin, 26 Feb 2026).

1. Terminological scope

The available literature uses the phrase in several mathematically distinct ways. In quiver representation theory, “arrow removal” is literal: one passes from $A=kQ/I$ to a quotient that kills an arrow or a removable set of arrows, typically to control finitistic or global dimension. In the median-algebraic Arrow-type setting of Couceiro–Foldes–Meletiou, “Arrow” refers to Arrow-style impossibility, and the relevant codomains are trees. In Meilhan–Yasuhara’s Arrow calculus, the generators are arrow and w-tree diagrams, and “removal” is encoded by a quotient by local cancellation relations. In the theory of arrow algebras for modified realizability, the paper explicitly states that it does not define a literal removal operation, but nuclei induce subtriposes through a closure/restriction mechanism. In the KLR setting, subdivision of an edge and ordered truncation produce a balanced algebra interpreted as an arrow-removal algebra (Couceiro et al., 2015, Meilhan et al., 2017, Tarantino, 2024, Qin, 26 Feb 2026).

Setting	Defining object	Structural effect
Bound quiver algebras	$\Lambda/K_A$ with $A$ removable	preserves finiteness of $\fpd$, $Fpd$, and $gd$
Median algebras	tree codomain $\Lambda/K_A$0	all median-homomorphisms from products are essentially unary
Arrow calculus	$\Lambda/K_A$1	models welded/classical equivalence and $\Lambda/K_A$2-equivalence
Arrow algebras	$\Lambda/K_A$3 from a nucleus $\Lambda/K_A$4	subtriposes correspond to nuclei
KLR subdivision	$\Lambda/K_A$5	partial categorification of runner removal

2. Bound quiver algebras: from classical arrow removal to generalized removal

For a bound quiver algebra $\Lambda/K_A$6, generalized arrow removal is a homological reduction technique that preserves finiteness of the little and big finitistic dimensions and the global dimension, and significantly extends the classical arrow removal of Green–Psaroudakis–Solberg (Giatagantzidis, 17 Jul 2025). The classical 2018 construction starts from an arrow $\Lambda/K_A$7 and forms the quotient

$\Lambda/K_A$8

More generally, if a set of arrows $\Lambda/K_A$9 does not occur in any minimal generating set of $A$0, and additionally $A$1 for all $A$2, then the simultaneous arrow removal is

$A$3

Under these hypotheses one has a trivial extension description $A$4 with

$A$5

and the finitistic dimensions satisfy

$A$6

hence $A$7 (Green et al., 2018).

The 2025 generalization replaces “an arrow that does not occur in minimal relations” by a removable set $A$8. A set of arrows $A$9 is pre-removable if any, equivalently all, of the following hold: the natural epimorphism $S_{\Phi_k(\alpha)}$0 admits a section algebra monomorphism whose image is the subalgebra generated by trivial paths and arrows in $S_{\Phi_k(\alpha)}$1; the $S_{\Phi_k(\alpha)}$2-space sum $S_{\Phi_k(\alpha)}$3 is direct; for every $S_{\Phi_k(\alpha)}$4, both $S_{\Phi_k(\alpha)}$5 and $S_{\Phi_k(\alpha)}$6 lie in $S_{\Phi_k(\alpha)}$7; $S_{\Phi_k(\alpha)}$8; and $S_{\Phi_k(\alpha)}$9 has a finite generating set $A=kQ/I$0 with $A=kQ/I$1. If $A=kQ/I$2 is pre-removable, set $A=kQ/I$3. Then $A=kQ/I$4 is two-sided removable if $A=kQ/I$5 as both a left and a right $A=kQ/I$6-module is finite; non-repetitive if $A=kQ/I$7; only left removable if $A=kQ/I$8 and $A=kQ/I$9 as a right $\Lambda/K_A$0-module is finite; and removable if it is either two-sided or only left removable. A generalized arrow removal algebra of $\Lambda/K_A$1 is any algebra isomorphic to $\Lambda/K_A$2 with $\Lambda/K_A$3 removable (Giatagantzidis, 17 Jul 2025).

