Definition 6

(conjugation) The family { μ π } subscript 𝜇 𝜋 \{\mu_{\pi}\} { italic_μ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT } of amplimorphisms admits a conjugation if there is for every μ 𝜇 \mu italic_μ an amplimorphism μ ¯ normal-¯ 𝜇 \bar{\mu} ¯ start_ARG italic_μ end_ARG and an intertwiner g ⁢ ( μ , μ ¯ ) ∈ 𝐸𝑛𝑑 ⁢ ( W μ ¯ ⊗ W μ , 𝐂 ) ⊗ 𝒦 𝑔 𝜇 normal-¯ 𝜇 tensor-product 𝐸𝑛𝑑 tensor-product subscript 𝑊 normal-¯ 𝜇 subscript 𝑊 𝜇 𝐂 𝒦 g(\mu,\bar{\mu})\in{\mbox{\it End\/}}(W_{\bar{\mu}}\otimes W_{\mu},{\bf C})% \otimes{\cal K} italic_g ( italic_μ , ¯ start_ARG italic_μ end_ARG ) ∈ End ( italic_W start_POSTSUBSCRIPT ¯ start_ARG italic_μ end_ARG end_POSTSUBSCRIPT ⊗ italic_W start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT , bold_C ) ⊗ caligraphic_K such that

g ⁢ ( μ , μ ¯ ) ⁢ μ ∘ μ ¯ ⁢ ( ξ ) = ξ ⁢ g ⁢ ( μ , μ ¯ ) 𝑔 𝜇 ¯ 𝜇 𝜇 ¯ 𝜇 𝜉 𝜉 𝑔 𝜇 ¯ 𝜇 g(\mu,\bar{\mu})\mu\circ{\bar{\mu}}(\xi)=\xi g(\mu,{\bar{\mu}}) italic_g ( italic_μ , ¯ start_ARG italic_μ end_ARG ) italic_μ ∘ ¯ start_ARG italic_μ end_ARG ( italic_ξ ) = italic_ξ italic_g ( italic_μ , ¯ start_ARG italic_μ end_ARG )
Definition 8

(partial derivatives) We adjoin partial derivatives ∂ = ∑ b ˙ ∂ b ˙ ⁡ e b ˙ subscript normal-˙ 𝑏 subscript normal-˙ 𝑏 superscript 𝑒 normal-˙ 𝑏 \partial=\sum_{\dot{b}}\partial_{\dot{b}}e^{\dot{b}} ∂ = ∑ start_POSTSUBSCRIPT ˙ start_ARG italic_b end_ARG end_POSTSUBSCRIPT ∂ start_POSTSUBSCRIPT ˙ start_ARG italic_b end_ARG end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT ˙ start_ARG italic_b end_ARG end_POSTSUPERSCRIPT to Ω normal-Ω \Omega roman_Ω . They are 𝒦 𝒦 {\cal K} caligraphic_K -covariant in the sense that

ξ ⁢ ∂ = ∂ ⁡ μ ¯ ⁢ ( ξ ) , 𝜉 ¯ 𝜇 𝜉 \xi\partial=\partial\bar{\mu}(\xi), italic_ξ ∂ = ∂ ¯ start_ARG italic_μ end_ARG ( italic_ξ ) , ( 2.22)

