| Preface |
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xi | |
| Introduction |
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1 | (16) |
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Part I Species and operads |
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17 | (188) |
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Chapter 1 Hyperplane arrangements |
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19 | (54) |
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1.1 Faces, bifaces, flats |
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19 | (6) |
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1.2 Nested faces and lunes |
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25 | (2) |
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27 | (2) |
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1.4 Minimal galleries, distance functions, Varchenko matrices |
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29 | (5) |
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1.5 Incidence algebras, and zeta and Mobius functions |
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34 | (6) |
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1.6 Bilune-incidence algebra |
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40 | (6) |
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1.7 Descent, lune, Witt identities |
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46 | (3) |
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1.8 Noncommutative Zaslavsky formula |
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49 | (3) |
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1.9 Birkhoff algebra, Tits algebra, Janus algebra |
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52 | (10) |
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62 | (2) |
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1.11 Orientation space and signature space |
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64 | (2) |
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1.12 Lie and Zie elements |
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66 | (4) |
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70 | (1) |
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71 | (2) |
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Chapter 2 Species and bimonoids |
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73 | (61) |
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74 | (3) |
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2.2 Monoids, comonoids. bimonoids |
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77 | (4) |
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2.3 (Co)commutative (co)monoids |
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81 | (7) |
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2.4 Deformed bimonoids and signed bimonoids |
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88 | (3) |
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2.5 Signed (co)commutative (co)monoids |
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91 | (3) |
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2.6 Subspecies and quotient species |
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94 | (2) |
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2.7 (Co)abelianizations of (co)monoids |
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96 | (3) |
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2.8 Generating subspecies of monoids |
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99 | (1) |
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2.9 Duality functor on species |
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100 | (2) |
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2.10 Op and cop constructions |
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102 | (3) |
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2.11 Monoids, comonoids, bimonoids as functor categories |
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105 | (8) |
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2.12 Presentation for (co)monoids using covering generators |
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113 | (5) |
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2.13 Partially commutative monoids |
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118 | (3) |
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2.14 Set-species and set-bimonoids |
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121 | (2) |
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2.15 Bimonoids for a rank-one arrangement |
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123 | (1) |
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2.16 Joyal species and Joyal bimonoids |
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124 | (3) |
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127 | (7) |
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Chapter 3 Bimonads on species |
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134 | (34) |
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3.1 Bimonoids as bialgebras over a bimonad |
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135 | (5) |
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3.2 Bicommutative bimonoids as bialgebras over a bimonad |
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140 | (5) |
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3.3 Duality as a bilax functor |
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145 | (2) |
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3.4 Opposite transformation |
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147 | (3) |
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3.5 Lifting of monads to comonoids |
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150 | (1) |
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3.6 Monad for partially commutative monoids |
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151 | (3) |
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3.7 Bimonad for set-species |
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154 | (4) |
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3.8 Symmetries, braidings, lax braidings |
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158 | (4) |
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162 | (2) |
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3.10 Mesablishvili-Wisbauer |
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164 | (3) |
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167 | (1) |
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168 | (37) |
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169 | (1) |
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170 | (4) |
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174 | (1) |
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4.4 Connected and positive operads |
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175 | (1) |
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4.5 Commutative, associative, Lie operads |
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176 | (2) |
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178 | (1) |
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4.7 Hadamard product. Hopf operads |
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179 | (2) |
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4.8 Orientation functor and signature functor |
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181 | (1) |
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182 | (5) |
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4.10 Black and white circle products |
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187 | (3) |
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4.11 Left modules over operads |
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190 | (6) |
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4.12 Bioperads. Mixed distributive laws |
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196 | (3) |
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4.13 Incidence algebra of an operad |
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199 | (3) |
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4.14 Operads for LRB species |
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202 | (1) |
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203 | (2) |
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Part II Basic theory of bimonoids |
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205 | (324) |
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Chapter 5 Primitive nitrations and decomposable nitrations |
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207 | (28) |
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5.1 Cauchy powers of a species |
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208 | (3) |
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5.2 Graded and filtered bimonoids |
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211 | (3) |
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5.3 Primitive nitrations of comonoids |
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214 | (4) |
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5.4 Decomposable nitrations of monoids |
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218 | (3) |
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5.5 Trivial (co)monoids and (co)derivations |
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221 | (1) |
