\( \gdef\badDispatch#1{\textbf{\textcolor{e1432d}{#1}}} \gdef\noKatexForm#1{\badDispatch{#1}} \gdef\largeDot{\raisebox{0.06em}{\tiny∙}} \gdef\rarrbar{ {\raisebox{-0.05em}{→}\mkern{-0.13em}{\large\shortmid}}} \gdef\larrbar{ { {\large\shortmid}\mkern{-0.13em}{\raisebox{-0.05em}{←}}}} \gdef\suptrans{^\mathsf{T}} \gdef\supdagger{^\dagger} \gdef\rawsymbol#1{\operatorname{#1}} \gdef\colorsymbol#1#2{\textcolor{#1}{\rawsymbol{#2}}} \gdef\lsymbol#1{\colorsymbol{262626}{#1}} \gdef\msymbol#1{\colorsymbol{b50800}{#1}} \gdef\osymbol#1{\colorsymbol{00427f}{#1}} \gdef\lstring#1{\textcolor{666666}{\textrm{\textquotedblleft{\text{#1}}\textquotedblright}}} \gdef\boldForm#1{\textbf{#1}} \gdef\lrform#1{ {\textcolor{e1432d}{#1}}} \gdef\lgform#1{ {\textcolor{4ea82a}{#1}}} \gdef\lbform#1{ {\textcolor{3e81c3}{#1}}} \gdef\rform#1{ {\textcolor{cb2b1a}{#1}}} \gdef\gform#1{ {\textcolor{399318}{#1}}} \gdef\bform#1{ {\textcolor{2b6fb0}{#1}}} \gdef\drform#1{ {\textcolor{9b0000}{#1}}} \gdef\dgform#1{ {\textcolor{006800}{#1}}} \gdef\dbform#1{ {\textcolor{004987}{#1}}} \gdef\drgform#1{ {\textcolor{934100}{#1}}} \gdef\drgform#1{ {\textcolor{ae5900}{#1}}} \gdef\dgbform#1{ {\textcolor{006567}{#1}}} \gdef\drbform#1{ {\textcolor{86004d}{#1}}} \gdef\rgform#1{ {\textcolor{dc841a}{#1}}} \gdef\gbform#1{ {\textcolor{47a5a7}{#1}}} \gdef\rbform#1{ {\textcolor{c74883}{#1}}} \gdef\pform#1{ {\textcolor{665996}{#1}}} \gdef\dpform#1{ {\textcolor{41326b}{#1}}} \gdef\waform#1{ {\textcolor{6b6b6b}{#1}}} \gdef\wbform#1{ {\textcolor{929292}{#1}}} \gdef\wcform#1{ {\textcolor{c5c5c5}{#1}}} \gdef\tick{\checkmark} \gdef\barToken{\mathbf{|}} \gdef\filledRectangleToken{▮} \gdef\emptyRectangleToken{▯} \gdef\filledSquareToken{■} \gdef\emptySquareToken{□} \gdef\filledToken{∙} \gdef\emptyToken{∘} \gdef\cardinalRewrite#1#2{\rewrite{#1}{#2}} \gdef\primed#1{ {#1}^{\prime}} \gdef\tinybullet{ {\tiny●}} \gdef\forwardFactor{\uparrow} \gdef\backwardFactor{\downarrow} \gdef\neutralFactor{\mid} \gdef\arrowhead{⏵} \gdef\upArrowhead{⏶} \gdef\deSymbol{ { { {\raisebox{1.1pt}{\tinybullet}\mkern{-1pt}{→}}}\,}} \gdef\ueSymbol{ {\raisebox{1pt}{\tinybullet\raisebox{-0.03em}{\mkern{-0.3pt}{\tiny──}\mkern{-0.3pt}}\tinybullet}\,}} \gdef\ldeSymbol{ { { {\raisebox{1.1pt}{\tinybullet}\mkern{-1pt}{\longrightarrow}}}\,}} \gdef\lueSymbol{ {\raisebox{1pt}{\tinybullet\raisebox{-0.03em}{\mkern{-0.3pt}{\tiny────}\mkern{-0.3pt}}\tinybullet}\,}} \gdef\de#1#2{ { {#1} \deSymbol {#2}}} \gdef\ue#1#2{ { {#1} \ueSymbol {#2}}} \gdef\tde#1#2#3{ { {#1} \overset{#3}{\deSymbol} {#2}}} \gdef\tue#1#2#3{ { {#1} \overset{#3}{\ueSymbol} {#2}}} \gdef\ltde#1#2#3{ { {#1} \overset{#3}{\ldeSymbol} {#2}}} \gdef\ltue#1#2#3{ { {#1} \overset{#3}{\lueSymbol} {#2}}} \gdef\mapsfrom{\htmlClass{hreflect}{\mapsto}} \gdef\longmapsfrom{\htmlClass{hreflect}{\longmapsto}} \gdef\diffd{𝕕} \gdef\partialdof#1{\partial {#1}} \gdef\emptySet{\empty} \gdef\unknown{\wbform{\text{?}}} \gdef\notApplicable{\wbform{\text{---}}} \gdef\textAnd{\text{\,and\,}} \gdef\identicallyEqualSymbol{\equiv} \gdef\congruentSymbol{\equiv} \gdef\isomorphicSymbol{\simeq} \gdef\homeomorphicSymbol{\cong} \gdef\homotopicSymbol{\simeq} \gdef\approxEqualSymbol{\approx} \gdef\bijectiveSymbol{\approx} \gdef\defeq{\mathrel{≝}} \gdef\defEqualSymbol{\mathrel{≝}} \gdef\syntaxEqualSymbol{\mathrel{\textcolor{888888}{\equiv}}} \gdef\tailEqualSymbol{\underdot{=}} \gdef\headEqualSymbol{\dot{=}} \gdef\ruledelayed{:\to} \gdef\mathdollar{\text{\textdollar}} \gdef\hyphen{\text{-}} \gdef\endash{\text{--}} \gdef\emdash{\text{---}} \gdef\updownarrows{\uparrow\!