''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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...osable) Lie algebra over $k$; by definition, the latter means that the Lie algebra $L$ has a split Cartan subalgebra, that is, a [[Cartan subalgebra|Cartan su
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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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