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5.6 Bimonoid axiom on the primitive part |
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222 | (4) |
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5.7 Primitively generated bimonoids and cocommutativity |
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226 | (3) |
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5.8 Browder-Sweedler and Milnor-Moore |
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229 | (3) |
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232 | (3) |
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Chapter 6 Universal constructions |
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235 | (48) |
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6.1 Free monoids on species |
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236 | (4) |
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6.2 Cofree comonoids on species |
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240 | (3) |
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6.3 (Co)free (co)commutative (co)monoids on species |
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243 | (7) |
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6.4 (Co)free bimonoids associated to species |
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250 | (5) |
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6.5 (Co)free (co)commutative bimonoids associated to species |
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255 | (4) |
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6.6 (Co)abelianizations of (co)free (co)monoids |
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259 | (2) |
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6.7 Primitive nitrations and decomposable nitrations |
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261 | (2) |
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6.8 Alternative descriptions of bimonoids |
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263 | (2) |
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265 | (5) |
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6.10 (Co)free graded (co)monoids on species |
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270 | (5) |
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6.11 Free partially bicommutative bimonoids |
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275 | (2) |
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277 | (6) |
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Chapter 7 Examples of bimonoids |
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283 | (52) |
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7.1 Species characteristic of chambers |
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284 | (1) |
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285 | (3) |
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288 | (6) |
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294 | (4) |
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7.5 Species of charts and dicharts |
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298 | (3) |
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301 | (11) |
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7.7 Species of top-nested faces and top-lunes |
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312 | (8) |
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320 | (6) |
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326 | (4) |
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330 | (5) |
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Chapter 8 Hadamard product |
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335 | (49) |
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336 | (6) |
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8.2 Internal horn for the Hadamard product |
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342 | (1) |
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8.3 Biconvolution bimonoids |
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343 | (4) |
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8.4 Internal horn for comonoids. Bimonoid of star families |
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347 | (8) |
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8.5 Species of chamber maps |
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355 | (3) |
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8.6 Universal measuring comonoids |
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358 | (5) |
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8.7 Enrichment of the category of monoids over comonoids |
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363 | (4) |
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8.8 Internal horn for monoids and bimonoids |
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367 | (8) |
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8.9 Hadamard product of set-species |
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375 | (1) |
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376 | (5) |
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381 | (3) |
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Chapter 9 Exponential and logarithm |
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384 | (58) |
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9.1 Exp-log correspondences |
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386 | (10) |
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9.2 Commutative exp-log correspondence |
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396 | (12) |
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9.3 Deformed exp-log correspondences |
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408 | (11) |
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9.4 0-exp-log correspondence |
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419 | (3) |
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9.5 Primitive and group-like series of bimonoids |
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422 | (5) |
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9.6 Primitive and group-like series of bicomm. bimonoids |
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427 | (4) |
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9.7 Comparisons between exp-log correspondences |
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431 | (2) |
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9.8 Formal power series. Series of Joyal species |
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433 | (6) |
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439 | (3) |
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Chapter 10 Characteristic operations |
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442 | (27) |
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10.1 Characteristic operations |
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443 | (9) |
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10.2 Commutative? characteristic operations |
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452 | (5) |
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10.3 Two-sided characteristic operations |
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457 | (2) |
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10.4 Set-theoretic characteristic operations |
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459 | (1) |
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10.5 Idempotent operators on bimonoid components |
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460 | (7) |
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467 | (2) |
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Chapter 11 Modules over monoid algebras and bimonoids in species |
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469 | (26) |
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11.1 Modules over the Tits algebra |
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470 | (2) |
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11.2 Modules over the Birkhoff algebra |
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472 | (1) |
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11.3 Modules over the Janus algebra |
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472 | (2) |
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474 | (2) |
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11.5 Duality and base change |
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476 | (3) |
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479 | (1) |
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11.7 A unified viewpoint via partial-biflats |
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480 | (1) |
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481 | (4) |
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11.9 Monoid-sets and bimonoids in set-species |
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485 | (1) |
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11.10 Bimonoids of h-faces and h-flats |
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486 | (7) |
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493 | (2) |
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495 | (34) |
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495 | (4) |
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12.2 Interaction with op and cop constructions |
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499 | (4) |
