代数格中链的长度
The length of chains in algebraic lattices
摘要：
我们研究如何存在于一个代数格$L$链的一个给定的类型是反映在联合半格$K(L)$的紧凑元素。我们证明了对于每个链$α$的大小$κ$，有一个集合$\B$最多$2^κ$连接半格，每个有一个最小的元素，使得一个代数格$L$不包含链的顺序类型$I(α)$当且仅当连接半格$K(L)$的紧凑的元素包含任何连接子半格同构的$\B$。我们证明了在属于$\B$的$[ω]^{<ω}$的并子半格中，有一个可嵌入到所有其他的并子半格中.我们猜想，如果$α$是可数的，则存在有限集$\B$ .
英文摘要：
We study how the existence in an algebraic lattice $L$ of a chain of a given type is reflected in the join-semilattice $K(L)$ of its compact elements. We show that for every chain $α$ of size $κ$, there is a set $\B$ of at most $2^κ$ join-semilattices, each one having a least element such that an algebraic lattice $L$ contains no chain of order type $I(α)$ if and only if the join-semilattice $K(L)$ of its compact elements contains no join-subsemilattice isomorphic to a member of $\B$. We show that among the join-subsemilattices of $[ω]^{<ω}$ belonging to $\B$, one is embeddable in all the others. We conjecture that if $α$ is countable, there is a finite set $\B$.
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