Kontodimopoulos N., Niakas D.'s A 12-year Analysis of Malmquist Total Factor Productivity in PDF

Kontodimopoulos N., Niakas D.'s A 12-year Analysis of Malmquist Total Factor Productivity in PDF

By Kontodimopoulos N., Niakas D.

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Extra info for A 12-year Analysis of Malmquist Total Factor Productivity in Dialysis Facilities

Example text

4. p (T ) ⊆ θ(Ult(B(T ))). Upper Semi-lattice Algebras and Combinatorics 33 Now, by 1. and 2. p (T ) ∪ {∅}. Moreover, by 1. we have ∅ = θu0 ∈ θ(Ult(B(T ))). p (T ) ∪ {∅}. 5. Let u ∈ Ult(B(T )) and set I = θu. 1. θu = ∅. By the same argument as in case i) 5. take t1 , . . , tn ∈ I, s1 , . . , sm ∈ / I and set / J}. p (T ) ∪ {∅} : t1 , . . , tn ∈ J and s1 , . . 2. θu = ∅. Let s1 , . . , sm ∈ T . p (T ) ∪ {∅} : s1 , . . , sm ∈ −1 (V(I)), where s is again the J}. Thus, b = m j=1 −bsj = 0 and u ∈ s(b) ⊆ θ Stone representation mapping.

Thus, there are t1 , . . , tn ∈ I0 , s1 , . . , sm ∈ / I0 so n m that, i=1 bti · − j=1 bsj = ∅. Hence, I0 ∈ U({t1 , . . , tn }, {s1 , . . p (T ) ∪ {∅}). p (T ) ∪ {∅} is closed in 2T . 2. p (T ) ∪ {∅}. p (T ) ∪ {∅}. i) implies ii). , V0 = {∅}. Thus, there are s1 , . . , sm ∈ T so that / I for i = 1, . . , m} = {∅}. p (T ) ∪ {∅} : si ∈ i=1 bsi . Otherwise, pick t ∈ T so that si t for all i = 1, . . , m. Let I :=↓ t. p (T ) ∪ {∅} with si ∈ / I for i = 1, . . , m. Thus, I ∈ V0 and I = ∅: Contradiction.

R \ ∅ is a disjunctive set: To this end we need to show: “For each n ≥ 1, for each r(ϕ, h), r(ϕ1 , h1 ), . . ” Let, n r(ϕ, h) ⊆ r(ϕi , hi ). (∗) i=1 Pick n minimal in (∗). If n = 1 we are done. , n ≥ 2), r(ϕ, h) n i=1,i=j r(ϕi , hi ) and r(ϕ, h) ∩ r(ϕj , hj ) = ∅, for every j ∈ {1, . . , n}. Hence, there are x1 , . . , xn ∈ r(ϕ, h) so that for each j ∈ {1, . . , n} xj ∈ r(ϕ, h) \ ∪ni=1,i=j r(ϕi , hi ) and since r(ϕ, h) ⊆ ∪ni=1 r(ϕi , hi ), it follows that xj ∈ r(ϕ, h) ∩ / r(ϕi , hi ) for i = j.

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A 12-year Analysis of Malmquist Total Factor Productivity in Dialysis Facilities by Kontodimopoulos N., Niakas D.


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