$$\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{#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{e1432d}{#1}}} \gdef\gform#1{ {\textcolor{4ea82a}{#1}}} \gdef\bform#1{ {\textcolor{3e81c3}{#1}}} \gdef\rgform#1{ {\textcolor{dc841a}{#1}}} \gdef\gbform#1{ {\textcolor{47a5a7}{#1}}} \gdef\rbform#1{ {\textcolor{c74883}{#1}}} \gdef\waform#1{ {\textcolor{6b6b6b}{#1}}} \gdef\wbform#1{ {\textcolor{929292}{#1}}} \gdef\wcform#1{ {\textcolor{c5c5c5}{#1}}} \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\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{\cong} \gdef\homotopicSymbol{\simeq} \gdef\approxEqualSymbol{\approx} \gdef\bijectiveSymbol{\cong} \gdef\defeq{\;≝\;} \gdef\defEqualSymbol{\;≝\;} \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\!\degree} \gdef\toroidalModifier#1{\supercirc{#1}} \gdef\modulo#1{\supercirc{#1}} \gdef\dividesSymbol{\,|\,} \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{ {\,∷\,}} \gdef\pathCompose#1#2{ {#1}\pathComposeSymbol{#2}} \gdef\translateSymbol{\uparrow} \gdef\backwardTranslateSymbol{\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\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\multiImageModifier#1{ {#1}^{\Rightarrow}} \gdef\multiPreimageModifier#1{ {#1}^{\Leftarrow}} \gdef\functionComposition#1{#1} \gdef\functionCompositionSymbol{∘} \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\realVectorSpace#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\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}|{#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\powerSet#1{\powerSetSymbol({#1})} \gdef\powerSetSymbol{\mathscr{P}} \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\latticeElement#1{#1} \gdef\latticeMeetSymbol{\wedge} \gdef\latticeJoinSymbol{\vee} \gdef\latticeTop{\top} \gdef\latticeBottom{\bot} \gdef\latticeGreaterSymbol{>} \gdef\latticeGreaterEqualSymbol{\ge} \gdef\latticeLessSymbol{<} \gdef\latticeLessEqualSymbol{\le} \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\groupoidElement#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\multisets#1{\mathscr{M}({#1})} \gdef\signedMultisets#1{\mathscr{M^*}({#1})} \gdef\setSymbol#1{#1} \gdef\multisetSymbol#1{#1} \gdef\setElementSymbol#1{#1} \gdef\multisetElementSymbol#1{#1} \gdef\multisetMultiplicitySymbol{\raisebox{.1em}{\small\#}\mkern{.1em}} \gdef\boundMultiplicityFunction#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\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\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{\mathsf{#1}} \gdef\genericRewritingSystem{\namedSystem{System}} \gdef\stringRewritingSystem{\namedSystem{Str}} \gdef\turingMachineRewritingSystem{\namedSystem{TM}} \gdef\cellularAutomatonRewritingSystem{\namedSystem{CA}} \gdef\graphRewritingSystem{\namedSystem{Gr}} \gdef\hypergraphRewritingSystem{\namedSystem{HGr}} \gdef\petriNetRewritingSystem{\namedSystem{Petri}} \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\set#1{\{ {#1} \}} \gdef\list#1{\{ {#1} \}} \gdef\multiset#1{\lBrace {#1} \rBrace} \gdef\permutationCycle#1{#1} \gdef\permutationCycleSymbol{\to} \gdef\permutationSet#1{#1} 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\gdef\cartesianProductSymbol{\times} \gdef\functionSignature#1#2#3{ { {#1} : {#2} \to {#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\dependentQuiverProductSymbol{\,\times\,} \gdef\independentQuiverProductSymbol{\,⨝\,} \gdef\graphUnionSymbol{\sqcup} \gdef\graphProductSymbol{\times} \gdef\inlineProdSymbol{\mid} \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\oneTo#1{1..{#1}} \gdef\zeroTo#1{0..{#1}} \gdef\qstring#1{\mathtt{"}{#1}\mathtt{"}} \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}}$$
Quiver products

Quiver products

Motivation

To this point we have left the questions of topology unexamined. Namely: what are the discrete analogs of continuous functions and open sets for quiver geometry? We won't examine these questions yet, but we will consider a topic that is similarly abstract and will ground future investigations into topology. That topic is of product spaces: how do we form products of quivers? Quiver products will be important in the next section for defining fibre bundles, but they'll also give us a new perspective on lattice quivers, which we will see later relates to irreducible and indecomposable representations.

Quiver products

We'll shortly introduce two kinds of quiver products, and then point out a straightforward generalization that gives us an infinite family of products on any number of quivers.

But first, what is a quiver product? Speaking informally, a quiver product between two quivers $$\quiver{R}$$ and $$\quiver{S}$$ is another quiver $$\quiver{Q}$$ with the following properties:

\begin{aligned} \vertexList(\quiver{Q})&\subseteq \vertexList(\quiver{R})\cartesianProductSymbol \vertexList(\quiver{S})\\ \edgeList(\quiver{Q})&\subseteq \edgeList(\quiver{R})\cartesianProductSymbol \edgeList(\quiver{S})\end{aligned}

Defining a particular quiver product involves specifying which subsets of the Cartesian products we choose. But we must also attach a cardinal structure to the product $$\quiver{Q}$$.