\downarrow} \gdef\vthinspace{\mkern{1pt}} \gdef\dlq{\text{\textquotedblleft}} \gdef\drq{\text{\textquotedblright}} \gdef\dprime{ {\prime\prime}} \gdef\inverse#1{ { {#1}^{-1}}} \gdef\inverseSymbol{\inverse{□}} \gdef\groupDirectProduct#1{#1} \gdef\groupDirectProductSymbol{\times} \gdef\groupInverse{\inverse} \gdef\groupPower#1#2{ {#1}^{#2}} \gdef\groupCommutator#1#2{\left[{#1},{#2}\right]} \gdef\groupoidInverse{\inverse} \gdef\latticeBFS#1{\textrm{bfs}({#1})} \gdef\pathHomomorphism#1{#1} \gdef\groupoidFunction#1{#1} \gdef\groupoidHomomorphism#1{#1} \gdef\affineModifier#1{\overrightharpoon{#1}} \gdef\supercirc#1{#1^{\circ}} \gdef\supercircb#1{#1^{\bullet}} \gdef\blackCircleModifier#1{\supercircb{#1}} \gdef\whiteCircleModifier#1{\supercirc{#1}} \gdef\totalSpaceStyle#1{\rbform{#1}} \gdef\baseSpaceStyle#1{\bform{#1}} \gdef\fiberSpaceStyle#1{\rform{#1}} \gdef\totalSpaceElementStyle#1{\drbform{#1}} \gdef\baseSpaceElementStyle#1{\dbform{#1}} \gdef\fiberSpaceElementStyle#1{\drform{#1}} \gdef\bundleProjectionStyle#1{\dgform{#1}} \gdef\bundleGraphStyle#1{\rbform{#1}} \gdef\bundleSectionStyle#1{\drgform{#1}} \gdef\bundleFunctionStyle#1{\drbform{#1}} \gdef\graphFunctionStyle#1{\totalSpaceStyle{#1}} \gdef\projectionFunctionStyle#1{\totalSpaceStyle{#1}} \gdef\sectionFunctionStyle#1{\totalSpaceStyle{#1}} \gdef\functionGraph#1{G_{#1}} \gdef\toroidalModifier#1{\supercirc{#1}} \gdef\modulo#1{\supercirc{#1}} \gdef\dividesSymbol{\mathrel{|}} \gdef\groupFunction#1{#1} \gdef\groupHomomorphism#1{#1} \gdef\automorphisms{\operatorname{Aut}} \gdef\endomorphisms{\operatorname{End}} \gdef\transportMap#1{\transportMapSymbol_{#1}} \gdef\transportMapSymbol{\tau} \gdef\action#1{#1} \gdef\selfAction#1{\hat{#1}} \gdef\actionGroupoid#1{\utilde{#1}} \gdef\cardinalGroup#1{G^*({#1})} \gdef\signed#1{ {#1}^*} \gdef\transportAtlas#1{T_{#1}} \gdef\inverted#1{\underline{#1}} \gdef\mirror#1{\overline{#1}} \gdef\pathComposeSymbol{\mathbin{∷}} \gdef\pathCompose#1#2{ {#1}\pathComposeSymbol{#2}} \gdef\translateSymbol{\mathbin{\uparrow}} \gdef\backwardTranslateSymbol{\mathbin{\downarrow}} \gdef\pathTranslate#1#2{ {#1}\translateSymbol{#2}} \gdef\pathLeftTranslate#1{ { {#1}\translateSymbol}} \gdef\pathBackwardTranslate#1#2{ {#1}{\backwardTranslateSymbol}{#2}} \gdef\pathLeftBackwardTranslate#1{ {#1}{\backwardTranslateSymbol}} \gdef\compactCovariantDifference#1#2{\Delta_{#1}\,{#2}} \gdef\covariantDifference{ {#1}\,\Delta\,{#2}} \gdef\forwardDifference{\Delta^{+}} \gdef\backwardDifference{\Delta^{-}} \gdef\centralDifference{\Delta} \gdef\pathForwardDifference#1{\forwardDifference_{#1}} \gdef\pathBackwardDifference#1{\backwardDifference_{#1}} \gdef\pathCentralDifference#1{\centralDifference_{#1}} \gdef\pathHead#1{\pathHeadVector{#1}} \gdef\pathTail#1{\pathTailVector{#1}} \gdef\pathHeadVector#1{ {#1}^{\bullet}} \gdef\pathTailVector#1{ {#1}_{\bullet}} \gdef\pathReverse#1{ {#1}^{\dagger}} \gdef\pathIntegral#1#2{ {#1} \int {#2}} \gdef\pathIntegralSymbol{ {\int}} \gdef\pathDot#1#2{ {#1} \cdot {#2}} \gdef\pathDotSymbol{ {\cdot}} \gdef\compactBasis#1{\mathscr{B}} \gdef\length{\operatorname{len}} \gdef\signedLength{\operatorname{len^*}} \gdef\andFn{\operatorname{and}} \gdef\orFn{\operatorname{or}} \gdef\notFn{\operatorname{not}} \gdef\vertexList{\operatorname{vertices}} \gdef\vertexList{\operatorname{vertices}} \gdef\edgeList{\operatorname{edges}} \gdef\pathList{\operatorname{paths}} \gdef\cardinalList{\operatorname{cards}} \gdef\signedCardinalList{\operatorname{cards^*}} \gdef\wordOf{\operatorname{word}} \gdef\headVertex{\operatorname{head}} \gdef\tailVertex{\operatorname{tail}} \gdef\basis{\operatorname{basis}} \gdef\split{\operatorname{split}} \gdef\lcm{\operatorname{lcm}} \gdef\minimalContractionSets{\operatorname{MCSets}} \gdef\minimalContractions{\operatorname{MC}} \gdef\grade{\operatorname{grade}} \gdef\support{\operatorname{support}} \gdef\coefficient{\operatorname{coeff}} \gdef\domain{\operatorname{domain}} \gdef\codomain{\operatorname{codomain}} \gdef\modFunction{\operatorname{mod}} \gdef\clip{\operatorname{clip}} \gdef\sign{\operatorname{sign}} \gdef\step{\operatorname{step}} \gdef\projection{\operatorname{proj}} \gdef\lift{\operatorname{lift}} \gdef\identity{\operatorname{id}} \gdef\total{\operatorname{total}} \gdef\torus{\operatorname{torus}} \gdef\mobius{\operatorname{mobius}} \gdef\stateCompose{\operatorname{glue}} \gdef\infixStateComposeSymbol{\_} \gdef\stateDecompose{\operatorname{melt}} \gdef\stateJoin{\operatorname{conj}} \gdef\stateMeet{\operatorname{disj}} \gdef\stateExtent{\operatorname{extent}} \gdef\stateIntent{\operatorname{intent}} \gdef\infixStateJoinSymbol{\sqcup} \gdef\infixStateMeetSymbol{\sqcap} \gdef\isPrime#1{#1\textrm{ prime}} \gdef\blank{\_} \gdef\emptyWord{} \gdef\multiwordSymbol#1{\mathbf{#1}} \gdef\wordSymbol#1{\mathtt{#1}} \gdef\word#1{#1} \gdef\pathMap#1{#1} \gdef\function#1{#1} \gdef\imageModifier#1{ {#1}^{\rightarrow}} \gdef\preimageModifier#1{ {#1}^{\leftarrow}} \gdef\multiImageColorModifier#1{\msetCol{#1}} \gdef\multiImageModifier#1{ {#1}^{\Rightarrow}} \gdef\multiPreimageModifier#1{ {#1}^{\Leftarrow}} \gdef\functionComposition#1{#1} \gdef\functionCompositionSymbol{\mathbin{\small ∘}} \gdef\route#1#2#3{[{#1}\!:\!