\( \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\lrgform#1{ {\textcolor{ffc87d}{#1}}} \gdef\lgbform#1{ {\textcolor{96e2e3}{#1}}} \gdef\lrbform#1{ {\textcolor{f989b8}{#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{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\factsym#1{\mathord{#1}} \gdef\forwardFactor{\factsym{\uparrow}} \gdef\backwardFactor{\factsym{\downarrow}} \gdef\forwardBackwardFactor{\factsym{\updownarrow}} \gdef\forwardBackwardNeutralFactor{\factsym{\mathrlap{\downarrow}{\mathrlap{\uparrow}{\endash} } }} \gdef\neutralFactor{\factsym{\shornarrow}} \gdef\arrowhead{⏵} \gdef\inverseArrowhead{⏴} \gdef\rightArrowhead{⏵} \gdef\leftArrowhead{⏴} \gdef\upArrowhead{⏶} \gdef\downArrowhead{⏷} \gdef\de#1#2{ { {#1}\xundirectededge{}{#2}}} \gdef\ue#1#2{ { {#1}\xrightedge{}{#2}}} \gdef\shifttag#1{\raisebox{-1em}{$#1$}} \gdef\shifttag#1{#1} \gdef\tde#1#2#3{ {#1}\:\xrightedge{\shifttag{#3}}\;{#2}} \gdef\tue#1#2#3{ {#1}\;\xundirectededge{\shifttag{#3}}\;{#2}} \gdef\shiftunderset#1#2{\underset{\raisebox{0.15em}{\scriptsize $#1$}}{#2}} \gdef\mdtde#1#2#3#4{ {#1}\,\;\shiftunderset{#4}{\xrightedge{#3}}\;{#2}} \gdef\mtue#1#2#3#4{ {#1}\,\;\shiftunderset{#4}{\xundirectededge{#3}}\;\,{#2}} \gdef\mtde#1#2#3#4{ {#1}\,\;\operatornamewithlimits{\xrightedge{#3}}\limits_{#4} \;{#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\parenPathWord#1#2#3{\left(\pathWord{#1}{#2}{#3}\right)} \gdef\nullPath{\bot} \gdef\nullElement{\bot} \gdef\path#1{#1} \gdef\vert#1{#1} \gdef\underdot#1{\underset{\raisebox{0.3em}{.}}{#1}} \gdef\headVertexSymbol{◨} \gdef\tailVertexSymbol{◧} \gdef\placeholderVertexSymbol{\mathrlap{◨}{◧}} \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\variable#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\wordRingElement#1{#1} \gdef\wordRingBasisElement#1{e_{#1}} \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\kroneckerDeltaForm#1{\kroneckerDeltaSymbol_{#1}} \gdef\kroneckerDeltaSymbol{\delta} \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\graphCovering#1#2#3{ {#2} \sqsupseteq_{#1} {#3}} \gdef\quiverCovering#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\groupoidIdentityElement#1#2{#1_{#2}} \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\rightIndependentQuiverProductSymbol{\mathbin{⋊}} \gdef\leftIndependentQuiverProductSymbol{\mathbin{⋉}} \gdef\rightStrongQuiverProductSymbol{\mathbin{⧒}} \gdef\leftStrongQuiverProductSymbol{\mathbin{⧑}} \gdef\rightFiberQuiverProductSymbol{\mathbin{⧕}} \gdef\leftFiberQuiverProductSymbol{\mathbin{⧔}} \gdef\lockedQuiverProductSymbol{\mathbin{\searrow}} \gdef\rightFreeQuiverProductSymbol{\mathbin{\smallerthan}} \gdef\leftFreeQuiverProductSymbol{\mathbin{\largerthan}} \gdef\strongIndependentQuiverProductSymbol{\mathbin{⨝}} \gdef\cartesianQuiverProductSymbol{\mathbin{□}} \gdef\strongQuiverProductSymbol{\mathbin{⊠}} \gdef\graphUnionSymbol{\sqcup} \gdef\graphProductSymbol{\times} \gdef\inlineProdSymbol{|} \gdef\serialCardSymbol{ {:}} \gdef\parallelCardSymbol{ {\mid}} \gdef\cardinalSequenceSymbol{ {:}} \gdef\cardinalProduct#1{(#1)} \gdef\vertexProduct#1{(#1)} \gdef\edgeProduct#1{(#1)} \gdef\cardinalProductSymbol{\inlineProdSymbol} \gdef\vertexProductSymbol{\inlineProdSymbol} \gdef\edgeProductSymbol{\inlineProdSymbol} \gdef\verticalVertexProduct#1#2{\cfrac{#1}{#2}} \gdef\verticalCardinalProduct#1#2{\cfrac{#1}{#2}} \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\indexGraphUnion#1#2#3{ {\bigcup_{#2}^{#3} #1}} \gdef\indexGraphDisjointUnion#1#2#3{ {\bigcup_{#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\starModifier#1{ {#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}} \)
Exceptional products

Exceptional products

In the previous section, quiver products, we defined a general family of products of quivers in which edges of the product quiver correspond to combinations of transitions in the factor quivers. We saw that we could obtain some of the familiar transitive quivers as simple products.