Terminology

To talk clearly about the vertices, edges, and cardinals of $$\quiver{Q}$$, and contrast them with the vertices, edges, and cardinals of $$\quiver{R}$$ and $$\quiver{S}$$, we will use the following terms:

zzz
factor quiversf-quivers$$\quiver{R}$$ and $$\quiver{S}$$
factor vertexf-vertexvertex of $$\quiver{R}$$ or $$\quiver{S}$$
factor edgef-edgeedge of $$\quiver{R}$$ or $$\quiver{S}$$
factor cardinalf-cardinalcardinal of $$\quiver{R}$$ or $$\quiver{S}$$
product quiverp-quiver$$\quiver{Q}$$
product vertexp-vertexvertex of $$\quiver{Q}$$, constructed from f-vertices
product edgep-edgeedge of $$\quiver{Q}$$, constructed from f-edges
product cadinalp-cardinalcardinal of $$\quiver{Q}$$, constructed from f-cardinals

We will not use tuples $$\tuple{\vert{u},\vert{v}}$$ to represent the results of product constructions as is typically done: when writing a product vertex that is constructed out of factor vertices $$\vert{u}$$ and $$\vert{v}$$, we'll use the syntax $$\vert{u} \vertexProductSymbol \vert{v}$$, and similarly for product edges and product cardinals.

Color

To ease the notational burden and avoid unnecessary use of subscripts, we'll use color to associate vertices, edges, and cardinals with their corresponding graphs in a product. For example, for quivers $$\quiver{\rform{R}},\quiver{\bform{S}}$$ we'll refer generically to particular vertices belonging to $$\quiver{\rform{R}}$$ and $$\quiver{\bform{S}}$$ with symbols $$\vert{\rform{r}}$$ and $$\vert{\bform{s}}$$. Likewise we'll refer to particulars edges with $$\edge{\rform{e}},\edge{\bform{f}}$$, and refer to particular cardinals with $$\card{\rform{c}},\card{\bform{d}}$$.

We'll also use color to relate the product vertices and edges with factor vertices and edges, similar to the section "Covers". Of course we'll announce when we switch the role of color switches between these two aspects.

When denoting vertices that serve as the head or tail of some edge, we'll use $$\hvert{v}$$ and $$\tvert{v}$$ respectively. Therefore an edge will be written $$\de{\tvert{v}}{\hvert{v}}$$ – allowing us to write the reversal of this edge as $$\de{\hvert{v}}{\tvert{v}}$$. This convention is similar to the convention of using $$\sym{x}$$ and $$\primed{\sym{x}}$$ to denote two distinct variables related by some transformation.

Dependent quiver product

We'll introduce a binary quiver product called the dependent quiver product, denoted $$\graphProductSymbol$$. This product is related to the notion of the tensor graph product $$\otimes$$ of undirected graphs.

The dependent quiver product $$\quiver{\rform{R}}\dependentQuiverProductSymbol \quiver{\bform{S}}$$ of two quivers $$\quiver{\rform{R}}$$, $$\quiver{\bform{S}}$$ is the graph constructed as follows:

\begin{aligned} \vertexList(\quiver{\rform{R}}\dependentQuiverProductSymbol \quiver{\bform{S}})&\defEqualSymbol \setConstructor{\vert{\rform{r}} \vertexProductSymbol \vert{\bform{s}}}{\begin{array}{c} \elemOf{\vert{\rform{r}}}{\vertexList(\quiver{\rform{R}})}\\ \elemOf{\vert{\bform{s}}}{\vertexList(\quiver{\bform{S}})} \end{array} }\\ \\ \edgeList(\quiver{\rform{R}}\dependentQuiverProductSymbol \quiver{\bform{S}})&\defEqualSymbol \setConstructor{\ltde{\tvert{\rform{r}} \vertexProductSymbol \tvert{\bform{s}}}{\hvert{\rform{r}} \vertexProductSymbol \hvert{\bform{s}}}{\card{\rform{c}} \cardinalProductSymbol \card{\bform{d}}}}{\begin{array}{c} \elemOf{\tde{\tvert{\rform{r}}}{\hvert{\rform{r}}}{\card{\rform{c}}}}{\edgeList(\quiver{\rform{R}})}\\ \elemOf{\tde{\tvert{\bform{s}}}{\hvert{\bform{s}}}{\card{\bform{d}}}}{\edgeList(\quiver{\bform{S}})} \end{array} }\end{aligned}

It's easy to check that the product has a number of vertices and edges given by:

\begin{aligned} \setCardinality{\vertices{\quiver{\rform{R}}\dependentQuiverProductSymbol \quiver{\bform{S}}}}&= \setCardinality{\vertices{\quiver{\rform{R}}}} \, \setCardinality{\vertices{\quiver{\bform{S}}}}\\ \setCardinality{\edges{\quiver{\rform{R}}\dependentQuiverProductSymbol \quiver{\bform{S}}}}&= \setCardinality{\edges{\quiver{\rform{R}}}} \, \setCardinality{\edges{\quiver{\bform{S}}}}\end{aligned}

Visualizing graph products

We can visualize the graph product in ways that help clarify the product structure. We'll use two methods to do this:

• We will color vertices of the product by additive blending of the constituent vertices. For example, the product vertex $$\rform{\filledToken } \vertexProductSymbol \gform{\filledToken }$$ will be displayed as $$\rgform{\filledToken }$$.