{#2}\!:\!{#3}]} \gdef\multiroute#1#2#3{[{#1}\!:\!{#2}\!:\!{#3}]} \gdef\pathWord#1#2#3{ {#1}\!:\!{#2}\!:\!{#3}} \gdef\nullPath{\bot} \gdef\nullElement{\bot} \gdef\path#1{#1} \gdef\vert#1{#1} \gdef\underdot#1{\underset{\raisebox{0.3em}{.}}{#1}} \gdef\tvert#1{\underdot{#1}} \gdef\hvert#1{\dot{#1}} \gdef\edge#1{#1} \gdef\card#1{\mathtt{#1}} \gdef\path#1{#1} \gdef\quiver#1{#1} \gdef\bindingRuleSymbol{\to} \gdef\compactBindingRuleSymbol{:} \gdef\cayleyQuiverSymbol#1{\selfAction{#1}} \gdef\bindCayleyQuiver#1#2{\selfAction{#1}[#2]} \gdef\bindActionQuiver#1#2{#1[#2]} \gdef\bindSize#1#2{#1(#2)} \gdef\bindCardSize#1#2{#1[#2]} \gdef\bindCards#1#2{#1[#2]} \gdef\subSize#1#2{#1_{#2}} \gdef\gridQuiver#1{\textrm{Grid}^{#1}} \gdef\treeQuiver#1{\textrm{Tree}^{#1}} \gdef\bouquetQuiver#1{\textrm{Bq}^{#1}} \gdef\lineQuiver{\textrm{L}} \gdef\cycleQuiver{\textrm{C}} \gdef\squareQuiver{\textrm{Sq}} \gdef\cubicQuiver{\textrm{Cbc}} \gdef\triangularQuiver{\textrm{Tri}} \gdef\hexagonalQuiver{\textrm{Hex}} \gdef\rhombilleQuiver{\textrm{Rmb}} \gdef\limit#1#2{\lim_{#2}\,#1} \gdef\realVectorSpace#1{\mathbb{R}^{#1}} \gdef\complexVectorSpace#1{\mathbb{C}^{#1}} \gdef\matrixRing#1#2{M_{#2}(#1)} \gdef\groupoid#1{#1} \gdef\group#1{#1} \gdef\field#1{#1} \gdef\ring#1{#1} \gdef\indeterminate#1{#1} \gdef\semiring#1{#1} \gdef\sym#1{#1} \gdef\matrix#1{#1} \gdef\polynomial#1{#1} \gdef\setLetter{\mathcal{S}} \gdef\signedSetLetter{\mathcal{S^*}} \gdef\multisetLetter{\mathcal{M}} \gdef\signedMultisetLetter{\mathcal{M^*}} \gdef\multisetSemiringSymbol#1{#1} \gdef\multisetSemiring#1#2{\multisetLetter\left[#1, #2\right]} \gdef\signedMultisetRingSymbol#1{#1} \gdef\signedMultisetRing#1#2{\signedMultisetLetter\left[#1, #2\right]} \gdef\polynomialRing#1#2{#1[{#2}]} \gdef\routeSymbol#1{#1} \gdef\multirouteSymbol#1{\mathbf{#1}} \gdef\planSymbol#1{\mathbf{#1}} \gdef\ringElement#1{#1} \gdef\matrixPart#1#2#3{#1\llbracket{#2,#3}\rrbracket} \gdef\matrixRowPart#1{#1} \gdef\matrixColumnPart#1{#1} \gdef\subMatrixPart#1#2#3{#1_{#2,#3}} \gdef\matrixDotSymbol{\cdot} \gdef\matrixPlusSymbol{+} \gdef\wordGroup#1{\wordGroupSymbol_{#1}} \gdef\wordGroupSymbol{\Omega} \gdef\wordRing#1{\wordRingSymbol_{#1}} \gdef\wordRingSymbol{\Omega\!\degree} \gdef\linearCombinationCoefficient#1{ {\textcolor{888888}{#1}}} \gdef\plan#1{#1} \gdef\planRing#1{\planRingSymbol_{#1}} \gdef\planRingSymbol{\Phi} \gdef\basisPath#1#2{\mathbf{#1}_{#2}} \gdef\basisPathWeight#1#2{ {#1}_{#2}} \gdef\unitSymbol{\mathbf{e}} \gdef\unitVertexField{\unitSymbol_1} \gdef\forwardSymbol{f} \gdef\backwardSymbol{b} \gdef\symmetricSymbol{s} \gdef\antisymmetricSymbol{a} \gdef\wordVector#1#2{\unitSymbol_{#1}^{#2}} \gdef\gradOf#1{\grad\,{#1}} \gdef\grad{\nabla} \gdef\divOf#1{\div\,{#1}} \gdef\div{\dot{\nabla}} \gdef\laplacianOf#1{\laplacian\,{#1}} \gdef\laplacian{\ddot{\nabla}} \gdef\suchThat#1#2{ {#1}\,\big|\,{#2}} \gdef\chart#1{\chartSymbol_{#1}} \gdef\chartSymbol{C} \gdef\graphRegionIntersectionSymbol{\cap} \gdef\graphRegionUnionSymbol{\cup} \gdef\pathIso{\simeq} \gdef\factorial#1{#1!} \gdef\power#1#2{ {#1}^{#2}} \gdef\repeatedPower#1#2{ {#1}^{#2}} \gdef\contractionLattice#1{\operatorname{Con}(#1)} \gdef\contractedRelation#1{\sim_{#1}} \gdef\isContracted#1#2{ {#1} \sim {#2}} \gdef\isContractedIn#1#2#3{ {#1} \sim_{#3} {#2}} \gdef\isNotContracted#1#2{ {#1} \not \sim {#2}} \gdef\isNotContractedIn#1#2#3{ {#1} \not \sim_{#3} {#2}} \gdef\contractionSum#1{#1} \gdef\contractionSumSymbol{\sqcup} \gdef\contractionProduct#1{#1} \gdef\contractionProductSymbol{ {\cdot}} \gdef\graph#1{#1} \gdef\graphHomomorphism#1{#1} \gdef\coversSymbol{\sqsupseteq} \gdef\coveredBySymbol{\sqsubseteq} \gdef\strictlyCoversSymbol{\sqsupset} \gdef\strictlyCoveredBySymbol{\sqsubset} \gdef\covering#1#2#3{ {#2} \sqsupseteq_{#1} {#3}} \gdef\powerSetSymbol{\mathcal{P}} \gdef\powerSet#1{\powerSetSymbol({#1})} \gdef\existsForm#1#2{\exists\,{#1}\,:\,{#2}} \gdef\forAllForm#1#2{\forall\,{#1}\,:\,{#2}} \gdef\notted#1{\notSymbol {#1}} \gdef\andSymbol{\land} \gdef\orSymbol{\lor} \gdef\notSymbol{\lnot} \gdef\latticeSymbol#1{#1} \gdef\meetSemilatticeSymbol#1{#1} \gdef\joinSemilatticeSymbol#1{#1} \gdef\posetSymbol#1{#1} \gdef\latticeElementSymbol#1{#1} \gdef\meetSemilatticeElementSymbol#1{#1} \gdef\joinSemilatticeElementSymbol#1{#1} \gdef\posetElementSymbol#1{#1} \gdef\latticeMeetSymbol{\wedge} \gdef\latticeJoinSymbol{\vee} \gdef\latticeTop{\top} \gdef\latticeBottom{\bot} \gdef\semilatticeMeetSymbol{\wedge} \gdef\semilatticeJoinSymbol{\vee} \gdef\semilatticeTop{\top} \gdef\semilatticeBottom{\bot} \gdef\semilatticeSemimeetSymbol{\wedge} \gdef\semilatticeSemijoinSymbol{\vee} \gdef\latticeGreaterSymbol{>} \gdef\latticeGreaterEqualSymbol{\ge} \gdef\latticeLessSymbol{<} \gdef\latticeLessEqualSymbol{\le} \gdef\posetGreaterSymbol{>} \gdef\posetGreaterEqualSymbol{\ge} \gdef\posetLessSymbol{<} \gdef\posetLessEqualSymbol{\le} \gdef\posetCoversSymbol{⋗} \gdef\posetCoveredBySymbol{⋖} \gdef\grpname#1{\operatorname{\mathsf{#1}}} \gdef\mkg#1#2#3{\grpname{#1}({#2},{#3})} \gdef\mka#1#2#3{\mathfrak{#1}({#2},{#3})} \gdef\generalLinearAlgebra#1#2{\mka{gl}{#1}{#2}} \gdef\generalLinearGroup#1#2{\mkg{GL}{#1}{#2}} \gdef\specialLinearAlgebra#1#2{\mka{sl}{#1}{#2}} \gdef\specialLinearGroup#1#2{\mkg{SL}{#1}{#2}} \gdef\projectiveGeneralLinearAlgebra#1#2{\mka{pgl}{#1}{#2}} \gdef\projectiveGeneralLinearGroup#1#2{\mkg{PGL}{#1}{#2}} \gdef\projectiveSpecialLinearAlgebra#1#2{\mka{psl}{#1}{#2}} \gdef\projectiveSpecialLinearGroup#1#2{\mkg{PSL}{#1}{#2}} \gdef\orthogonalAlgebra#1#2{\mka{o}{#1}{#2}} \gdef\orthogonalGroup#1#2{\mkg{O}{#1}{#2}} \gdef\specialOrthogonalAlgebra#1#2{\mka{so}{#1}{#2}} \gdef\specialOrthogonalGroup#1#2{\mkg{SO}{#1}{#2}} \gdef\unitaryAlgebra#1{\mka{u}{#1}{#2}} \gdef\unitaryGroup#1{\mkg{U}{#1}{#2}} \gdef\specialUnitaryAlgebra#1#2{\mka{su}{#1}{#2}} \gdef\specialUnitaryGroup#1#2{\mkg{su}{#1}{#2}} \gdef\spinAlgebra#1#2{\mka{spin}{#1}{#2}} \gdef\spinGroup#1#2{\mkg{Spin}{#1}{#2}} \gdef\pinAlgebra#1#2{\mka{pin}{#1}{#2}} \gdef\pinGroup#1#2{\mkg{Pin}{#1}{#2}} \gdef\symmetricGroup#1{\grpname{Sym}({#1})} \gdef\presentation#1{#1} \gdef\groupPresentation#1#2{\left\langle\,{#1}\,\,\middle|\mathstrut\,\,{#2}\,\right\rangle} \gdef\groupRelationIso{=} \gdef\groupGenerator#1{#1} \gdef\groupRelator#1{#1} \gdef\groupElement#1{#1} \gdef\identityElement#1{#1} \gdef\groupoidElement#1{#1} \gdef\groupIdentity#1{#1} \gdef\groupoidIdentity#1{#1} \gdef\ringIdentity#1{#1} \gdef\iconstruct#1#2{ {#1}\,\,\middle|{\large\mathstrut}\,\,{#2}} \gdef\setConstructor#1#2{\left\{\,\iconstruct{#1}{#2}\,\right\}} \gdef\multisetConstructor#1#2{\left\{\mkern{-2.3pt}\left|\,\,\iconstruct{#1}{#2}\,\right|\mkern{-2.3pt}\right\}} \gdef\cardinalityConstructor#1#2{\left|\,\iconstruct{#1}{#2}\,\right|} \gdef\setToMultiset#1{ {#1}^\uparrow} \gdef\multisetToSet#1{ {#1}^\downarrow} \gdef\subsets#1{\setLetter({#1})} \gdef\signedSubsets#1{\signedSetLetter({#1})} \gdef\multisets#1{\multisetLetter({#1})} \gdef\signedMultisets#1{\signedMultisetLetter({#1})} \gdef\circleSpaceSymbol{S} \gdef\topologicalSpace#1{#1} \gdef\bundleSection#1{#1} \gdef\bundleProjection#1{#1} \gdef\setSymbol#1{#1} \gdef\signedSetSymbol#1{#1} \gdef\multisetSymbol#1{#1} \gdef\signedMultisetSymbol#1{#1} \gdef\setElementSymbol#1{#1} \gdef\signedSetElementSymbol#1{#1} \gdef\multisetElementSymbol#1{#1} \gdef\signedMultisetElementSymbol#1{#1} \gdef\negated#1{\bar{#1}} \gdef\positiveSignedPart#1{ {#1}^+} \gdef\negativeSignedPart#1{ {#1}^-} \gdef\multisetMultiplicitySymbol{\,\raisebox{.1em}{\small\#}\mkern{.1em}\,} \gdef\signedMultisetMultiplicitySymbol{\,\raisebox{.1em}{\small\#}\mkern{.1em}\,} \gdef\boundMultiplicityFunction#1{ {#1}^{\sharp}} \gdef\boundSignedMultiplicityFunction#1{\operatorname{ {#1}^{\sharp}}} \gdef\constructor#1#2{\left.{#1}\,\,\middle|\mathstrut\,\,{#2}\right.