Every principal derivation d of L satisfies that the set d − 1 ( b ) = { x | d x ≤ b } has a greatest element for any b ∈ L. Every principal derivation d of L satisfies that the set d − 1 ( b ) = { x | d x ≤ b } has a greatest element for any b ∈ Fix d ( L ).
Assume, for every player i, the following conditions hold: 1. P i ( S i, x − i ) is a subset of ( U, ⪰ U ) and has a greatest element, for every x − i ∈ S − i ; 2.
(2) Every principal derivation d of L satisfies that the set d − 1 ( b ) = { x | d x ≤ b } has a greatest element for any b ∈ L. (3) Every principal derivation d of L satisfies that the set d − 1 ( b ) = { x | d x ≤ b } has a greatest element for any b ∈ Fix d ( L ). (4) Every principal derivation d of L satisfies that the set d − 1 ( b ) is a principal ideal of L for any b ∈ Fix d ( L ). .
Assume, for every player i, the following conditions hold: 1. P i ( S i, x − i ) is a subset of ( U, ⪰ U ) possessing its greatest element, for every x − i ∈ S − i ; 2. the best response function β i : S − i → 2 S i ∖ is order-increasing upward on ( S − i, ⪰ − i ) with compact values; 3.
P i ( S i, x − i ) is a subset of ( U, ⪰ U ) and has a greatest element, for every x − i ∈ S − i ; The best response function β i : S − i → 2 S i ∖ is order-increasing upward on ( S − i, ⪰ − i ) with universally inductive values; There are elements a = ( a 1, a 2, …, a n ) and b = ( b 1, b 2, …, b n ) in S with a ⪯ S b satisfying a i ∈ β i ( b − i ), for i = 1, 2, …, n.
P i ( S i, x − i ) is a subset of ( U, ⪰ U ) possessing its greatest element, for every x − i ∈ S − i ; the best response function β i : S − i → 2 S i ∖ is order-increasing upward on ( S − i, ⪰ − i ) with compact values; there are elements a = ( a 1, a 2, …, a n ) and b = ( b 1, b 2, …, b n ) in S with a ⪯ S b satisfying a i ∈ β i ( b − i ), for i = 1, 2, …, n.
Let b ∗ be the greatest element of d − 1 ( b ) for b ∈ Fix d ( L ).
For the Vaibhāṣikas, secondary properties such as color (varṇa) are in fact substances (dravya) similar to the four great elements.
But there is a greater element.
L has no greatest element.