• For graphs that can be layed out in one dimension, we will derive the $$\sym{y}$$ coordinate of $$\vert{u} \vertexProductSymbol \vert{v}$$ from the one-dimensional coordinate of $$\vert{u}$$ and the $$\sym{x}$$ coordinate of $$\vert{u} \vertexProductSymbol \vert{v}$$ from the one-dimensional coordinate of $$\vert{v}$$.

Let's examine the graph product $$\quiver{S}\dependentQuiverProductSymbol \quiver{R}$$, where $$\quiver{S} = \quiver{R} = \subSize{\lineQuiver }{2}$$:

Notice that the p-vertices $$\emptyToken \vertexProductSymbol \rform{\filledToken }$$ and $$\filledToken \vertexProductSymbol \bform{\filledToken }$$ are disconnected from the other vertices. The only p-edge is $$\de{\filledToken \vertexProductSymbol \rform{\filledToken }}{\emptyToken \vertexProductSymbol \bform{\filledToken }}$$, corresponding to the f-edges $$\de{\filledToken }{\emptyToken }$$ and $$\de{\rform{\filledToken }}{\bform{\filledToken }}$$.

Next we enlarge $$\quiver{S}$$, setting $$\quiver{S} = \subSize{\lineQuiver }{3}$$:

Negation

We'll now illustrate the four products $$\quiver{R}\graphProductSymbol \quiver{S},\inverted{\quiver{R}}\graphProductSymbol \quiver{S},\quiver{R}\graphProductSymbol \inverted{\quiver{S}},\inverted{\quiver{R}}\graphProductSymbol \inverted{\quiver{S}}$$, where $$\inverted{\quiver{Q}}$$ indicates the quiver derived from $$\quiver{Q}$$ whose edges have been reversed, or equivalently, whose cardinals have been inverted.

From this table we can notice a couple things:

• the top-left and bottom-right graphs are inverses: $$\inverted{\quiver{R}\graphProductSymbol \quiver{S}} = \inverted{\quiver{R}}\graphProductSymbol \inverted{\quiver{S}}$$

• the bottom-left and top-right graphs are inverses: $$\inverted{\inverted{\quiver{R}}\graphProductSymbol \quiver{S}} = \quiver{R}\graphProductSymbol \inverted{\quiver{S}}$$

These properties are general identities, easily verified.

Independent graph product

We now introduce the independent graph product of $$\quiver{\rform{R}}$$ and $$\quiver{\bform{S}}$$, written $$\quiver{\rform{R}}\independentQuiverProductSymbol \quiver{\bform{S}}$$, which is the graph constructed as follows:

\begin{aligned} \vertexList(\quiver{\rform{R}}\independentQuiverProductSymbol \quiver{\bform{S}})&\defEqualSymbol \setConstructor{\vert{\rform{r}} \vertexProductSymbol \vert{\bform{s}}}{\begin{array}{c} \elemOf{\vert{\rform{r}}}{\vertexList(\quiver{\rform{R}})}\\ \elemOf{\vert{\bform{s}}}{\vertexList(\quiver{\bform{S}})} \end{array} }\\ \\ \edgeList(\quiver{\rform{R}}\independentQuiverProductSymbol \quiver{\bform{S}})&\defEqualSymbol \setConstructor{\begin{array}{c} \ltde{\tvert{\rform{r}} \vertexProductSymbol \tvert{\bform{s}}}{\hvert{\rform{r}} \vertexProductSymbol \hvert{\bform{s}}}{\card{\rform{c}} \cardinalProductSymbol \card{\bform{d}}}\\ \ltde{\tvert{\rform{r}} \vertexProductSymbol \hvert{\bform{s}}}{\hvert{\rform{r}} \vertexProductSymbol \tvert{\bform{s}}}{\card{\rform{c}} \cardinalProductSymbol \inverted{\card{\bform{d}}}} \end{array} }{\begin{array}{c} \elemOf{\tde{\tvert{\rform{r}}}{\hvert{\rform{r}}}{\card{\rform{c}}}}{\edgeList(\quiver{\rform{R}})}\\ \elemOf{\tde{\tvert{\bform{s}}}{\hvert{\bform{s}}}{\card{\bform{d}}}}{\edgeList(\quiver{\bform{S}})} \end{array} }\end{aligned}

The intuitive meaning of this is: we can transition from a p-vertex to another p-vertex by choosing transitions for its f-vertices. We choose such transitions by choosing f-cardinals in the corresponding graphs. These f-cardinals can be inverted, or not. For the dependent quiver product, these choices were not indepenent: we could either take the p-cardinal $$\card{\rform{c}} \cardinalProductSymbol \card{\bform{d}}$$ or its p-cardinal $$\inverted{\card{\rform{c}} \cardinalProductSymbol \card{\bform{d}}} = \inverted{\card{\rform{c}}} \cardinalProductSymbol \inverted{\card{\bform{d}}}$$.

For the independent quiver product, these choices are independent: we can take any of the p-cardinals $$\list{\card{\rform{c}} \cardinalProductSymbol \card{\bform{d}},\card{\rform{c}} \cardinalProductSymbol \inverted{\card{\bform{d}}},\inverted{\card{\rform{c}}} \cardinalProductSymbol \card{\bform{d}},\inverted{\card{\rform{c}}} \cardinalProductSymbol \inverted{\card{\bform{d}}}}$$ – however, up to inversion there are only two choices, being (for example) $$\list{\card{\rform{c}} \cardinalProductSymbol \card{\bform{d}},\card{\rform{c}} \cardinalProductSymbol \inverted{\card{\bform{d}}}}$$, corresponding to the two f-edges in the definition seen above.