} \gdef\elemOf#1#2{ { {#1} \in {#2} }} \gdef\notElemOf#1#2{ { {#1} \notin {#2} }} \gdef\edgeOf#1#2{ { {#1} {\in}_E {#2} }} \gdef\vertOf#1#2{ { {#1} {\in}_V {#2} }} \gdef\pathOf#1#2{ { {#1} {\in}_P {#2} }} \gdef\pathType#1{\operatorName{path}{#1}} \gdef\vertexType#1{\operatorName{vertex}{#1}} \gdef\edgeType#1{\operatorName{edge}{#1}} \gdef\multisetType#1{\operatorName{mset}{#1}} \gdef\signedMultisetType#1{\operatorName{mset^*}{#1}} \gdef\unitInterval{\mathbb{I}} \gdef\fromTo#1{#1} \gdef\fromToSymbol{\mapsto} \gdef\vertexCountOf#1{|{#1}|} \gdef\vertices#1{ V_{#1} } \gdef\edges#1{ E_{#1} } \gdef\vertexField#1{#1} \gdef\edgeField#1{#1} \gdef\pathVector#1{\mathbf{#1}} \gdef\pathVectorSpace#1{\mathscr{#1}} \gdef\baseField#1{#1} \gdef\finiteField#1{\mathbb{F}_{#1}} \gdef\functionSpace#1#2{#2^{#1}} \gdef\finiteTotalFunctionSpace#1#2{#2^{\sub #1}} \gdef\pathGroupoid#1{ { \Gamma_{#1} }} \gdef\forwardPathQuiver#1#2{ {\overrightharpoon{#1}_{#2}}} \gdef\backwardPathQuiver#1#2{ {\overrightharpoon{#1}^{#2}}} \gdef\pathQuiver#1{ {\overrightharpoon{#1}}} \gdef\mto#1#2{ {#1}\mapsto{#2}} \gdef\mtoSymbol{\mapsto} \gdef\groupWordRewriting#1{\langle{#1}\rangle} \gdef\rewrite#1#2{ {#1}\mapsto{#2}} \gdef\rewritingRule#1#2{ {#1}\mapsto{#2}} \gdef\rewritingSystem#1{\mathcal{#1}} \gdef\multiwayBFS#1{\textrm{bfs}({#1})} \gdef\rewritingStateBinding#1#2{ {\left.{#1}\!\mid\!{#2}\right.}} \gdef\rewritingRuleBinding#1#2{#1{\left[\,{#2}\,\right]}} \gdef\namedSystem#1{\mathtt{#1}} \gdef\genericRewritingSystem{\namedSystem{Sys}} \gdef\stringRewritingSystem{\namedSystem{Str}} \gdef\circularStringRewritingSystem{\namedSystem{CStr}} \gdef\turingMachineRewritingSystem{\namedSystem{TM}} \gdef\cellularAutomatonRewritingSystem{\namedSystem{CA}} \gdef\graphRewritingSystem{\namedSystem{Gr}} \gdef\hypergraphRewritingSystem{\namedSystem{HGr}} \gdef\petriNetRewritingSystem{\namedSystem{Petri}} \gdef\localStates#1{\localStatesSymbol^#1} \gdef\regionalStates#1{\regionalStatesSymbol^#1} \gdef\globalStates#1{\globalStatesSymbol^#1} \gdef\keySubStates#1{\keySubStatesSymbol^#1} \gdef\valueSubStates#1{\valueSubStatesSymbol^#1} \gdef\localState#1#2{#2_{#1}} \gdef\regionalState#1{#1} \gdef\globalState#1{#1} \gdef\keySubState#1{#1} \gdef\valueSubState#1{#1} \gdef\lhsState#1{#1_{L}} \gdef\rhsState#1{#1_{R}} \gdef\rewriteLHSRegionalState#1{#1_{L}} \gdef\rewriteRHSRegionalState#1{#1_{R}} \gdef\regionalStateForm#1{\waform{(}#1\waform{)}} \gdef\invalidRegionalState{\rform{\times}} \gdef\emptyRegionalState{\regionalStateForm{}} \gdef\regionalSubstateSymbol{\sqsubseteq} \gdef\regionalSuperstateSymbol{\sqsupseteq} \gdef\comparableRegionalStatesSymbol{\mathbin{\square}} \gdef\incomparableRegionalStatesSymbol{\mathbin{\boxtimes}} \gdef\namedStateSet#1{\mathbf #1} \gdef\localStatesSymbol{\namedStateSet L} \gdef\regionalStatesSymbol{\namedStateSet R} \gdef\globalStatesSymbol{\namedStateSet G} \gdef\keySubStatesSymbol{\namedStateSet L_K} \gdef\valueSubStatesSymbol{\namedStateSet L_V} \gdef\localStateSymbol#1{#1} \gdef\regionalStateSymbol#1{#1} \gdef\globalStateSymbol#1{#1} \gdef\keySubStateSymbol#1{#1} \gdef\valueSubStateSymbol#1{#1} \gdef\infixComposeLocalStatesSymbol{\_} \gdef\composeLocalStatesSymbol{\operatorname{glue}} \gdef\composeLocalStatesForm#1{\composeLocalStatesSymbol(#1)} \gdef\ncard#1{\card{\inverted{#1}}} \gdef\mcard#1{\card{\mirror{#1}}} \gdef\nmcard#1{\card{\inverted{\mirror{#1}}}} \gdef\assocArray#1{\left\langle {#1} \right\rangle} \gdef\openMultiset{\lBrace} \gdef\closeMultiset{\rBrace} \gdef\set#1{\{ {#1} \}} \gdef\signedSet#1{\{ {#1} \}} \gdef\list#1{\{ {#1} \}} \gdef\multiset#1{\openMultiset {#1} \closeMultiset} \gdef\signedMultiset#1{\openMultiset {#1} \closeMultiset} \gdef\styledSet#1#2{#1\{ {#1} #1\}} \gdef\styledList#1#2{#1\{ {#1} #1\}} \gdef\styledMultiset#1#2{#1\openMultiset {#2} #1\closeMultiset} \gdef\styledSignedMultiset#1#2{\openMultiset {#2} \closeMultiset} \gdef\permutationCycle#1{#1} \gdef\permutationCycleSymbol{\to} \gdef\permutationSet#1{#1} \gdef\permutationSetSymbol{;} \gdef\transposition#1#2{ {#1} \leftrightarrow {#2}} \gdef\tuple#1{( {#1} )} \gdef\concat#1{ {#1} } \gdef\paren#1{\left( {#1} \right)} \gdef\ceiling#1{\lceil{#1}\rceil} \gdef\floor#1{\lfloor{#1}\rfloor} \gdef\translationVector#1{\left[{#1}\right]_{\textrm{T}}} \gdef\quotient#1#2{ {#1} / {#2}} \gdef\compactQuotient#1#2#3{ {#1}\;{\upharpoonright_{#2}}\;{#3}} \gdef\multilineQuotient#1#2{\frac{#1}{#2}} \gdef\idElem#1{e_{#1}} \gdef\minus#1{-{#1}} \gdef\elem{\ \in\ } \gdef\vsp{\mkern{0.4pt}} \gdef\iGmult{\vsp} \gdef\gmult{\vsp\ast\vsp} \gdef\Gmult{\vsp\ast\vsp} \gdef\gdot{\vsp\cdot\vsp} \gdef\gDot{\vsp\mathbin{\largeDot}\vsp} \gdef\mdot{\vsp\cdot\vsp} \gdef\smallblackcirc{\vsp\raisebox{0.15em}{\tiny∙}\vsp} \gdef\smallwhitecirc{\vsp\raisebox{0.15em}{\tiny∘}\vsp} \gdef\sgdot{\mathbin{\smallwhitecirc}} \gdef\srdot{\mathbin{\smallblackcirc}} \gdef\srplus{+} \gdef\ellipsis{\,\wbform{...}\,} \gdef\verticalEllipsis{\vdots} \gdef\appliedRelation#1{#1} \gdef\setUnionSymbol{\cup} \gdef\setIntersectionSymbol{\cap} \gdef\setRelativeComplementSymbol{-} \gdef\msetCol{\textcolor{bb4444}} \gdef\repeatedMultiset#1#2{#1\,#2} \gdef\msrdot{\mathbin{\smallblackcirc}} \gdef\msrplus{+} \gdef\smrdot{\mathbin{\smallblackcirc}} \gdef\smrplus{+} \gdef\dotminus{\mathbin{\dot{-}}} \gdef\dotcap{\mathbin{\dot{\cap}}} \gdef\dotcup{\mathbin{\dot{\cup}}} \gdef\multisetUnionSymbol{\dotcup} \gdef\multisetIntersectionSymbol{\dotcap} \gdef\multisetRelativeComplementSymbol{\dotminus} \gdef\multisetSumSymbol{\dotplus} \gdef\cartesianProductSymbol{\times} \gdef\functionType#1#2{ {#1} \to {#2}} \gdef\functionSignature#1#2#3{ { {#1} : {#2} \to {#3}}} \gdef\partialFunctionSignature#1#2#3{ { {#1} : {#2} \rightharpoonup {#3}}} \gdef\poly#1{#1} \gdef\quiverProdPoly#1{#1} \gdef\quiverProdPower#1#2{#1^{#2}} \gdef\quiverProdTimes#1{#1} \gdef\parenLabeled#1#2{#1\ ({#2})} \gdef\parenRepeated#1#2{\parenLabeled{#1}{ {#2}\text{ times}}} \gdef\underLabeled#1#2{\underbrace{#1}_{#2}} \gdef\underRepeated#1#2{\underbrace{#1}_{#2\text{ times}}} \gdef\overLabeled#1#2{\overbrace{#1}^{#2}} \gdef\overRepeated#1#2{\overbrace{#1}^{#2\text{ times}}} \gdef\modLabeled#1#2{ {#1 }\textrm{ mod }{#2}} \gdef\freeGroup#1{F_{#1}} \gdef\cyclicGroup#1{\mathbb{Z}_{#1}} \gdef\componentSuperQuiverOfSymbol{\succ} \gdef\setCardinality#1{|{#1}|} \gdef\multisetCardinality#1{|{#1}|} \gdef\dependentQuiverProductSymbol{\mathbin{\times}} \gdef\independentQuiverProductSymbol{\mathbin{⨝}} \gdef\graphUnionSymbol{\sqcup} \gdef\graphProductSymbol{\times} \gdef\inlineProdSymbol{|} \gdef\serialCardSymbol{ {:}} \gdef\parallelCardSymbol{ {\mid}} \gdef\cardinalSequenceSymbol{ {:}} \gdef\cardinalProductSymbol{\inlineProdSymbol} \gdef\vertexProductSymbol{\inlineProdSymbol} \gdef\edgeProductSymbol{\inlineProdSymbol} \gdef\indexSum#1#2#3{ {\sum_{#2}^{#3} #1}} \gdef\indexProd#1#2#3{ {\prod_{#2}^{#3} #1}} \gdef\indexMax#1#2#3{ {\max_{#2}^{#3} #1}} \gdef\indexMin#1#2#3{ {\min_{#2}^{#3} #1}} \gdef\indexUnion#1#2#3{ {\bigcup_{#2}^{#3} #1}} \gdef\indexIntersection#1#2#3{ {\bigcap_{#2}^{#3} #1}} \gdef\styledIndexSum#1#2#3#4{ {#1\sum_{#3}^{#4} #2}} \gdef\styledIndexProd#1#2#3#4{ {#1\prod_{#3}^{#4} #2}} \gdef\styledIndexMax#1#2#3#4{ {#1\max_{#3}^{#4} #2}} \gdef\styledIndexMin#1#2#3#4{ {#1\min_{#3}^{#4} #2}} \gdef\indexSumSymbol{\sum} \gdef\indexProdSymbol{\prod} \gdef\indexMaxSymbol{\max} \gdef\indexMinSymbol{\min} \gdef\openInterval#1#2{(#1,#2)} \gdef\closedInterval#1#2{[#1,#2]} \gdef\openClosedInterval#1#2{(#1,#2]} \gdef\closedOpenInterval#1#2{[#1,#2)} \gdef\oneTo#1{1..{#1}} \gdef\zeroTo#1{0..{#1}} \gdef\qstring#1{\mathtt{"}{#1}\mathtt{"}} \gdef\wstring#1{\textcolor{6b6b6b}{#1}} \gdef\qchar#1{\mathtt{'}{#1}\mathtt{'}} \gdef\lstr#1{\mathtt{#1}} \gdef\lchar#1{\mathtt{#1}} \gdef\string#1{ {#1}} \gdef\character#1{ {#1}} \gdef\homomorphismMapping#1{\assocArray{#1}} \gdef\translationPresentation#1{\textrm{Z}_{#1}} \gdef\starTranslationPresentation#1{\textrm{Z}^*_{#1}} \gdef\translationPathValuation#1{\mathcal{\overrightharpoon Z}_{#1}} \gdef\starTranslationPathValuation#1{\overrightharpoon{\mathcal{Z}^*_{#1}}} \gdef\translationWordHomomorphism#1{\mathcal{Z}_{#1}} \gdef\starTranslationWordHomomorphism#1{\mathcal{Z}^*_{#1}} \gdef\translationCardinalValuation#1{\textrm{T}_{#1}} \gdef\starTranslationCardinalValuation#1{\textrm{T}^*_{#1}} \)
Word groups