If $\Lambda/K_A$4 is pre-removable, the quotient $\Lambda/K_A$5 has a canonical bound quiver presentation $\Lambda/K_A$6 with $\Lambda/K_A$7 obtained by deleting $\Lambda/K_A$8 and $\Lambda/K_A$9. The central equivalence is Theorem B: $A$0 and the equivalence holds for $A$1 and $A$2 as well. The generalized method allows removal even when arrows appear in every generating set; one can still remove them if $A$3 is finite or $A$4 (Giatagantzidis, 17 Jul 2025).

3. Homological invariants, canonical reduction, and inverse operations

The quiver-theoretic theory is not restricted to finitistic dimension. In the arrow-removal setting $A$5, the restriction functor $A$6 is a $A$7-eventually homological isomorphism. From this, one obtains three preservation theorems: $A$8 is Gorenstein if and only if $A$9 is Gorenstein; $\fpd$0 induces a triangle equivalence

$\fpd$1

and $\fpd$2 satisfies $\fpd$3 if and only if $\fpd$4 satisfies $\fpd$5. The same paper defines a generalized arrow removal algebra as one obtained from $\fpd$6 by a finite sequence of admissible arrow removals, so that these invariants are preserved along the entire chain (Erdmann et al., 2021).

The 2025 framework also introduces a canonical maximal reduction. A subset $\fpd$7 is eventually removable if it admits an ordered partition $\fpd$8 such that each $\fpd$9 is removable in the successive quotient $Fpd$0. Every bound quiver algebra $Fpd$1 has a unique maximal eventually removable set $Fpd$2, independent of removal order, yielding the arrow reduced version

$Fpd$3

For any $Fpd$4,

$Fpd$5

with the same equivalences for $Fpd$6 and $Fpd$7. Moreover,

$Fpd$8

equivalently, iff all arrows are removable (Giatagantzidis, 17 Jul 2025).

A further extension proceeds in the opposite direction. A multiplicative bimodule is a pair $Fpd$9 with $gd$0 a $gd$1–$gd$2-bimodule and $gd$3 an associative bimodule map; the split extension $gd$4 has multiplication

$gd$5

A finite-dimensional $gd$6-algebra $gd$7 is a generalized arrow removal of $gd$8 iff $gd$9 is isomorphic to a split extension $\Lambda/K_A$00 by a removable multiplicative bimodule $\Lambda/K_A$01. The same framework introduces trivial one-arrow extensions $\Lambda/K_A$02, and for such $\Lambda/K_A$03,

$\Lambda/K_A$04

and similarly for $\Lambda/K_A$05 and $\Lambda/K_A$06 (Giatagantzidis, 17 Jul 2025).

4. Strict monomial arrow removal and Gröbner-basis control

A different generalization allows the removed arrow to occur in relations, provided it occurs in a controlled monomial way. Fix an arrow $\Lambda/K_A$07. A path avoids $\Lambda/K_A$08 if $\Lambda/K_A$09 does not occur as a subpath. Let $\Lambda/K_A$10 be a finite generating set of relations for $\Lambda/K_A$11, and let $\Lambda/K_A$12 be paths avoiding $\Lambda/K_A$13 with $\Lambda/K_A$14 and $\Lambda/K_A$15. Then $\Lambda/K_A$16 is an $\Lambda/K_A$17-monomial generating set if every $\Lambda/K_A$18 that is not a single path avoids $\Lambda/K_A$19. It is single $\Lambda/K_A$20-monomial if there exist $\Lambda/K_A$21 such that $\Lambda/K_A$22, all paths occurring in $\Lambda/K_A$23 avoid $\Lambda/K_A$24, no proper subpath of $\Lambda/K_A$25 lies in the ideal generated by $\Lambda/K_A$26, and at most one of $\Lambda/K_A$27 is trivial. It is strict $\Lambda/K_A$28-monomial if it is single $\Lambda/K_A$29-monomial and, additionally, no path occurring in $\Lambda/K_A$30 overlaps with $\Lambda/K_A$31 from the right, overlaps with $\Lambda/K_A$32 from the left, or divides $\Lambda/K_A$33 or $\Lambda/K_A$34 (Erdmann et al., 30 Jun 2025).

