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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.

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