A Generalized Next-Closure Algorithm -- Enumerating Semilattice Elements from a Generating Set

多参数特殊化半格

Multi-argument specialization semilattices

本文研究多参数特殊化半格，给出其公理系统并证明对子结构的完备性，同时提供嵌入闭包半格的典范嵌入方法。

泛特异化半格

Universal specialization semilattices

本文研究特异化半格中闭包不可表达的问题，证明每个特异化半格都可典范嵌入一个主特异化半格，使闭包在其中可被定义。

Various types of completeness in topologized semilattices

A topologized semilattice $X$ is called complete if each non-empty chain $C\subset X$ has $\inf C$ and $\sup C$ that belong to the closure $C$ of the chain $C$ in $X$. In this paper, we introduce various concepts of completeness of topologized semilattices in the context of operators that generalize the closure operator, and study their basic properties. In addition, examples of specific topologized semilattices are given, showing that these classes do not coincide with each other. Also in this paper, we prove theorems that allow us to generalize the available results on complete semilattices …

Universal extensions of specialization semilattices

A specialization semilattice is a join semilattice together with a coarser preorder $ \sqsubseteq $ satisfying an appropriate compatibility condition. If $X$ is a topological space, then $(\mathcal P(X), \cup, \sqsubseteq )$ is a specialization semilattice, where $ x \sqsubseteq y$ if $x \subseteq Ky$, for $x,y \subseteq X$, and $K$ is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. For short, the notion is useful since it allows us to consider a relation of "being generated by" with no need to requ…

半格序半群上的闭包算子与诱导运算的谱性

Closure operators on semilattice-ordered semigroups and spectrality of induced operations

研究半格序半群上闭包算子相关的空间及诱导运算，证明\(X\)是谱空间等条件，给出诱导运算\(\star\)谱性的等价刻画与充分条件，还引入相关闭包算子并研究其性质及应用。

Lattice of closure endomorphisms of a Hilbert algebra

A closure endomorphism of a Hilbert algebra A is a mapping that is simultaneously an endomorphism of and a closure operator on A. It is known that the set CE of all closure endomorphisms of A is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of A, anti-isomorphic to the lattice of certain closure retracts of A, and compactly generated. The set of compact elements of CE coincides with the adjoint semilattice of A, conditions under which …