Topology of closure systems in algebraic lattices

IF 0.6 4区 数学 Q3 MATHEMATICS
Niels Schwartz
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引用次数: 0

Abstract

Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice P generates a closure system \(\langle X \rangle _P\). The closure system \(\langle Y \rangle _P\) generated by the patch closure Y of X is the patch closure of \(\langle X \rangle _P\). If X is contained in the set of nontrivial prime elements of P then \(\langle X \rangle _P\) is a frame and is a coherent algebraic frame if X is patch closed in P. Conversely, if the algebraic lattice P is coherent then its set of nontrivial prime elements is patch closed.

代数格中闭包系统的拓扑
代数格是粗糙下拓扑的谱空间。代数格中的闭包系统被研究为子空间。探讨了闭系统的序理论性质和子空间拓扑性质之间的联系。闭系统是代数的,当且仅当它是环境代数格的补闭子集。代数格P中的每个子集X都生成一个闭包系统\(\langle X\rangle _P\)。由X的补丁闭包Y生成的闭包系统\(\langle Y\rangle _P\)是\(\langel X\rangle _P\)的补丁闭包。如果X包含在P的非平凡素元集合中,则\(\langle X\rangle _P\)是一个框架,并且如果X在P中是补闭的,则是相干代数框架。相反,如果代数格P是相干的,则其非平凡素元素集合是补闭。
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来源期刊
Algebra Universalis
Algebra Universalis 数学-数学
CiteScore
1.00
自引率
16.70%
发文量
34
审稿时长
3 months
期刊介绍: Algebra Universalis publishes papers in universal algebra, lattice theory, and related fields. In a pragmatic way, one could define the areas of interest of the journal as the union of the areas of interest of the members of the Editorial Board. In addition to research papers, we are also interested in publishing high quality survey articles.
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