良基代数格的一个刻画
A characterization of well-founded algebraic lattices
摘要：
利用良基代数格的紧元构成的并半格的禁子半格刻画良基代数格。更具体地说，我们证明了一个代数格$L$是良基的当且仅当$L$的紧元素的并半格$K(L)$是良基的，并且既不包含$[ω]^{<ω}$，也不包含$\underlineΩ(ω^*)$作为并子半格。作为直接推论，我们得到代数模格$L$是良基的当且仅当$K(L)$是良基的且不含无限独立集.如果$K(L)$是$I_{<ω}(Q)$的并子半格，$L$是良基偏序集$Q$的初始段的集合，则$L$是良基的当且仅当$K(L)$是良准序的。
英文摘要：
We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice $L$ is well-founded if and only if $K(L)$, the join-semilattice of compact elements of $L$, is well-founded and contains neither $[ω]^{<ω}$, nor $\underlineΩ(ω^*)$ as a join-subsemilattice. As an immediate corollary, we get that an algebraic modular lattice $L$ is well-founded if and only if $K(L)$ is well-founded and contains no infinite independent set. If $K(L)$ is a join-subsemilattice of $I_{<ω}(Q)$, the set of finitely generated initial segments of a well-founded poset $Q$, then $L$ is well-founded if and only if $K(L)$ is well-quasi-ordered.
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