Visualization

Let's visualize the independent quiver product and contrast it with the dependent quiver product. Let's start with the same products we considered earlier:

Notice that we no longer have vertices that are totally disconnected. However this graph still has two weakly-connected components: the ones associated with the non-inverted "forward" f-cardinals, and the other with a choice of f-cardinals in which one of them is inverted.

Let's enlarge $$\quiver{S}$$, setting $$\quiver{S} = \subSize{\lineQuiver }{3}$$:

We're now using multiple arrowheads to illustrate the f-cardinal structure of the edges of $$\quiver{R}\independentQuiverProductSymbol \quiver{S}$$. Again we see two connected components, which correspond in a certain sense to the "parity" of the f-vertex in $$\quiver{R}$$ relative to some f-vertex in $$\quiver{S}$$. For example, if we consider a component of $$\quiver{R}\independentQuiverProductSymbol \quiver{S}$$ that includes p-vertex $$\filledToken \vertexProductSymbol \rform{\filledToken }$$, then we can only ever reach the f-vertex $$\emptyToken$$ after an odd number of transitions, but we can only ever reach f-vertex $$\rform{\filledToken }$$ after an even number of transitions. Hence this component is disconnected from the component that includes p-vertex $$\emptyToken \vertexProductSymbol \rform{\filledToken }$$.

Let's compare with the dependent product:

Now let's enlarge $$\quiver{R}$$ to contain 3 vertices:

Again we see two weakly connected components with the same underlying explanation.

Product notation

We notice that in the constructions of the dependent and independent quiver products, the vertices were always given by the Cartesian product of f-vertices. The only freedom came in how to construct edges: when choosing edges, we were able to decide how one should be allowed to choose f-cardinals from each of the corresponding graphs.

We can encode such choices in a compact notation we'll call arrow notation.

For a binary quiver product between $$\quiver{\rform{R}}$$ and $$\quiver{\bform{S}}$$, we write a kind of polynomial in variables $$\list{\rform{\forwardFactor },\rform{\backwardFactor },\bform{\forwardFactor },\bform{\backwardFactor }}$$. Each term of the polynomial represents a p-edge constructor, with the variable $$\rform{\forwardFactor }$$ representing a choice of f-cardinal from $$\quiver{\rform{R}}$$, and $$\rform{\backwardFactor }$$ a inverted f-cardinal. For example, the dependent product $$\rform{\quiver{R}}\dependentQuiverProductSymbol \bform{\quiver{S}}$$ is represented by the monomial $$\quiverProdPoly{\bform{\forwardFactor }\,\rform{\forwardFactor }}$$, because the f-cardinals from both graphs have the same sign in the p-edge.

What about the independent quiver product? It is $$\quiverProdPoly{\rform{\forwardFactor }\,\bform{\backwardFactor }+\rform{\forwardFactor }\,\bform{\forwardFactor }}$$, which we can also write as $$\quiverProdPoly{\rform{\forwardFactor }\,\paren{\bform{\backwardFactor }+\bform{\forwardFactor }}}$$.

We can also use this notation to describe quiver versions of two traditional graph products form the literature, the Cartesian product and the strong product.

Cartesian product

The p-edges of the Cartesian product are constructed by choosing only one f-edge, taken from either $$\quiver{\rform{R}}$$ or $$\quiver{\bform{S}}$$. If $$\rform{ \uarr }$$ is chosen, then the f-vertex for $$\quiver{\bform{S}}$$ does not experience a transition, and vice versa. In explicit notation, in other words:

$\setConstructor{\begin{array}{c} \ltde{\tvert{\rform{r}} \vertexProductSymbol \vert{\bform{s}}}{\hvert{\rform{r}} \vertexProductSymbol \vert{\bform{s}}}{\card{\rform{c}} \cardinalProductSymbol 1}\\ \ltde{\vert{\rform{r}} \vertexProductSymbol \tvert{\bform{s}}}{\vert{\rform{r}} \vertexProductSymbol \hvert{\bform{s}}}{1 \cardinalProductSymbol \card{\bform{d}}} \end{array} }{\begin{array}{c} \elemOf{\tde{\tvert{\rform{r}}}{\hvert{\rform{r}}}{\card{\rform{c}}}}{\edgeList(\quiver{\rform{R}})}\\ \elemOf{\tde{\tvert{\bform{s}}}{\hvert{\bform{s}}}{\card{\bform{d}}}}{\edgeList(\quiver{\bform{S}})} \end{array} }$

Using our abbreviated notation, this is written $$\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }}$$.

Strong product

The p-edges of the strong product are constructed by combining the p-edges of the Cartesian product and the dependent (or tensor) product:

$\setConstructor{\begin{array}{c} \ltde{\tvert{\rform{r}} \vertexProductSymbol \vert{\bform{s}}}{\hvert{\rform{r}} \vertexProductSymbol \vert{\bform{s}}}{\card{\rform{c}} \cardinalProductSymbol 1}\\ \ltde{\vert{\rform{r}} \vertexProductSymbol \tvert{\bform{s}}}{\vert{\rform{r}} \vertexProductSymbol \hvert{\bform{s}}}{1 \cardinalProductSymbol \card{\bform{d}}}\\ \ltde{\tvert{\rform{r}} \vertexProductSymbol \tvert{\bform{s}}}{\hvert{\bform{s}} \vertexProductSymbol \hvert{\bform{s}}}{\card{\rform{c}} \cardinalProductSymbol \card{\bform{d}}} \end{array} }{\begin{array}{c} \elemOf{\tde{\tvert{\rform{r}}}{\hvert{\rform{r}}}{\card{\rform{c}}}}{\edgeList(\quiver{\rform{R}})}\\ \elemOf{\tde{\tvert{\bform{s}}}{\hvert{\bform{s}}}{\card{\bform{d}}}}{\edgeList(\quiver{\bform{S}})} \end{array} }$

In our abbreviated notation, this is written $$\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }+\rform{\forwardFactor }\,\bform{\forwardFactor }}$$.

