''composition sequence''
A composition series is a finite subset $\{a_0,\ldots,a_n\}$ of a partially ordered set w
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...nger decomposable into a direct sum. A module has a [[Composition sequence|composition series]] if and only if it is both Artinian and [[Noetherian module|Noether
|valign="top"|{{Ref|Fa}}||valign="top"| C. Faith, "Algebra: rings, modules, and categories" , '''1''' , Springer (1973)
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...d only if it is finitely generated. A module has a [[
Composition sequence|
composition series]] if and only if it is both [[Artinian module|Artinian]] and Noether
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lang, "
Algebra" , Addison-Wesley (1974)</TD></TR></table>
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...Quaternion|Quaternion]]) over the same field. Namely, corresponding to the algebra with basis $ [ 1 , i _ {1} , i _ {2} , i _ {3} ] $,
...quadratic forms corresponding to quaternion algebras, and only for these, composition of quaternary quadratic forms is defined:
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...other words, it was proved that the sequence of orders of quotients of two composition (chief) series are the same up to a permutation. Hölder proved that the co
...ies of groups with simple factors are isomorphic (these series need not be composition series) (see [[#References|[5]]]). 2) The Jordan–Hölder theorem was exte
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...P$ is a set of functions and $\Omega$ is a set of operations equivalent to composition operations with different types of restrictions. Finite-valued and countabl
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and has a composition series (cf. [[Composition sequence|Composition sequence]]) $ B = B _{0} \supset B _{1} \supset \dots \supset B _{t} = \{
if and only if its Lie algebra $ L $
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that is closed with respect to composition and contains all the projections $ \omega _ {n} ^ {i} : A ^ {n} \right
is an arbitrary fixed set. By a composition of operations $ \omega _ {1} ( x _ {1}, \dots, x _ {j}, \dots, x _ {n} )
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with a trilinear composition
is a [[Lie algebra|Lie algebra]] and $ \mathfrak m \subset \mathfrak g $
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...asi-regular elements forms a group with respect to the operation of cyclic composition: $x\cdot y=x+y+xy$. An important example of a quasi-regular ring is the rin
...E. Sasiada, P.M. Cohn, "An example of a simple radical ring" ''J. of Algebra'' , '''5''' : 3 (1967) pp. 373–377</TD></TR>
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...e category whose objects are all sets belonging to $U$, with morphisms and composition as above. The category of sets may be denoted by $\mathfrak S$, ENS, $\math
...="top">[1]</TD> <TD valign="top"> P. Freyd, "Concreteness" ''J. Pure Appl. Algebra'' , '''3''' (1973) pp. 171–191 {{MR|0322006}} {{ZBL|0277.18002}} </TD></T
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is defined (an external law of composition), such that the following axioms are satisfied:
operator ring, can also be treated as a [[Universal algebra|universal algebra]] with two binary operations (addition and multiplication) and with a set
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...a of sets|$\sigma$-algebra]] of Borel sets of $X$ is the smallest $\sigma$-algebra containing the open sets).
====Closure under composition====
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...]]s of $G$ over $K$ are precisely the irreducible modules over the [[group algebra]] $R = k[G]$.
...endomorphism ring of an irreducible module is a skew-field). If $R$ is an algebra over an algebraically closed field $K$ and if $A$ and $B$ are irreducible m
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...een applied to periodic links and to the building of a [[Hopf algebra|Hopf algebra]] structure on the Jones–Conway [[Skein module|skein module]] of the prod
To define the Jaeger composition product it is convenient to work with the following regular isotopy variant
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into itself is a monoid relative to the operation of successive application (composition) of mappings. The identity mapping is the identity.
2) The set of endomorphisms of a [[universal algebra]] $ A $
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...c\in A$, the conditions $aRb$ and $bRc$ imply $aRc$: equivalently if the [[composition]] $R \circ R \subseteq R$. [[Equivalence relation]]s and [[Order (on a set)
<TR><TD valign="top">[a3]</TD> <TD valign="top"> P.M. Cohn, "Universal algebra", Reidel (1981) {{ISBN|90-277-1213-1}} {{MR|0620952}} {{ZBL|0461.08001}}</T
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''function of the algebra of logic''
...ematics in logic in the middle of the 19th century; cf. [[Algebra of logic|Algebra of logic]].
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of invertible elements of a Banach algebra $ A $
...res of the Banach manifold and the group (with respect to the operation of composition) are not compatible.
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...on [[#References|[a1]]] to characterize the interpolating sequences in the algebra $H ^ { \infty }$ of bounded analytic functions in the open unit disc and to
...used to give complete characterizations of boundedness and compactness of composition operators on various spaces of analytic functions, such as Hardy and Bergma
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