and are subject to the relations

∂ ⁡ ∂ ⁡ Π ¯ γ = 0 subscript ¯ Π 𝛾 0 \displaystyle\partial\partial\bar{\Pi}_{\gamma}=0 ∂ ∂ ¯ start_ARG roman_Π end_ARG start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT = 0 , absent \displaystyle, , ξ ⁢ ∂ = ∂ ⁡ μ ¯ ⁢ ( ξ ) 𝜉 ¯ 𝜇 𝜉 \displaystyle\ \ \ \xi\partial=\partial{\bar{\mu}}(\xi) italic_ξ ∂ = ∂ ¯ start_ARG italic_μ end_ARG ( italic_ξ ) ( 2.23)
Θ ⁢ ∂ Θ \displaystyle\Theta\partial roman_Θ ∂ = \displaystyle= = - γ ⁢ ∂ ⁡ Θ ⁢ ℛ ^ ⁢ ( μ ¯ , μ ) 𝛾 Θ ^ ℛ ¯ 𝜇 𝜇 \displaystyle-\gamma\partial\Theta\hat{\cal R}({\bar{\mu}},\mu) - italic_γ ∂ roman_Θ ^ start_ARG caligraphic_R end_ARG ( ¯ start_ARG italic_μ end_ARG , italic_μ ) ( 2.24)
∂ ⁡ Z 𝑍 \displaystyle\partial Z ∂ italic_Z = \displaystyle= = g ⁢ ( μ , μ ¯ ) - γ ⁢ Z ⁢ ∂ ⁡ R ^ ⁢ ( μ , μ ¯ ) 𝑔 𝜇 ¯ 𝜇 𝛾 𝑍 ^ 𝑅 𝜇 ¯ 𝜇 \displaystyle g(\mu,{\bar{\mu}})-\gamma Z\partial\hat{R}(\mu,{\bar{\mu}}) italic_g ( italic_μ , ¯ start_ARG italic_μ end_ARG ) - italic_γ italic_Z ∂ ^ start_ARG italic_R end_ARG ( italic_μ , ¯ start_ARG italic_μ end_ARG ) ( 2.25)
Definition 2

(1) The set of all functions

η : n = { 0 , … , n - 1 } → λ : 𝜂 𝑛 0 … 𝑛 1 → 𝜆 \eta\colon n=\{0,\ldots,n-1\}\rightarrow\lambda italic_η : italic_n = { 0 , … , italic_n - 1 } → italic_λ

( n ∈ ω ) 𝑛 𝜔 (n\in\omega) ( italic_n ∈ italic_ω ) is denoted λ < ω superscript 𝜆 absent 𝜔 {}^{<\omega}\!\lambda start_FLOATSUPERSCRIPT < italic_ω end_FLOATSUPERSCRIPT italic_λ ; the domain of η 𝜂 \eta italic_η is denoted ℓ ⁢ ( η ) normal-ℓ 𝜂 \ell(\eta) roman_ℓ ( italic_η ) and called the length of η 𝜂 \eta italic_η ; we identify η 𝜂 \eta italic_η with the sequence

⟨ η ⁢ ( 0 ) , η ⁢ ( 1 ) , … , η ⁢ ( n - 1 ) ⟩ . 𝜂 0 𝜂 1 … 𝜂 𝑛 1 \langle\eta(0),\eta(1),\ldots,\eta(n-1)\rangle. ⟨ italic_η ( 0 ) , italic_η ( 1 ) , … , italic_η ( italic_n - 1 ) ⟩ .