Application

When computing a product polynomial in the context of particular quivers, we'll use the following notation to indicate which arrow factors come from which quivers:

$\frac{\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }}} {\rform{\quiver{R}},\bform{\quiver{S}}}$

Summary

At this point we can summarize the above products in the following table:

zz
dependent quiver product$$\quiverProdPoly{\rform{\forwardFactor }\,\bform{\forwardFactor }}$$
independent quiver product$$\quiverProdPoly{\rform{\forwardFactor }\,\bform{\backwardFactor }+\rform{\forwardFactor }\,\bform{\forwardFactor }}$$
Cartesian product$$\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }}$$
strong product$$\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }+\rform{\forwardFactor }\,\bform{\forwardFactor }}$$

Algebraic properties

We can identify some algebraic properties of the generalized product, verifiable from the definition. Note that in all these cases we make the identification $$\paren{\vert{u} \vertexProductSymbol \vert{v}} \vertexProductSymbol \vert{w} = \vert{u} \vertexProductSymbol \paren{\vert{v} \vertexProductSymbol \vert{w}} = \vert{u} \vertexProductSymbol \vert{v} \vertexProductSymbol \vert{w}$$ for product vertices and cardinals.

Associativity

$\quiverProdPoly{\rform{\forwardFactor }\,\paren{\gform{\forwardFactor }\,\bform{\forwardFactor }}} = \quiverProdPoly{\paren{\rform{\forwardFactor }\,\gform{\forwardFactor }}\,\bform{\forwardFactor }} = \quiverProdPoly{\rform{\forwardFactor }\,\gform{\forwardFactor }\,\bform{\forwardFactor }}$

Distributivity

\begin{aligned} \quiverProdPoly{\paren{\gform{\forwardFactor }+\bform{\forwardFactor }}\,\rform{\forwardFactor }}&= \quiverProdPoly{\gform{\forwardFactor }\,\rform{\forwardFactor }+\bform{\forwardFactor }\,\rform{\forwardFactor }}\\ \quiverProdPoly{\rform{\forwardFactor }\,\paren{\gform{\forwardFactor }+\bform{\forwardFactor }}}&= \quiverProdPoly{\rform{\forwardFactor }\,\gform{\forwardFactor }+\rform{\forwardFactor }\,\bform{\forwardFactor }}\end{aligned}

Negation

\begin{aligned} \quiverProdPoly{\rform{\inverted{\forwardFactor }}}&= \rform{\backwardFactor }\\ \inverted{\quiverProdPoly{\rform{\forwardFactor }\,\bform{\forwardFactor }}}&= \quiverProdPoly{\rform{\backwardFactor }\,\bform{\backwardFactor }}\\ \inverted{\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }}}&= \quiverProdPoly{\rform{\backwardFactor }+\bform{\backwardFactor }}\end{aligned}

Lattice quiver products

We've now able to understand an interesting new perspective on the lattice quivers in terms of products.

Square lattice

Let's start with the Cartesian product $$\rform{ \uarr } + \bform{ \uarr }$$ between two finite line lattices $$\subSize{\lineQuiver }{\sym{n}}$$:

We recover a fragment of the square lattice quiver. We can write this formally as:

$\frac{\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }}} {\bindCards{\subSize{\lineQuiver }{\sym{n}}}{\rform{\card{r}}},\bindCards{\subSize{\lineQuiver }{\sym{n}}}{\bform{\card{b}}}} = \bindCards{\subSize{\squareQuiver }{\sym{n}}}{\rform{\card{r}},\bform{\card{b}}}$

Setting $$\sym{n} = \infty$$ in the above equation states that Cartesian product of two infinite line lattices gives in the infinite square lattice.

Triangular lattice

Here we show the product $$\rform{ \uarr } \, \gform{ \darr } + \gform{ \uarr } \, \bform{ \darr } + \bform{ \uarr } \, \rform{ \darr }$$ between three infinite line lattices $$\rform{\quiver{R}},\gform{\quiver{G}},\bform{\quiver{B}}$$. The product is a graph whose connected components are subgraphs of the triangular lattice. Here we show the product between three finite line lattices of length 5, and take the component containing the "center" product vertex $$0 \vertexProductSymbol 0 \vertexProductSymbol 0$$:

Here we color the p-cardinals by the additive blend of the graphs involved, so that $$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }}$$ is represented by the arrowhead $$\rgform{\arrowhead }$$, $$\poly{\gform{\forwardFactor } \, \bform{\backwardFactor }}$$ by $$\gbform{\arrowhead }$$, and $$\quiverProdPoly{\bform{\forwardFactor }\,\rform{\backwardFactor }}$$ by $$\rbform{\arrowhead }$$. The symbol $$\componentSuperQuiverOfSymbol$$ means "contains the connected component".