Word groups

In this short section we introduce objects that nevertheless plays a vital role in understanding paths on quivers. These are word groups that describe possible path words, but divorced from any particular path in a quiver.

Notation

We write the word group on a set of cardinals as \(\bindCards{\wordGroupSymbol }{\card{\card{c}_1},\card{\card{c}_2},\ellipsis ,\card{\card{c}_{\sym{n}}}}\).

The word group of a quiver \(\quiver{Q}\), written \(\wordGroup{\quiver{Q}}\), is just the word group on the cardinals of that quiver.

If we wish only to specify how many cardinals are present, we will write this as \(\wordGroup{\quiver{\sym{n}}}\).

If it is clear from context what cardinals we are talking about, we'll just write \(\wordGroupSymbol\).

Elements

A group elements \(\elemOf{\groupElement{ \omega }}{\wordGroupSymbol }\) is a word, which is a finite sequence of (possibly inverted) cardinals. For example:

\[ \bindCards{\wordGroupSymbol }{\rform{\card{r}},\bform{\card{b}}} = \list{\word{1},\word{\rform{\card{r}}},\word{\rform{\ncard{r}}},\word{\bform{\card{b}}},\word{\bform{\ncard{b}}},\word{\rform{\card{r}}}{\rform{\card{r}}},\word{\rform{\ncard{r}}}{\rform{\ncard{r}}},\word{\rform{\card{r}}}{\bform{\card{b}}},\word{\rform{\card{r}}}{\bform{\ncard{b}}},\word{\rform{\ncard{r}}}{\bform{\card{b}}},\word{\rform{\ncard{r}}}{\bform{\ncard{b}}},\word{\bform{\card{b}}}{\rform{\card{r}}},\word{\bform{\card{b}}}{\rform{\ncard{r}}},\word{\bform{\ncard{b}}}{\rform{\card{r}}},\word{\bform{\ncard{b}}}{\rform{\ncard{r}}},\word{\bform{\card{b}}}{\bform{\card{b}}},\word{\bform{\ncard{b}}}{\bform{\ncard{b}}},\word{\rform{\card{r}}}{\rform{\card{r}}}{\rform{\card{r}}},\word{\rform{\card{r}}}{\rform{\card{r}}}{\bform{\card{b}}},\ellipsis } \]

We reserve the symbol \(\card{1}\) to refer to the group identity, the empty word consisting of zero cardinals, which would otherwise be hard to indicate textually, since it naturally be written as a blank space.