Define a partial ordering on λ < ω superscript 𝜆 absent 𝜔 {}^{<\omega}\!\lambda start_FLOATSUPERSCRIPT < italic_ω end_FLOATSUPERSCRIPT italic_λ by: η 1 ≤ η 2 subscript 𝜂 1 subscript 𝜂 2 \eta_{1}\leq\eta_{2} italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT if and only if η 1 subscript 𝜂 1 \eta_{1} italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is a restriction of η 2 subscript 𝜂 2 \eta_{2} italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT . This makes λ < ω superscript 𝜆 absent 𝜔 {}^{<\omega}\!\lambda start_FLOATSUPERSCRIPT < italic_ω end_FLOATSUPERSCRIPT italic_λ into a tree . For any η = ⟨ α 0 , … , α n - 1 ⟩ ∈ λ < ω 𝜂 subscript 𝛼 0 … subscript 𝛼 𝑛 1 superscript 𝜆 absent 𝜔 \eta=\langle\alpha_{0},\ldots,\alpha_{n-1}\rangle\in{}^{<\omega}\!\lambda italic_η = ⟨ italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_α start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ⟩ ∈ start_FLOATSUPERSCRIPT < italic_ω end_FLOATSUPERSCRIPT italic_λ , η ⌢ ⟨ β ⟩ ⌢ 𝜂 delimited-⟨⟩ 𝛽 \eta\smallfrown\langle\beta\rangle italic_η ⌢ ⟨ italic_β ⟩ denotes the sequence ⟨ α 0 , … , α n - 1 , β ⟩ subscript 𝛼 0 … subscript 𝛼 𝑛 1 𝛽 \langle\alpha_{0},\ldots,\alpha_{n-1},\beta\rangle ⟨ italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_α start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT , italic_β ⟩ . If S 𝑆 S italic_S is a subtree of λ < ω superscript 𝜆 absent 𝜔 {}^{<\omega}\!\lambda start_FLOATSUPERSCRIPT < italic_ω end_FLOATSUPERSCRIPT italic_λ , an element η 𝜂 \eta italic_η of S 𝑆 S italic_S is called a final node of S 𝑆 S italic_S if no η ⌢ ⟨ β ⟩ ⌢ 𝜂 delimited-⟨⟩ 𝛽 \eta\smallfrown\langle\beta\rangle italic_η ⌢ ⟨ italic_β ⟩ belongs to S 𝑆 S italic_S . Denote the set of final nodes of S 𝑆 S italic_S by S f subscript 𝑆 𝑓 S_{f} italic_S start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT .

(2) A λ 𝜆 \lambda italic_λ - set is a subtree S 𝑆 S italic_S of λ < ω superscript 𝜆 absent 𝜔 {}^{<\omega}\!\lambda start_FLOATSUPERSCRIPT < italic_ω end_FLOATSUPERSCRIPT italic_λ together with a cardinal λ η subscript 𝜆 𝜂 \lambda_{\eta} italic_λ start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT for every η ∈ S 𝜂 𝑆 \eta\in S italic_η ∈ italic_S such that λ ∅ = λ subscript 𝜆 𝜆 \lambda_{\emptyset}=\lambda italic_λ start_POSTSUBSCRIPT ∅ end_POSTSUBSCRIPT = italic_λ , and:

(a) for all η ∈ S 𝜂 𝑆 \eta\in S italic_η ∈ italic_S , η 𝜂 \eta italic_η is a final node of S 𝑆 S italic_S if and only if λ η = ℵ 0 ; subscript 𝜆 𝜂 subscript normal-ℵ 0 \lambda_{\eta}=\aleph_{0}; italic_λ start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT = roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ;

(b) if η ∈ S ∖ S f 𝜂 𝑆 subscript 𝑆 𝑓 \eta\in S\setminus S_{f}{} italic_η ∈ italic_S ∖ italic_S start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT , then η ⌢ ⟨ β ⟩ ∈ S normal-⌢ 𝜂 delimited-⟨⟩ 𝛽 𝑆 \eta\smallfrown\langle\beta\rangle\in S italic_η ⌢ ⟨ italic_β ⟩ ∈ italic_S implies β ∈ λ η 𝛽 subscript 𝜆 𝜂 \beta\in\lambda_{\eta} italic_β ∈ italic_λ start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT , λ η ⌢ ⟨ β ⟩ < λ η subscript 𝜆 normal-⌢ 𝜂 delimited-⟨⟩ 𝛽 subscript 𝜆 𝜂 \lambda_{\eta\smallfrown\langle\beta\rangle}<\lambda_{\eta} italic_λ start_POSTSUBSCRIPT italic_η ⌢ ⟨ italic_β ⟩ end_POSTSUBSCRIPT < italic_λ start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT and E η = { β < λ η : η ⌢ ⟨ β ⟩ ∈ S } subscript 𝐸 𝜂 conditional-set 𝛽 subscript 𝜆 𝜂 normal-⌢ 𝜂 delimited-⟨⟩ 𝛽 𝑆 E_{\eta}=\{\beta<\lambda_{\eta}\colon\eta\smallfrown\langle\beta\rangle\in S\} italic_E start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT = { italic_β < italic_λ start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT : italic_η ⌢ ⟨ italic_β ⟩ ∈ italic_S } is stationary in λ η subscript 𝜆 𝜂 \lambda_{\eta} italic_λ start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT .

(3) A λ 𝜆 \lambda italic_λ -system is a λ 𝜆 \lambda italic_λ -set together with a set B η subscript 𝐵 𝜂 B_{\eta} italic_B start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT for each η ∈ S 𝜂 𝑆 \eta\in S italic_η ∈ italic_S such that B ∅ = ∅ subscript 𝐵 B_{\emptyset}=\emptyset italic_B start_POSTSUBSCRIPT ∅ end_POSTSUBSCRIPT = ∅ , and for all η ∈ S ∖ S f 𝜂 𝑆 subscript 𝑆 𝑓 \eta\in S\setminus S_{f} italic_η ∈ italic_S ∖ italic_S start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT :

(a) for all β ∈ E η 𝛽 subscript 𝐸 𝜂 \beta\in E_{\eta} italic_β ∈ italic_E start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT , λ η ⌢ ⟨ β ⟩ ≤ | B η ⌢ ⟨ β ⟩ | < λ η ; subscript 𝜆 normal-⌢ 𝜂 delimited-⟨⟩ 𝛽 subscript 𝐵 normal-⌢ 𝜂 delimited-⟨⟩ 𝛽 subscript 𝜆 𝜂 \lambda_{\eta\smallfrown\langle\beta\rangle}\leq|B_{\eta\smallfrown\langle% \beta\rangle}|<\lambda_{\eta}; italic_λ start_POSTSUBSCRIPT italic_η ⌢ ⟨ italic_β ⟩ end_POSTSUBSCRIPT ≤ | italic_B start_POSTSUBSCRIPT italic_η ⌢ ⟨ italic_β ⟩ end_POSTSUBSCRIPT | < italic_λ start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ;

(b) { B η ⌢ ⟨ β ⟩ : β ∈ E η } conditional-set subscript 𝐵 normal-⌢ 𝜂 delimited-⟨⟩ 𝛽 𝛽 subscript 𝐸 𝜂 \{B_{\eta\smallfrown\langle\beta\rangle}\colon\beta\in E_{\eta}\} { italic_B start_POSTSUBSCRIPT italic_η ⌢ ⟨ italic_β ⟩ end_POSTSUBSCRIPT : italic_β ∈ italic_E start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT } is a continuous chain of sets, i.e.  if β ≤ β ′ 𝛽 superscript 𝛽 normal-′ \beta\leq\beta^{\prime} italic_β ≤ italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are in E η subscript 𝐸 𝜂 E_{\eta} italic_E start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT , then B η ⌢ ⟨ β ⟩ ⊆ B η ⌢ ⟨ β ′ ⟩ subscript 𝐵 normal-⌢ 𝜂 delimited-⟨⟩ 𝛽 subscript 𝐵 normal-⌢ 𝜂 delimited-⟨⟩ superscript 𝛽 normal-′ B_{\eta\smallfrown\langle\beta\rangle}\subseteq B_{\eta\smallfrown\langle\beta% ^{\prime}\rangle} italic_B start_POSTSUBSCRIPT italic_η ⌢ ⟨ italic_β ⟩ end_POSTSUBSCRIPT ⊆ italic_B start_POSTSUBSCRIPT italic_η ⌢ ⟨ italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟩ end_POSTSUBSCRIPT , and if σ 𝜎 \sigma italic_σ is a limit point of E η subscript 𝐸 𝜂 E_{\eta} italic_E start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT , then B η ⌢ ⟨ σ ⟩ = ∪ { B η ⌢ ⟨ β ⟩ : β < σ fragments subscript 𝐵 normal-⌢ 𝜂 delimited-⟨⟩ 𝜎 fragments normal-{ subscript 𝐵 normal-⌢ 𝜂 delimited-⟨⟩ 𝛽 normal-: β σ B_{\eta\smallfrown\langle\sigma\rangle}=\cup\{B_{\eta\smallfrown\langle\beta% \rangle}\colon\beta<\sigma italic_B start_POSTSUBSCRIPT italic_η ⌢ ⟨ italic_σ ⟩ end_POSTSUBSCRIPT = ∪ { italic_B start_POSTSUBSCRIPT italic_η ⌢ ⟨ italic_β ⟩ end_POSTSUBSCRIPT : italic_β < italic_σ , β ∈ E η } fragments β subscript 𝐸 𝜂 normal-} \beta\in E_{\eta}\} italic_β ∈ italic_E start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT } ;