The full set of connected components correspond to slices through a larger object which we will describe at a later stage:

Hexagonal lattice

It might seem that we cannot decompose non-transitive lattice quivers, such as the hexagonal lattice, as products of transitive lattices. And this is true. But finite line lattices are not transitive: their two end vertices are different from the others. This leads to the following representation of the hexagonal lattice quiver:

$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }\,\waform{\forwardFactor }+\gform{\forwardFactor }\,\bform{\backwardFactor }\,\waform{\forwardFactor }+\bform{\forwardFactor }\,\rform{\backwardFactor }\,\waform{\forwardFactor }}$

Here $$\rform{\quiver{R}},\gform{\quiver{G}},\bform{\quiver{B}}$$ are copies of $$\subSize{\lineQuiver }{\sym{n}}$$ as before, but $$\wbform{\quiver{X}}$$ is $$\subSize{\lineQuiver }{2}$$.Taking $$\sym{n} = 8$$ we obtain:

Here is the full set of connected graph components, this time for $$\sym{n} = 6$$.

We can also interpret the factorization $$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }\,\waform{\forwardFactor }+\gform{\forwardFactor }\,\bform{\backwardFactor }\,\waform{\forwardFactor }+\bform{\forwardFactor }\,\rform{\backwardFactor }\,\waform{\forwardFactor }} = \quiverProdPoly{\paren{\rform{\forwardFactor }\,\gform{\backwardFactor }+\gform{\forwardFactor }\,\bform{\backwardFactor }+\bform{\forwardFactor }\,\rform{\backwardFactor }}\,\waform{\forwardFactor }}$$ as stating that we can obtain the hexagonal lattice quiver as the product of the triangular lattice quiver with the 2-line quiver. In fact, three copies of the triangular lattice are produced, shown below:

Note that the number of connected components (three) does not depend on the size of the $$\rform{\quiver{R}},\gform{\quiver{G}},\bform{\quiver{B}}$$ line lattices.

In general we can write:

$\frac{\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }\,\waform{\forwardFactor }+\gform{\forwardFactor }\,\bform{\backwardFactor }\,\waform{\forwardFactor }+\bform{\forwardFactor }\,\rform{\backwardFactor }\,\waform{\forwardFactor }}} {\bindCards{\subSize{\lineQuiver }{ \infty }}{\rform{\card{r}}},\bindCards{\subSize{\lineQuiver }{ \infty }}{\gform{\card{g}}},\bindCards{\subSize{\lineQuiver }{ \infty }}{\bform{\card{b}}},\bindCards{\subSize{\lineQuiver }{2}}{\wbform{\card{w}}}}\componentSuperQuiverOfSymbol \bindCards{\subSize{\hexagonalQuiver }{ \infty }}{\rform{\card{r}},\gform{\card{g}},\bform{\card{b}}}$

Rhombille lattice

Extending $$\wbform{\quiver{X}}$$ to be a 3-line lattice yields the rhombille lattice:

Again, the factorization gives us an alternative way of building the rhombille lattice, again showing the three connected components:

Extending $$\wbform{\quiver{X}}$$ to be a 4-line lattice yields the "alternating rhombille lattice", which involves a similar motif to the rhombille lattice, in which vertices alternate between degree 3 and degree 6:

Returning to the square lattice, we apply the same technique to obtain non-transitive versions of the square lattice. Here, we compute:

$\quiverProdPoly{\rform{\forwardFactor }\,\waform{\forwardFactor }+\bform{\forwardFactor }\,\waform{\forwardFactor }}$

varying $$\wbform{\quiver{X}}$$ between a 2-line lattice and a 5-line lattice:

Decomposition of products

In the previous section we decomposed the non-transitive hexagonal and rhombille lattices into their connected components. We now extend this technique to the earlier products we examined, so we can better understand how they produce fragments of corresponding lattice quivers.

As before, we'll visualize these as by superimposing each connected component on top of the full union of all conneced components, shown dimmed. Note that it may appear that the union itself is fully connected, but when this occurs it is an artefact of the projection onto two dimensions.

Square decomposition

The Cartesian product $$\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }}$$ decomposition is trivial, since the Cartesian product of two line lattices yields a single connected graph.

Triangular decomposition

The triangular product $$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }+\gform{\forwardFactor }\,\bform{\backwardFactor }+\bform{\forwardFactor }\,\rform{\backwardFactor }}$$ is more interesting, yielding a "stack" of disconnected components:

These components correspond to angled slices through the vertices of a cubic grid. Here we show a smaller fragment to avoid clutter, from an angle to emphasize the separate planes that yield the connected components:

And from another angle to emphasize the hexagonal structure of each plane:

Hexagonal decomposition

The hexagonal product $$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }\,\waform{\forwardFactor }+\gform{\forwardFactor }\,\bform{\backwardFactor }\,\waform{\forwardFactor }+\bform{\forwardFactor }\,\rform{\backwardFactor }\,\waform{\forwardFactor }}$$ is similar to the triangular product, except with more slices possible:

The corresponding higher-dimensional object of which the connected components are slices is 4-dimensional, and cannot easily be visualized. Notice that the appearance of two single vertex connected components above is a reflection of the fact that there are two p-vertices (corresponding to the two $$\wbform{\quiver{X}}$$-vertices) whose whose projections into 2 (and 3) dimensions co-incide.