In this section, we will apply use products to obtain some intransitive lattices, as well as other more exotic objects.

Intransitive lattices

Triangular quiver

Recall from the previous section we could obtain countably many disjoint copies of the triangular quiver using the triangular product:

The triangular product is defined to be:

\[ \function{ \Delta }(\quiver{R},\quiver{G},\quiver{B})\defEqualSymbol \frac{\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }\,\bform{\neutralFactor }+\rform{\neutralFactor }\,\gform{\forwardFactor }\,\bform{\backwardFactor }+\rform{\backwardFactor }\,\gform{\neutralFactor }\,\bform{\forwardFactor }}} {\quiver{\rform{R}},\quiver{\gform{G}},\quiver{\bform{B}}} \]

The statement that we obtain countably many copies of the triangular lattice is then:

\[ \function{ \Delta }(\bindCardSize{\subSize{\lineQuiver }{ \infty }}{\rform{\card{r}}},\bindCardSize{\subSize{\lineQuiver }{ \infty }}{\gform{\card{g}}},\bindCardSize{\subSize{\lineQuiver }{ \infty }}{\bform{\card{b}}})\homeomorphicSymbol \indexGraphDisjointUnion{\bindCardSize{\subSize{\triangularQuiver }{ \infty }}{\rform{\card{r}},\gform{\card{g}},\bform{\card{b}}}}{\sym{i}}{ \infty } \]

Hexagonal quiver

It might seem that we cannot construct intransitive quivers, such as the hexagonal lattice, as products of transitive quivers. And this is true. But finite line lattices are not transitive: their two end vertices are different from the others. Surprisingly, we can construct the hexagonal lattice by including one additional factor quiver, being the 2-line lattice. Here is the product that accomplishes this, which we will call the hexagonal product:

\[ \function{\starModifier{ \Delta }}(\quiver{R},\quiver{G},\quiver{B},\quiver{X})\defEqualSymbol \function{ \Delta }(\quiver{R},\quiver{G},\quiver{B})\lockedQuiverProductSymbol \quiver{X} = \frac{\quiverProdPoly{\rform{\forwardFactor }\,\gform{\backwardFactor }\,\bform{\neutralFactor }\,\waform{\forwardFactor }+\rform{\neutralFactor }\,\gform{\forwardFactor }\,\bform{\backwardFactor }\,\waform{\forwardFactor }+\rform{\backwardFactor }\,\gform{\neutralFactor }\,\bform{\forwardFactor }\,\waform{\forwardFactor }}} {\quiver{\rform{R}},\quiver{\gform{G}},\quiver{\bform{B}},\wbform{\quiver{X}}} \]

Let's examine an example, taking \(\rform{\quiver{R}},\gform{\quiver{G}},\bform{\quiver{B}}\) to be copies of \(\subSize{\lineQuiver }{\sym{n}}\) as before, with \(\wbform{\quiver{X}}\) being \(\subSize{\lineQuiver }{2}\). For \(\sym{n} = 8\) we obtain:

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

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

An important fact is 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, which leads to the theorem:

\[ \bindCardSize{\subSize{\triangularQuiver }{ \infty }}{\rform{\card{r}},\gform{\card{g}},\bform{\card{b}}}\lockedQuiverProductSymbol \bindCardSize{\subSize{\lineQuiver }{2}}{\waform{\card{x}}}\homeomorphicSymbol \indexGraphDisjointUnion{\bindCardSize{\subSize{\hexagonalQuiver }{ \infty }}{\rform{\card{r}},\gform{\card{g}},\bform{\card{b}}}}{\sym{i}}{3} \]

Since the triangular product produces countably infinite copies of the triangular lattice, we also have this property of the hexagonal product:

\[ \function{\starModifier{ \Delta }}(\bindCardSize{\subSize{\lineQuiver }{ \infty }}{\rform{\card{c}}},\bindCardSize{\subSize{\lineQuiver }{ \infty }}{\gform{\card{g}}},\bindCardSize{\subSize{\lineQuiver }{ \infty }}{\bform{\card{b}}},\bindCardSize{\subSize{\lineQuiver }{2}}{\waform{\card{w}}})\homeomorphicSymbol \indexGraphDisjointUnion{\bindCardSize{\subSize{\hexagonalQuiver }{ \infty }}{\rform{\card{r}},\gform{\card{g}},\bform{\card{b}}}}{\sym{i}}{ \infty } \]

Rhombille quiver

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 connected 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 product vertices (corresponding to the two \(\wbform{\quiver{X}}\)-vertices) whose whose projections into 2 (and

  1. dimensions coincide.

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

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:

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

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 product 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.

Interaction of \(\forwardFactor\) and \(\backwardFactor\)

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 \(\list{\tuple{1,0},\tuple{0,1},\tuple{1,1},\tuple{1,-1}}\).