Chain condition

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A finiteness condition for ascending or descending chains in a partially ordered set. The descending chain condition for a partially ordered set $P$ states: For any chain $a_1\geq\dots\geq a_k\geq\dots$ of elements there is a number $n$ such that $a_n=a_{n+1}=\dots$. This condition is equivalent to each of the following properties of $P$:

1) every non-empty subset $M\subseteq P$ has at least one minimal element in $M$ (the minimum condition);

2) all elements of $P$ have a given property $\epsilon$ if all minimal elements of $P$ have this property and if the validity of $\epsilon$ for any $a\in P$ can be deduced from the fact that $\epsilon$ is valid for all $x<a$, $x\in P$ (the inductiveness condition).

The inductiveness condition enables one to carry out proofs and constructions by induction for sets with the descending chain condition. Here if $P$ is totally ordered (and hence well-ordered) one obtains transfinite induction, and if $P$ is isomorphic to the set of natural numbers, one obtains ordinary mathematical induction (see Induction axiom).

The ascending chain condition (and assertions equivalent to it) is formulated in a dual way (see Duality principle in partially ordered sets); it therefore states that if $a_1\leq\ldots\leq a_k\leq\dots$ is an ascending chain in a partially ordered set $P$, then for $n$ large enough $a_n=a_{n+1}=\dots$. In a lattice with the ascending chain condition every ideal is principal. Every finite set obviously satisfies both chain conditions, but the converse assertion (that a set satisfying both these conditions is finite) is false. A lattice satisfying the descending and ascending chain conditions is complete.

In algebra, chain conditions are mainly applied to sets of subsystems of various algebraic systems ordered by inclusion (see for example, Artinian module; Artinian group; Artinian ring; Composition sequence; Noetherian module; Noetherian group; Noetherian ring).

References

 [1] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973) [2] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) [3] L.A. Skornyakov, "Elements of lattice theory" , A. Hilger & Hindustan Publ. Comp. (1977) (Translated from Russian)

The phrase "chain condition" is used in a different sense in the theory of Boolean algebras and in set theory (a Boolean algebra, considered as a partially ordered set, satisfies the descending chain condition if and only if it is finite, so this condition is not of interest in the context of Boolean algebras). A Boolean algebra $B$ is said to satisfy the $\kappa$-chain condition, where $\kappa$ is an infinite cardinal number, if every chain (i.e. totally ordered subset) in $B$ has cardinality less then $\kappa$. In particular, the $\aleph_1$-chain condition is commonly called the countable chain condition (abbreviated ccc); this is an important condition in the set-theoretic forcing method (see [a1], Section 3, for example). For a complete Boolean algebra, the $\kappa$-chain condition is equivalent to the $\kappa$-antichain condition, i.e. the condition that every discretely ordered subset has cardinality less than $\kappa$. In the theory of forcing, set-theorists frequently work not with complete Boolean algebras of forcing conditions, but with partially ordered sets which generate them (as algebras of regular open sets); such a set satisfies the $\kappa$-antichain condition if and only if the same condition holds for the Boolean algebra which it generates. Unfortunately, set-theorists tend to use the term "$\kappa$-chain condition" in this context, when they really mean "$\kappa$-antichain condition" .

References

 [a1] J.P. Burgess, "Forcing" J. Barwise (ed.) , Handbook of mathematical logic , North-Holland (1977) pp. 403–452