The number of words is obviously infinite if there is at least one cardinal.

Reduced form

The words are subject to the identity that we can rewrite any subword (any contiguous subsequence of cardinals) according to \(\concat{\card{c} \inverted{\card{c}}} = \concat{\inverted{\card{c}} \card{c}} = \card{1}\). Removing such adjacent inverses is called reduction.

We typically prefer to write these words in reduced form, so that the word \(\word{\gform{\card{g}}}{\rform{\card{r}}}{\rform{\inverted{\card{r}}}}{\bform{\card{b}}}\) is reduced to \(\word{\gform{\card{g}}}{\bform{\card{b}}}\).

Concatenation

The group multiplication of two words \(\elemOf{\groupElement{ \upsilon },\groupElement{ \omega }}{\wordGroupSymbol }\) is simply their concatenation, which we will write by putting words next to each other with a small gap: \(\concat{\groupElement{ \upsilon }\,\groupElement{ \omega }}\).

For the the case \(\groupElement{ \upsilon } = \word{\gform{\card{g}}}{\bform{\card{b}}}{\rform{\ncard{r}}},\groupElement{ \omega } = \word{\rform{\card{r}}}\), this looks like \(\concat{\groupElement{ \upsilon }\,\groupElement{ \omega }} = \concat{\word{\gform{\card{g}}}{\bform{\card{b}}}{\rform{\ncard{r}}}\,\word{\rform{\card{r}}}}\), which reduces to \(\word{\gform{\card{g}}}{\bform{\card{b}}}\).

Notice that if there is more than one cardinal, the group is not Abelian, since \(\concat{\groupElement{ \upsilon }\,\groupElement{ \omega }} \neq \concat{\groupElement{ \omega }\,\groupElement{ \upsilon }}\) in general. The order of cardinals in a word matters!

Here we list some concatenations of words in \(\bindCards{\wordGroupSymbol }{\rform{\card{r}},\gform{\card{g}},\bform{\card{b}}}\):

\[ \begin{csarray}{rlcl}{abeeb} \word{\card{1}} & \word{\card{1}} & = & \word{\card{1}}\\ \word{\card{1}} & \word{\rform{\card{r}}} & = & \word{\rform{\card{r}}}\\ \word{\rform{\card{r}}} & \word{\rform{\ncard{r}}} & = & \word{\card{1}}\\ \word{\rform{\card{r}}}{\bform{\card{b}}} & \word{\gform{\card{g}}} & = & \word{\rform{\card{r}}}{\bform{\card{b}}}{\gform{\card{g}}}\\ \word{\rform{\card{r}}}{\rform{\card{r}}}{\bform{\card{b}}} & \word{\bform{\ncard{b}}}{\gform{\card{g}}}{\gform{\card{g}}} & = & \word{\rform{\card{r}}}{\rform{\card{r}}}{\gform{\card{g}}}{\gform{\card{g}}} \end{csarray} \]

Inverses

To invert a word we reverse its individual letters and invert them.

Here we list a small number of examples of words from \(\wordGroup{\quiver{Q}}\) for \(\cardinalList(\quiver{Q}) = \{\rform{\card{r}},\bform{\card{b}}\}\), side-by-side with their inverses:

\[ \begin{csarray}{rlrlrlrl}{aeieieiei} \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }} & \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }} & \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }} & \groupElement{ \omega } & \groupInverse{\groupElement{ \omega }}\\ \word{\card{1}} & \word{\card{1}} & \word{\rform{\card{r}}} & \word{\rform{\ncard{r}}} & \word{\rform{\card{r}}}{\rform{\card{r}}} & \word{\rform{\ncard{r}}}{\rform{\ncard{r}}} & \word{\rform{\card{r}}}{\gform{\card{g}}}{\bform{\card{b}}} & \word{\bform{\ncard{b}}}{\gform{\ncard{g}}}{\rform{\ncard{r}}}\\ & & \word{\gform{\card{g}}} & \word{\gform{\ncard{g}}} & \word{\rform{\card{r}}}{\gform{\card{g}}} & \word{\gform{\ncard{g}}}{\rform{\ncard{r}}} & & \\ & & \word{\bform{\card{b}}} & \word{\bform{\ncard{b}}} & \word{\gform{\card{g}}}{\bform{\ncard{b}}} & \word{\bform{\card{b}}}{\gform{\ncard{g}}} & & \end{csarray} \]

Being free

Beyond the identity \(\concat{\card{c} \inverted{\card{c}}} = \concat{\inverted{\card{c}} \card{c}} = \card{1}\), which reflects in some sense the most generic property of a group, we do not impose any further relations (hence the term "free"). This implies that if two elements of \(\wordGroupSymbol\) "look different" in reduced form – that is, they contain a difference sequence of cardinals – they are different elements of the group.

Relationship to paths

As the name suggests, the path word of a path in \(\quiver{Q}\) is an element of the word group \(\wordGroup{\quiver{Q}}\), and path composition (when defined) will yield a path whose word is the concatenation (the group operation of \(\wordGroupSymbol\)) of the path words:

\[ \wordOf(\pathCompose{\path{P_1}}{\path{P_2}}) = \concat{\wordOf(\path{P_1})\,\wordOf(\path{P_2})} \]

A choice of vertex and a word will uniquely identify a path in a quiver, if one exists. Due to the local uniqueness property, there cannot be more than one path starting at a given vertex that posses a given word. However, there may be zero paths starting at a vertex with a given word.

It should be obvious that \(\functionSignature{\wordOf}{\pathGroupoid{\quiver{Q}}}{\wordGroup{\quiver{Q}}}\) is a groupoid homomorphism from the path groupoid to the word group (as \(\wordGroup{\quiver{Q}}\) is a group it is naturally also a groupoid).