Semilinear Topological Spaces And Applications
Semilinear Topological Spaces And Applications
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Then (denoting ^f^(S) = N) the following are equivalent: 1. (0, 0) e D(S); 2. N is a commutative band (in fact an "unsealed" semivector subspace); 3. {T = ^^ (n) I n e N} is a partition of S into a semi- lattice of semivector subspaces T each of which distributes '^ n at (0, 0). Furthermore, if F is the field of reals, then each of the above is equivalent to: 4. N is pointwise convex.
Proof ; lit was noted earlier in this section that N is an "unsealed" semivector subspace of S].
ad (1 => 2) ; ...For any Os = n e N, Os « Os = (0 +0)s = Os = n, so that N consists of idempotents.
ad (2 => 3) ; [The T 's clearly partition S] .
Writing Sup(m, n) = m t n in (the semilattice) N, the blocks T form a semilattice by setting Sup(T, T ) = T^, .. N -^ * ^^ m' n^ Sup(m, n) Defining T «T ={t #t|t eT, t el}, also note m n m n'm mn n T * T ^ '^Cnr. Rm r>^*^ Let t, t' £ T . T is a semivector m n bup(jn, n; m m m m subspace, as 0(7it ) = m, whereby At e T ; and as 0(t ^ t') = m by idempotence of m, so that T is closed mm m under ^.
What to read after Semilinear Topological Spaces And Applications?
You can find similar books in the "Read Also" column, or choose other free books by Prem Prakash to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Then (denoting ^f^(S) = N) the following are equivalent: 1. (0, 0) e D(S); 2. N is a commutative band (in fact an "unsealed" semivector subspace); 3. {T = ^^ (n) I n e N} is a partition of S into a semi- lattice of semivector subspaces T each of which distributes '^ n at (0, 0). Furthermore, if F is the field of reals, then each of the above is equivalent to: 4. N is pointwise convex.
Proof ; lit was noted earlier in this section that N is an "unsealed" semivector subspace of S].
ad (1 => 2) ; ...For any Os = n e N, Os « Os = (0 +0)s = Os = n, so that N consists of idempotents.
ad (2 => 3) ; [The T 's clearly partition S] .
Writing Sup(m, n) = m t n in (the semilattice) N, the blocks T form a semilattice by setting Sup(T, T ) = T^, .. N -^ * ^^ m' n^ Sup(m, n) Defining T «T ={t #t|t eT, t el}, also note m n m n'm mn n T * T ^ '^Cnr. Rm r>^*^ Let t, t' £ T . T is a semivector m n bup(jn, n; m m m m subspace, as 0(7it ) = m, whereby At e T ; and as 0(t ^ t') = m by idempotence of m, so that T is closed mm m under ^.
What to read after Semilinear Topological Spaces And Applications?
You can find similar books in the "Read Also" column, or choose other free books by Prem Prakash to read onlineMoreLess
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