If $\Lambda/K_A$35 has an $\Lambda/K_A$36-monomial generating set, then

$\Lambda/K_A$37

is called a monomial arrow removal of $\Lambda/K_A$38. If $\Lambda/K_A$39 admits a strict $\Lambda/K_A$40-monomial Gröbner basis, the homological control is explicit. The quotient satisfies

$\Lambda/K_A$41

where $\Lambda/K_A$42 is the quiver obtained from $\Lambda/K_A$43 by removing $\Lambda/K_A$44 and $\Lambda/K_A$45. The inclusion $\Lambda/K_A$46 induces a monomorphism $\Lambda/K_A$47, and the projection $\Lambda/K_A$48 induces an epimorphism $\Lambda/K_A$49 with $\Lambda/K_A$50; this realizes $\Lambda/K_A$51 and $\Lambda/K_A$52 as a ring cleft extension (Erdmann et al., 30 Jun 2025).

The cleft-extension formalism uses endofunctors $\Lambda/K_A$53 and the quantities

$\Lambda/K_A$54

$\Lambda/K_A$55

In the strict monomial arrow removal situation, explicit two-term projective resolutions are available: $\Lambda/K_A$56 for $\Lambda/K_A$57, and

$\Lambda/K_A$58

for $\Lambda/K_A$59. These imply

$\Lambda/K_A$60

and hence the main estimate

$\Lambda/K_A$61

In particular, if $\Lambda/K_A$62 then $\Lambda/K_A$63 (Erdmann et al., 30 Jun 2025).

The paper also emphasizes that the Gröbner-basis criterion is genuinely stronger than the existence of a strict generating set, and that the strictness conditions are essential. Example 5.5 shows that when the relevant divisibility property fails, the minimal projective resolution of $\Lambda/K_A$64 has length $\Lambda/K_A$65, the Strong No Loop Theorem forces infinite projective dimension for a simple module, and the $\Lambda/K_A$66 bound cannot be applied (Erdmann et al., 30 Jun 2025).

5. Median, diagrammatic, and logical variants

In the median-algebraic setting, a map

$\Lambda/K_A$67

is a median-homomorphism if

$\Lambda/K_A$68

The central Arrow-type impossibility theorem states that for median algebras $\Lambda/K_A$69 and $\Lambda/K_A$70, every median-homomorphism $\Lambda/K_A$71 is essentially unary if and only if $\Lambda/K_A$72 is a tree when viewed as an ordered $\Lambda/K_A$73-semilattice. Equivalently, Arrow-type impossibility holds precisely for tree codomains. Theorem 3.2 identifies trees with the relaxed $\Lambda/K_A$74-median semilattice condition and with the requirement that every interval $\Lambda/K_A$75 be a chain. In the terminology explicitly proposed in the synthesis, a codomain median algebra $\Lambda/K_A$76 “removes” nontrivial Arrow-style aggregation for all products of median algebras if and only if $\Lambda/K_A$77 is a tree; in this sense, trees are precisely the “Generalized Arrow Removal Algebras” (Couceiro et al., 2015).

In Meilhan–Yasuhara’s Arrow calculus for welded and classical links, the generalized arrow removal algebra is defined diagrammatically. Let $\Lambda/K_A$78 be generated by formal arrow diagrams and w-tree diagrams on oriented $\Lambda/K_A$79-manifolds. Let $\Lambda/K_A$80 be generated by Expansion, the six Arrow moves, the w-tree moves, and twist involutivity. Let $\Lambda/K_A$81 be the ideal generated by Isolated Arrow, Inverse cancellation, Fork, and, in the homotopy version, repeated w-tree deletion. Then

$\Lambda/K_A$82

For $\Lambda/K_A$83, the ideal $\Lambda/K_A$84 generated by all $\Lambda/K_A$85-moves with $\Lambda/K_A$86 models $\Lambda/K_A$87-equivalence, and finite type invariants of degree $\Lambda/K_A$88 factor through $\Lambda/K_A$89. In this setting, “removal” means passage to a quotient by topologically trivial or canceling arrow and w-tree configurations (Meilhan et al., 2017).