Square products

Finally, we can decompose the product of the square lattice with a 2-line lattice:

Unlike the case with the product of the triangular lattice with the 2-line, we have a large number of connected components that depends on the size of the finite square lattice. We can see these components as slices of the following three-dimensional "slab" in which the $$\wbform{\quiver{X}}$$-axis has length 2, depicted going into the page:

Here the $$\wbform{\quiver{X}}$$-axis is depicted vertically, and the connected components are more easily seen:

For a 3-line lattice we obtain a "thicker stripe" that scans across the square:

This corresponds to slices of a three-dimensional slab where the $$\wbform{\quiver{X}}$$-axis has length 3, depicted going into the page.

Here the $$\wbform{\quiver{X}}$$-axis is depicted vertically, and the connected components are more easily seen as being intersections of the slab vertices with particular three-dimensional planes:

Extended product notation

Neutral factor

We've used notation like $$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }+\gform{\forwardFactor }\,\bform{\backwardFactor }+\bform{\forwardFactor }\,\rform{\backwardFactor }}$$ to describe generalized products between three distinct factor quivers. We've mentioned that product edges generated by monomials with two factors, like $$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }}$$, correspond to transitions that apply only to factor vertices from e.g. $$\rform{\quiver{R}}$$ and $$\gform{\quiver{G}}$$, leaving the factor vertex from $$\bform{\quiver{B}}$$ unchanged. We write the corresponding product cardinals as $$\rform{\card{r}} \cardinalProductSymbol \gform{\card{g}} \cardinalProductSymbol \bform{\card{1}}$$, where $$\bform{\card{1}}$$ indicates the absense of a factor cardinal from $$\bform{\quiver{B}}$$. But the notation makes it difficult to interpret the meaning of a single monomial like $$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }}$$ in isolation, because it leaves ambiguous how many factor quivers are actually involved.

To remedy this ambiguity, we will use the "shorn" arrow $$\neutralFactor$$ to indicate such "neutral factor" edge constructors explicitly, so that for a product of three quivers all monomials are elements of the set $$\list{\rform{\forwardFactor },\rform{\neutralFactor },\rform{\backwardFactor }}\cartesianProductSymbol \list{\gform{\forwardFactor },\gform{\neutralFactor },\gform{\backwardFactor }}\cartesianProductSymbol \list{\bform{\forwardFactor },\bform{\neutralFactor },\bform{\backwardFactor }}$$. Using this convention, the Cartesian product $$\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }}$$ is written more explicitly as $$\quiverProdPoly{\rform{\forwardFactor }\,\bform{\neutralFactor }+\bform{\forwardFactor }\,\rform{\neutralFactor }}$$, and the triangular product $$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }+\gform{\forwardFactor }\,\bform{\backwardFactor }+\bform{\forwardFactor }\,\rform{\backwardFactor }}$$ as $$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }\,\bform{\neutralFactor }+\gform{\forwardFactor }\,\bform{\backwardFactor }\,\rform{\neutralFactor }+\bform{\forwardFactor }\,\rform{\backwardFactor }\,\gform{\neutralFactor }}$$.

Single-quiver products

It is interesting to examine the products we can form from a single quiver $$\rform{\quiver{R}}$$ using the constructors $$\list{\rform{\forwardFactor },\rform{\neutralFactor },\rform{\backwardFactor }}$$. They are shown below for a finite line quiver:

The action of the neutral factor constructor $$\rform{\neutralFactor }$$ is to entirely disconnect the quiver.

Two-quiver products

Let's examine the behavior of the neutral factor constructor for two-product quivers:

Three-quiver products

Finally we can look at the terms of the triangular product. Recall that the product $$\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }\,\bform{\neutralFactor }+\gform{\forwardFactor }\,\bform{\backwardFactor }\,\rform{\neutralFactor }+\bform{\forwardFactor }\,\rform{\backwardFactor }\,\gform{\neutralFactor }}$$ on line lattices $$\rform{\quiver{R}},\gform{\quiver{G}},\bform{\quiver{B}}$$ produces distinct connected components, again we will show only the component containing the “center” product vertex $$0 \vertexProductSymbol 0 \vertexProductSymbol 0$$. We will examine the products as we add terms one-by-one:

Clearly the one-product terms are line lattices, and the two-term products are square lattices. We will see shortly how to make this more manifest by a suitable factorization.

Powers

We are ready now to consider powers of factors like $$\quiverProdPoly{\rform{\forwardFactor }},\quiverProdPoly{\rform{\backwardFactor }}$$ allowing terms such as $$\quiverProdPoly{\rform{\forwardFactor }\,\rform{\forwardFactor }},\quiverProdPoly{\rform{\forwardFactor }\,\rform{\forwardFactor }\,\rform{\forwardFactor }},\ellipsis$$ and $$\quiverProdPoly{\rform{\backwardFactor }\,\rform{\backwardFactor }},\quiverProdPoly{\rform{\backwardFactor }\,\rform{\backwardFactor }\,\rform{\backwardFactor }},\ellipsis$$. We first describe $$\quiverProdPoly{\rform{\forwardFactor }\,\rform{\forwardFactor }}$$ and consider an example.

