Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.

Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows.

Any cyclotomic extension ( for either definition ) is abelian.

Any abelian variety is isogenous to a product of simple abelian varieties.

A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ ( y )= 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra and hence descends to a weight on the abelian Lie algebra g /.

* Any finite abelian group, with the discrete topology.

* Any ring R can be considered as a one-object preadditive category ; the category of left modules over R is the same as the additive functor category Add ( R, Ab ) ( where Ab denotes the category of abelian groups ), and the category of right R-modules is Add ( R < sup > op </ sup >, Ab ).

Any semisimple Lie algebra or abelian Lie algebra is a fortiori reductive.

The theorem states: Any linearly ordered abelian group can be embedded as an ordered subgroup of the additive group ℝ < sup > Ω </ sup > endowed with a lexicographical order, where ℝ is the additive group of real numbers ( with its standard order ), and Ω is the set of Archimedean equivalence classes of.

Any abelian von Neumann algebra of operators on a separable Hilbert space is *- isomorphic to exactly one of the following

Any abelian group can be turned into a zero ring by defining the product of any two elements to be 0.

Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j if and only if i ≤ j.

For 2009, the winner in the " least likely to succeed " category was " Any name of the decade 2000 – 2009, such as: Naughties, Aughties, Oughties, Pot stickers, etc.

:- Any agreement or category of agreements between undertakings

:- Any decision or category of decisions by associations of undertakings

:- Any concerted practice or category of concerted practices that improve the production or distribution of goods, or promotes technical or economic progress, while allowing consumers a fair share of the resulting benefit, and that does not:

Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

Any class can be viewed as a category whose only morphisms are the identity morphisms.

Any preordered set ( P, ≤) forms a small category, where the objects are the members of P, the morphisms are arrows pointing from x to y when x ≤ y.

Any ordinal number can be seen as a category when viewed as an ordered set.

Any monoid ( any algebraic structure with a single associative binary operation and an identity element ) forms a small category with a single object x.

Any group can be seen as a category with a single object in which every morphism is invertible ( for every morphism f there is a morphism g that is both left and right inverse to f under composition ) by viewing the group as acting on itself by left multiplication.

Any directed graph generates a small category: the objects are the vertices of the graph, and the morphisms are the paths in the graph ( augmented with loops as needed ) where composition of morphisms is concatenation of paths.

* quantification Any morphism f: X → Y in a category with pullbacks induces a monotonous map acting by pullbacks ( A monotonous map is a functor if we consider the preorders as categories ).

* Any partially ordered set can be interpreted as a category: the objects are the elements of, and there is a single morphism from to if and only if.

* Any topological space can be viewed as a category by taking the open sets as objects, and a single morphism between two open sets and if and only if.

* Any ring, thought of as a category with only one object, is a preadditive category.

Any finite product in a preadditive category must also be a coproduct, and conversely.

Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other.

# Any group G may be regarded as an " abstract " category with one object,, and one morphism for each element of the group.

Any directed poset can be considered as a small category where the morphisms consist of arrows if and only if.

Any ring can be seen as a preadditive category with a single object.

Any Other Weapons ( AOWs )— this is a broad " catch-all " category used to regulate any number of firearms which the BATFE under the NFA enforces registration and taxation.

Any organism that falters or misperceives the signals or weakens is done.

Any information we have here in Taiwan is at your disposal ''.

Any claimant whose claim is denied, or is approved for less than the full amount of such claim, shall be entitled, under such regulations as the Commission may prescribe, to a hearing before the Commission, or its duly authorized representatives, with respect to such claim.

Any alteration of one of these factors is distortion, although we generally use that word only for effects so pronounced that they can be stated quantitatively on the basis of standard tests.

Any musician playing Beethoven here, where Beethoven was born, is likely to examine his own interpretations with special care.

Any action is understood to create " seeds " in the mind that sprout into the appropriate results ( Pāli vipaka ) when they meet the right conditions.

** Any union of countably many countable sets is itself countable.

Any information may be conveyed by an analog signal ; often such a signal is a measured response to changes in physical phenomena, such as sound, light, temperature, position, or pressure.

Any quantity of energy absorbed by the electron in excess of this amount is converted to kinetic energy according to the conservation of energy.

Any breach of these laws is a criminal offence and can lead to fine of up to £ 5, 000 or six months in prison.

The prohibition against retaliation or coercion applies broadly to any individual or entity that seeks to prevent an individual from exercising his or her rights or to retaliate against him or her for having exercised those rights ... Any form of retaliation or coercion, including threats, intimidation, or interference, is prohibited if it is intended to interfere.

Any word or phrase that exactly reproduces the letters in another order is an anagram.

Any work ( δW ) done must be done at the expense of internal energy U, since no heat δQ is being supplied from the surroundings.

" Any threat of war Clovis could make would only be effective if they were neighbors ; " it is nowhere written that Syagrius was handed over in 486 or 487.

Any player may declare the game over at any time during his turn if either of two conditions is true: one chain has 41 or more tiles, or there is at least one chain on the board and every chain on the board has 11 or more tiles.

Any person who is seen as a threat to civil aviation is banned from flying.

* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.

* Any ring A is an algebra over its center Z ( A ), or over any subring of its center.

* Any commutative ring R is an algebra over itself, or any subring of R.

Any form of intelligent life would do ; so, specifying carbon-based life, per se, is irrelevant.

* Any base that is water-soluble and forms hydroxide ions or the solution of a base in water.