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Some Related Sentences
Every and maximal

** Zorn's lemma
: Every non-empty partially ordered set
in which
every chain ( i
. e
. totally ordered subset )
has an upper bound contains at least one
maximal element
.

The restricted principle "
Every partially ordered set
has a maximal totally ordered subset "
is also equivalent
to AC over ZF
.

** Tukey's lemma
: Every non-empty collection
of finite character
has a maximal element with respect
to inclusion
.

** Antichain principle
: Every partially ordered set
has a maximal antichain
.

**
Every unital ring other
than the trivial ring contains
a maximal ideal
.
Every character
is automatically continuous
from A to C
, since
the kernel
of a character
is a maximal ideal
, which
is closed
.

*
Every cofinal subset
of a partially ordered set must contain all
maximal elements
of that set
.

* In any ring R
, a maximal ideal
is an ideal M
that is maximal in the set
of all proper ideals
of R
, i
. e
. M
is contained
in exactly 2 ideals
of R
, namely M itself
and the entire ring R
. Every maximal ideal
is in fact prime
.
Every simple R-module
is isomorphic
to a quotient R / m where m
is a maximal right ideal
of R
. By
the above paragraph
, any quotient R / m
is a simple module
.

* Krull's theorem ( 1929 ):
Every ring with
a multiplicative identity
has a maximal ideal
.
Every prime ideal P
in a Boolean ring R
is maximal: the quotient ring R / P
is an integral domain
and also
a Boolean ring
, so
it is isomorphic
to the field F < sub > 2 </ sub >, which shows
the maximality
of P
. Since
maximal ideals are always prime
, prime ideals
and maximal ideals coincide
in Boolean rings
.

*
Every non-empty set
of left ideals
of R
, partially ordered
by inclusion
, has a maximal element with respect
to set inclusion
.
Every maximal outerplanar graph with n
vertices has exactly 2n − 3 edges
, and every bounded
face of a maximal outerplanar graph is a triangle.
Every maximal outerplanar graph is the visibility
graph of a simple polygon
.
Every torus
is contained
in a maximal torus simply
by dimensional considerations
.

We call
a field E
a splitting field
for A if
A ⊗ E
is isomorphic
to a matrix ring over E
. Every finite dimensional CSA
has a splitting field
: indeed
, in the case when
A is a division algebra
, then
a maximal subfield
of A is a splitting field
.
Every graph contains at most 3 < sup > n / 3 </ sup >
maximal independent sets
, but many graphs have far fewer
.
Every perfect matching
is maximum
and hence
maximal.

*
Every localization
of R at
a maximal ideal
is a field
Every and outerplanar
Every outerplanar graph is a planar
graph.
Every outerplanar graph is also
a subgraph
of a series-parallel
graph.
Every forest
, and every cactus
graph is outerplanar.
Every outerplanar graph is a circle
graph, the intersection
graph of a set
of chords
of a circle
.
Every outerplanar graph can
be represented as an intersection
graph of axis-aligned rectangles
in the plane
, so
outerplanar graphs have boxicity at most two
.
Every and graph
Every connected graph is an expander ; however
, different
connected graphs have different expansion parameters
.

:"[...]
Every invariant
and co-variant thus becomes expressible
by a graph precisely identical with
a Kekuléan diagram or chemicograph
.

*
Every tree
is a bipartite
graph and a median
graph.
Every tree with only countably many
vertices is a planar
graph.

*
Every connected graph G admits
a spanning tree
, which
is a tree
that contains
every vertex of G
and whose edges are edges
of G
.

*
Every connected graph with only countably many
vertices admits
a normal spanning tree
.

In mathematics
, Tait's conjecture states
that "
Every 3-connected planar cubic
graph has a Hamiltonian
cycle ( along
the edges ) through all its
vertices ".
Every homomorphism
of the Petersen
graph to itself
that doesn't identify adjacent
vertices is an automorphism
.
Every directed acyclic
graph has a topological ordering
, an ordering
of the vertices such that the starting endpoint
of every edge occurs earlier
in the ordering
than the ending endpoint
of the edge.

*
Every planar
graph whose faces all have even
length is bipartite
.
Every vertex of this graph has an even degree
, therefore
this is an Eulerian
graph.
Every Eulerian orientation
of a connected graph is a strong orientation
, an orientation
that makes
the resulting directed
graph strongly
connected.
Every and satisfies

*
Every continuous functor on
a small-complete category which
satisfies the appropriate solution set
condition has a left-adjoint (
the Freyd adjoint functor theorem ).

#
Every finitely generated ideal
of A is principal ( i
. e.,
A is a Bézout domain )
and A satisfies the ascending chain
condition on principal ideals
.
Every odd
number q
satisfies for.
Every Boolean ring R
satisfies x ⊕ x
Every real x
satisfies the inequality
Every bounded positive-definite measure μ on G
satisfies μ ( 1 ) ≥ 0
. improved
this criterion
by showing
that it is sufficient
to ask
that, for every continuous positive-definite compactly supported function f on G
, the function Δ < sup >– ½ </ sup > f
has non-negative integral with respect
to Haar measure
, where Δ denotes
the modular function
.

* Let K < sup >
a </ sup >
be an algebraic closure
of K
containing L
. Every embedding σ
of L
in K < sup >
a </ sup > which restricts
to the identity on K
, satisfies σ ( L ) = L
. In other words
, σ
is an automorphism
of L over K
.

Lemma ( Singleton bound ):
Every linear code C
satisfies.

#
Every elementary substructure
of a model
of a theory T also
satisfies T ; hence
it is a submodel
.
0.302 seconds.