- theory, but also topology,
whence they originate. The
notion dual to a
filter is an
order ideal.
Special cases of
filters include ultrafilters,
which are...
- =
DualIdeals ( X ) ∖ { ℘ ( X ) } ⊆
Prefilters ( X ) ⊆
FilterSubbases ( X ) . {\displaystyle {\textrm {Filters}}(X)\quad =\quad {\textrm {
DualIdeals}}(X)\...
- {S} ,\leq ).} Let ∅ ≠ F ⊆
DualIdeals ( X ) {\displaystyle \varnothing \neq \mathbb {F} \subseteq \operatorname {
DualIdeals} (X)} and let ∪ F = ⋃ F ∈...
- are
exactly ideals in the ring-theoretic
sense on the
Boolean ring
formed by the
powerset of the
underlying set. The
dual notion of an
ideal is a filter...
-
Maximal right/left/two-sided
ideals are the
dual notion to that of
minimal ideals. If F is a field, then the only
maximal ideal is {0}. In the ring Z of integers...
- Briefly, a sigma-
ideal must
contain the
empty set and
contain subsets and
countable unions of its elements. The
concept of 𝜎-
ideal is
dual to that of a countably...
-
nonreal dual numbers. Indeed, they are (trivially) zero
divisors and
clearly form an
ideal of the ****ociative
algebra (and thus ring) of the
dual numbers...
-
number theory, the
different ideal (sometimes
simply the different) is
defined to
measure the (possible) lack of
duality in the ring of
integers of an...
-
simply a
lower set. Similarly, an
ideal can also be
defined as a "directed
lower set". The
dual notion of an
ideal, i.e., the
concept obtained by reversing...
-
Never 𝜎-Ring
Never Algebra (Field)
Never 𝜎-Algebra (𝜎-Field)
Never Dual ideal Filter Never Never ∅ ∉ F {\displaystyle \varnothing \not \in {\mathcal...