For quiver $$\rform{\quiver{R}}$$ we define the monomial $$\quiverProdPoly{\rform{\forwardFactor }\,\rform{\forwardFactor }}$$ to be the quiver $$\rform{\quiver{R}_2}$$ defined as follows:

\begin{aligned} \vertexList(\rform{\quiver{R}_2})&\defEqualSymbol \vertexList(\rform{\quiver{R}})\\ \\ \edgeList(\rform{\quiver{R}_2})&\defEqualSymbol \setConstructor{\ltde{\tvert{r}}{\hvert{r}}{\card{c_1} \cardinalSequenceSymbol \card{c_2}}}{\elemOf{\tde{\tvert{r}}{\vert{m}}{\card{c_1}},\tde{\vert{m}}{\hvert{r}}{\card{c_2}}}{\edgeList(\rform{\quiver{R}})}}\end{aligned}

In other words, the product edges of $$\rform{\quiver{R}_2}$$ are 2-paths formed from (non-inverted) factor cardinals in $$\rform{\quiver{R}}$$. The corresponding product cardinals are ordered 2-lists $$\card{c_1} \cardinalSequenceSymbol \card{c_2}$$ of the corresponding factor cardinals. This is essentially identical to taking the square of the cardinal adjacency matrix.

Line lattice

Here is $$\rform{\quiver{R}_2}$$ visualized for $$\rform{\quiver{R}} = \bindCards{\subSize{\lineQuiver }{6}}{\rform{\card{r}}}$$. It splits into two isomorphic connected components:

For length $$\rform{\quiver{R}} = \bindCards{\subSize{\lineQuiver }{5}}{\rform{\card{r}}}$$, the two components are not isomorphic:

For $$\rform{\quiver{R}} = \bindCards{\subSize{\cycleQuiver }{6}}{\rform{\card{r}}}$$, we again obtain two connected components:

For $$\rform{\quiver{R}} = \bindCards{\subSize{\cycleQuiver }{5}}{\rform{\card{r}}}$$, we again obtain one connected component:

This general pattern is easy to state:

zzz
finite line quiver$$\subSize{\lineQuiver }{\sym{n}}$$$$\subSize{\lineQuiver }{\ceiling{\sym{n} / 2}}\graphUnionSymbol \subSize{\lineQuiver }{\floor{\sym{n} / 2}}$$
infinite line quiver$$\subSize{\lineQuiver }{ \infty }$$$$\subSize{\lineQuiver }{ \infty }\graphUnionSymbol \subSize{\lineQuiver }{ \infty }$$
even cycle quiver$$\subSize{\cycleQuiver }{2 \, \sym{n}}$$$$\subSize{\cycleQuiver }{2 \, \sym{n}}$$
odd cycle quiver$$\subSize{\cycleQuiver }{2 \, \sym{n} + 1}$$$$\subSize{\cycleQuiver }{\sym{n}}\graphUnionSymbol \subSize{\cycleQuiver }{\sym{n}}$$

Square lattice

For $$\quiver{Q}$$ a square quiver, which has multiple cardinals $$\cardinalList(\quiver{Q}) = \list{\rform{\card{r}},\bform{\card{b}}}$$, the construction for $$\quiver{Q}_2$$ is a little more complex. Here, the product cardinals are constructed from all possible pairs of non-inverted cardinals:

$\cardinalList(\quiver{Q_2}) = \list{\rform{\card{r}} \cardinalSequenceSymbol \rform{\card{r}},\rform{\card{r}} \cardinalSequenceSymbol \bform{\card{b}},\bform{\card{b}} \cardinalSequenceSymbol \rform{\card{r}},\bform{\card{b}} \cardinalSequenceSymbol \bform{\card{b}}}$

We visualize $$\quiver{Q}_2$$ for a 5,5-square lattice:

The long horizontal and vertical edges above are the p-cardinals $$\rform{\card{r}} \cardinalSequenceSymbol \rform{\card{r}}$$ and $$\bform{\card{b}} \cardinalSequenceSymbol \bform{\card{b}}$$, and the short pairs of diagonal edges with identical head and tail are $$\rform{\card{r}} \cardinalSequenceSymbol \bform{\card{b}}$$ and $$\bform{\card{b}} \cardinalSequenceSymbol \rform{\card{r}}$$.

Triangular lattice

The structure of the second power of the triangular lattice is complex and interesting, but will not be described further here.

Higher powers

We can define the general $$n^{\textrm{th}}$$ forward power of a quiver similarly, written as $$\quiverProdPoly{\quiverProdPower{\rform{\forwardFactor }}{\sym{n}}}\defEqualSymbol \parenLabeled{\quiverProdPoly{\rform{\forwardFactor }\,\rform{\forwardFactor }\,\quiver{\ellipsis }\,\rform{\forwardFactor }}}{\sym{n} \textrm{ times}}$$. The same constructions work as you would imagine for $$\quiverProdPoly{\quiverProdPower{\rform{\backwardFactor }}{\sym{n}}}$$.

Summary

Here are the four simple products we introduced in the last section, as applied to two line lattices. We also include some obvious generalizations for reference:

These examples make it clear that in the case of the products of infinite line lattices we choose to read the terms of the product as describing the roots of the resulting product, when see as a point lattice, in the sense of a subset of $$\mathbb{R}^2$$, rather than as a quiver. For example, the Cartesian lattice quiver $$\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }}$$ corresponds to the roots $$\list{\tuple{1,0},\tuple{0,1}}$$, whereas $$\quiverProdPoly{\rform{\forwardFactor }+\bform{\forwardFactor }+\rform{\forwardFactor }\,\bform{\forwardFactor }+\rform{\forwardFactor }\,\bform{\backwardFactor }}$$ corresponds to the roots $$\list{\tuple{1,0},\tuple{0,1},\tuple{1,1},\tuple{1,-1}}$$.