The theory of arrow algebras for modified realizability uses “arrow” in the implicative sense. An arrow algebra is a quadruple $\Lambda/K_A$90, and a nucleus $\Lambda/K_A$91 induces a new arrow algebra

$\Lambda/K_A$92

The paper proves

$\Lambda/K_A$93

and the identity $\Lambda/K_A$94 yields a geometric inclusion with right adjoint $\Lambda/K_A$95. The paper’s final remark is explicit: it does not define an operation of “removing arrows.” Instead, nuclei serve as closure operators that restrict entailment and produce subtriposes, which the paper compares with a principled restriction analogous to “removal” (Tarantino, 2024).

6. Subdivision, runner removal, and balanced KLR algebras

In affine type $\Lambda/K_A$96, subdivision replaces an edge $\Lambda/K_A$97 by $\Lambda/K_A$98, producing a new quiver $\Lambda/K_A$99 of type $A$00. For $A$01, subdivision on roots is

$A$02

Ordered sequences in the target are those in which every $A$03 is immediately followed by $A$04. If $A$05 and $A$06 is the ideal generated by unordered sequences, the balanced KLR algebra is

$A$07

The diagrammatic subdivision map $A$08 inserts an extra strand immediately to the right of every $A$09-strand. Although $A$10 is not an algebra homomorphism on the full target algebra, it induces the graded $A$11-algebra isomorphism

$A$12

The synthesis explicitly interprets $A$13 as the “Generalized Arrow Removal Algebra” associated to subdividing the arrow $A$14 (Qin, 26 Feb 2026).

This construction is compatible with cyclotomic quotients and preserves the defect: $A$15 For $A$16-horizontal $A$17, subdivision respects idempotents,

$A$18

and, after the splitting map $A$19, it yields degree-$A$20 isomorphisms on permutation and Specht modules: $A$21

$A$22

If $A$23 is standard, then $A$24. These results provide a partial categorification of runner addition/removal. The paper does not prove exactness of the subdivision functor on the entire module category, nor full equality of graded decomposition numbers across $A$25 and $A$26; that limitation is stated explicitly (Qin, 26 Feb 2026).

7. Structural themes and limitations

Taken together, these constructions indicate several non-equivalent notions of “arrow removal.” In the quiver-theoretic papers, removal is literal quotienting by an ideal generated by arrows, and the central issue is preservation of homological finiteness or of invariants such as Gorensteinness, singularity categories, and $A$27. In the median-algebraic paper, “Arrow” refers to Arrow-style aggregation, and tree-likeness eliminates nontrivial multi-coordinate aggregators. In Arrow calculus, removal is a quotient by local cancellation relations. In the tripos-theoretic arrow-algebra paper, “removal” is only an analogy for restriction by nuclei. In the KLR paper, removal is realized by ordered truncation and a quotient by the bad ideal (Giatagantzidis, 17 Jul 2025, Couceiro et al., 2015, Meilhan et al., 2017, Tarantino, 2024, Qin, 26 Feb 2026).

The literature also imposes sharp hypotheses. In quiver theory, the preservation theorems require pre-removability/removability, trivial-extension structure, Hom-vanishing, or strict Gröbner-basis conditions; arbitrary factoring by an arrow is not covered, and Example 5.8 in the homological-invariants paper shows that factoring out an arrow in a different context can yield $A$28 satisfying $A$29 while $A$30 does not (Erdmann et al., 2021). In the strict monomial theory, the divisibility and overlap restrictions are essential; without them, the two-term bimodule resolutions need not exist, and the $A$31 estimate fails (Erdmann et al., 30 Jun 2025). In the KLR setting, the present results are partial categorification results rather than full exactness or full decomposition-number equalities (Qin, 26 Feb 2026). In the tripos-theoretic setting, the paper expressly warns that there is no literal arrow-removal operation (Tarantino, 2024).

This plurality of meanings suggests a common methodological pattern rather than a single definition: arrow removal is repeatedly used to pass from a larger or less rigid structure to a smaller, ordered, truncated, or quotient structure while preserving a chosen class of invariants or equivalence relations. The invariant to be preserved, however, depends entirely on context: $A$32, $A$33, and $A$34 for bound quiver algebras; essentially unarity for median aggregators; $A$35-equivalence and finite type information for Arrow calculus; subtriposes for arrow algebras; and module-theoretic and combinatorial data for KLR subdivision (Giatagantzidis, 17 Jul 2025, Couceiro et al., 2015, Meilhan et al., 2017, Tarantino, 2024, Qin, 26 Feb 2026).