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12.3 Commutative Takeuchi formula |
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503 | (2) |
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12.4 Logarithm of the antipode |
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505 | (2) |
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507 | (5) |
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12.6 Antipodes of (co)free bimonoids |
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512 | (3) |
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12.7 Antipodes of (co)free (co)commutative bimonoids |
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515 | (3) |
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12.8 Takeuchi element and characteristic operations |
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518 | (5) |
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523 | (2) |
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525 | (4) |
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Part III Structure theory for bimonoids |
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529 | (188) |
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Chapter 13 Loday-Ronco, Leray-Samelson, Borel-Hopf |
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531 | (44) |
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13.1 Loday-Ronco for 0-bimonoids |
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534 | (4) |
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13.2 Leray-Samelson for bicommutative bimonoids |
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538 | (8) |
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13.3 Borel-Hopf for cocommutative bimonoids |
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546 | (10) |
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13.4 Borel-Hopf for commutative bimonoids |
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556 | (4) |
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13.5 Unification using partially bicommutative bimonoids |
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560 | (2) |
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13.6 Rigidity of q-bimonoids for q not a root of unity |
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562 | (8) |
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13.7 Monad for Lie monoids |
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570 | (1) |
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571 | (4) |
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Chapter 14 Hoffman-Newman-Radford |
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575 | (34) |
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14.1 Free O-bimonoids on comonoids |
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576 | (4) |
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14.2 Free bicomm. bimonoids on cocomm. comonoids |
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580 | (7) |
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14.3 Free O-~-bicommutative bimonoids |
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587 | (1) |
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14.4 Free bimonoids on cocommutative comonoids |
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588 | (9) |
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14.5 Free a-bimonoids on comonoids |
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597 | (8) |
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14.6 Zeta and Mobius as inverses |
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605 | (2) |
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607 | (2) |
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Chapter 15 Freeness under Hadamard products |
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609 | (29) |
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15.1 Freeness under Hadamard products |
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609 | (4) |
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15.2 Product of free and cofree bimonoids |
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613 | (10) |
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15.3 Product of free comm. and cofree cocomm. bimonoids |
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623 | (4) |
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15.4 Product of bimonoids with one free factor |
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627 | (2) |
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15.5 Species of pairs of chambers |
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629 | (7) |
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636 | (2) |
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638 | (40) |
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639 | (3) |
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16.2 Commutator bracket and primitive part of bimonoids |
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642 | (3) |
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16.3 Free Lie monoids on species |
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645 | (3) |
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16.4 Lie species and Zie species as Lie monoids |
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648 | (1) |
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16.5 Universal enveloping monoids |
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649 | (10) |
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659 | (1) |
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660 | (4) |
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664 | (7) |
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671 | (7) |
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Chapter 17 Poincare-Birkhoff-Witt and Cartier-Milnor-Moore |
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678 | (39) |
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17.1 Comonoid sections to the abelianization map |
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679 | (3) |
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17.2 Poincare-Birkhoff-Witt |
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682 | (5) |
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17.3 Projecting the free monoid onto the free Lie monoid |
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687 | (5) |
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692 | (6) |
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17.5 Cartier-Milnor-Moore |
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698 | (5) |
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17.6 Lie monoids for a rank-one arrangement |
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703 | (3) |
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706 | (4) |
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17.8 Lie monoids in LRB species |
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710 | (1) |
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711 | (6) |
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717 | (46) |
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719 | (4) |
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A.1 Kernel, cokernel, image, coimage |
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719 | (1) |
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A.2 Duality functor on vector spaces |
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719 | (1) |
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A.3 Internal hom for the tensor product |
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720 | (1) |
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A.4 Linear maps between direct sums of vector spaces |
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720 | (1) |
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721 | (2) |
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Appendix B Internal horn for monoidal categories |
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723 | (15) |
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B.1 Monoidal and 2-monoidal categories |
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723 | (2) |
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725 | (3) |
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728 | (2) |
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B.4 Internal horn for functor categories |
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730 | (3) |
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B.5 Modules over a monoid algebra |
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733 | (2) |
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735 | (3) |
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738 | (25) |
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738 | (14) |
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C.2 Higher monad algebras |
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752 | (6) |
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758 | (2) |
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760 | (3) |
| References |
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763 | (39) |
| List of Notations |
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802 | (10) |
| List of Tables |
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812 | (2) |
| Author Index |
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814 | (9) |
| Subject Index |
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823 | |
Daugiau apie elektronines knygas