(4) For any λ 𝜆 \lambda italic_λ -system Λ = ( S fragments Λ fragments normal-( S \Lambda=(S roman_Λ = ( italic_S , λ η subscript 𝜆 𝜂 \lambda_{\eta} italic_λ start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT , B η : η ∈ S ) fragments subscript 𝐵 𝜂 normal-: η S normal-) B_{\eta}\colon\eta\in S) italic_B start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT : italic_η ∈ italic_S ) , and any η ∈ S 𝜂 𝑆 \eta\in S italic_η ∈ italic_S , let B ¯ η = ∪ { B η ↾ m : m ≤ ℓ ⁢ ( η ) } subscript normal-¯ 𝐵 𝜂 conditional-set subscript 𝐵 normal-↾ 𝜂 𝑚 𝑚 normal-ℓ 𝜂 \bar{B}_{\eta}=\cup\{B_{\eta\restriction m}\colon m\leq\ell(\eta)\} ¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT = ∪ { italic_B start_POSTSUBSCRIPT italic_η ↾ italic_m end_POSTSUBSCRIPT : italic_m ≤ roman_ℓ ( italic_η ) } . Say that a family 𝒮 = { s ζ : ζ ∈ S f } 𝒮 conditional-set subscript 𝑠 𝜁 𝜁 subscript 𝑆 𝑓 {\cal S}=\{s_{\zeta}\colon\zeta\in S_{f}\} caligraphic_S = { italic_s start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT : italic_ζ ∈ italic_S start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT } of countable sets is based on Λ normal-Λ \Lambda roman_Λ if 𝒮 𝒮 {\cal S} caligraphic_S is indexed by S f subscript 𝑆 𝑓 S_{f}{} italic_S start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT and for every ζ ∈ S f 𝜁 subscript 𝑆 𝑓 \zeta\in S_{f}{} italic_ζ ∈ italic_S start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT , s ζ ⊆ B ¯ ζ subscript 𝑠 𝜁 subscript normal-¯ 𝐵 𝜁 s_{\zeta}\subseteq\bar{B}_{\zeta} italic_s start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ⊆ ¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT .

(5) A family 𝒮 = { s i : i ∈ I } 𝒮 conditional-set subscript 𝑠 𝑖 𝑖 𝐼 {\cal S}=\{s_{i}:i\in I\} caligraphic_S = { italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_i ∈ italic_I } is said to be free if it has a transversal, that is, a one-one function T : I → ∪ 𝒮 normal-: 𝑇 normal-→ 𝐼 𝒮 T:I\rightarrow\cup{\cal S} italic_T : italic_I → ∪ caligraphic_S such that for all i ∈ I 𝑖 𝐼 i\in I italic_i ∈ italic_I , T ⁢ ( i ) ∈ s i 𝑇 𝑖 subscript 𝑠 𝑖 T(i)\in s_{i} italic_T ( italic_i ) ∈ italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT . We say 𝒮 𝒮 {\cal S\ } caligraphic_S is λ 𝜆 \lambda italic_λ -free if every subset of 𝒮 𝒮 {\cal S} caligraphic_S of cardinality < λ absent 𝜆 <\lambda < italic_λ has a transversal.