Recent zbMATH articles in MSC 06A06
https://www.zbmath.org/atom/cc/06A06
2022-05-16T20:40:13.078697Z
Werkzeug
The Scott topology on posets and continuous posets
https://www.zbmath.org/1483.06001
2022-05-16T20:40:13.078697Z
"Fan, Lihong"
https://www.zbmath.org/authors/?q=ai:fan.lihong
"He, Wei"
https://www.zbmath.org/authors/?q=ai:he.wei|he.wei.3|he.wei.1|he.wei.2
(no abstract)
Local directed complete sets and properties of their categories
https://www.zbmath.org/1483.06002
2022-05-16T20:40:13.078697Z
"Guan, Xue Chong"
https://www.zbmath.org/authors/?q=ai:guan.xuechong
"Wang, Ge Ping"
https://www.zbmath.org/authors/?q=ai:wang.geping
(no abstract)
Meet continuity of posets via lim-inf-convergence
https://www.zbmath.org/1483.06003
2022-05-16T20:40:13.078697Z
"Li, Qingguo"
https://www.zbmath.org/authors/?q=ai:li.qingguo
"Li, Jibo"
https://www.zbmath.org/authors/?q=ai:li.jibo
(no abstract)
On a characterization theorem for continuous posets
https://www.zbmath.org/1483.06004
2022-05-16T20:40:13.078697Z
"Wang, Xijuan"
https://www.zbmath.org/authors/?q=ai:wang.xijuan
"Lu, Tao"
https://www.zbmath.org/authors/?q=ai:lu.tao
"He, Wei"
https://www.zbmath.org/authors/?q=ai:he.wei.2
(no abstract)
Cartesian closeness of the categories of algebraic local complete posets and FS-local directed complete posets
https://www.zbmath.org/1483.06005
2022-05-16T20:40:13.078697Z
"Xu, Ai Jun"
https://www.zbmath.org/authors/?q=ai:xu.ai-jun
"Wang, Ge Ping"
https://www.zbmath.org/authors/?q=ai:wang.geping
(no abstract)
\( \mathcal{Z} \)-quasidistributive and \(\mathcal{Z} \)-meet-distributive posets
https://www.zbmath.org/1483.06006
2022-05-16T20:40:13.078697Z
"Zhang, Wenfeng"
https://www.zbmath.org/authors/?q=ai:zhang.wenfeng.1|zhang.wenfeng
"Xu, Xiaoquan"
https://www.zbmath.org/authors/?q=ai:xu.xiaoquan.1|xu.xiaoquan
In domain theory, one fundamental result states that a poset is continuous if and only if it is quasicontinuous and meet-continuous. In this paper, the authors study two kinds of distributivity: \(Z\)-quasidistributivity and \(Z\)-meet-distributivity, which are the generalizations of quasicontinuity and meet-continuity. Here, \(Z\) is a subset system, it becomes meaningful when \(Z\) is replaced by adjectives such as ``directed'', ``chain'', ``finite'', etc. Analogous to the above fundamental result, the authors prove that, under some conditions, a poset is \(Z\)-predistributive iff it is \(Z\)-quasidistributive and \(Z\)-meet-distributive.
In order theory, the Dedekind-MacNeille completion is the most well-known completion, which embeds a poset into a complete lattice. The order-theoretical properties which are invariant under the Dedekind-MacNeille completion are called completion-invariant. In this paper, one main result states that \(Z\)-quasidistributivity is a completion-invariant property whenever \(Z\) is completion-stable.
The way-below relation is a fundamental concept in domain theory. Replacing directed sets by \(Z\)-sets, one has the concept of \(Z\)-below. The last main result of this paper: if the \(Z\)-below relation on the subsets of a poset \(P\) has the interpolation property, then \(P\) is embeddable in a cube.
Reviewer: Zhongxi Zhang (Yantai)
Convexity in topological betweenness structures
https://www.zbmath.org/1483.54017
2022-05-16T20:40:13.078697Z
"Anderson, Daron"
https://www.zbmath.org/authors/?q=ai:anderson.daron
"Bankston, Paul"
https://www.zbmath.org/authors/?q=ai:bankston.paul
"McCluskey, Aisling"
https://www.zbmath.org/authors/?q=ai:mccluskey.aisling-e
A betweenness structure is a pair \(\langle X,[\cdot,\cdot,\cdot] \rangle\), where \(X\) is a set and \([\cdot,\cdot,\cdot]\subset X^{3}\) is a ternary relation satisfying that
\begin{itemize}
\item[(B1)] Inclusivity: \((\forall\ xy )\) \(([x,y,y] \wedge [x,x,y])\)
\item[(B2)] Symmetry: \((\forall\ xzy )\) \(([x,z,y] \rightarrow [y,z,x])\)
\item[(B3)] Uniqueness: \((\forall\ xz)\) \(([x,z,x]\rightarrow x=z)\)
\end{itemize}
Given a betweenness structure \(\langle X,[\cdot,\cdot,\cdot] \rangle\), an interval is defined as \([a,b]=\{c\in X:[a,c,b]\}\), a convex subset of \(X\) is a subset \(C\) of \(X\) such that \(a,b\in C\), implies \([a,b]\subset C\). The span of a subset \(A\) of \(X\) is defined as \([A]=\bigcup\{[a,b]:a,b\in A\}\). The convex hull of \(A\) is defined as \([A]^{\omega}=\bigcup\{[A]^{n}:n\in\omega\}\), where \([A]^{0}=A\) and \([A]^{n+1}=[[A]^{n}]\).
With a detailed analysis of examples, in this paper the authors show how the notion of betweenness is related to several important concepts in mathematics. In particular if besides a betweenness structure, \(X\) has a topology, it is possible to define interesting relations between the two structures, starting by asking that intervals are closed. In this sense, the authors define local convexity, upper (and lower) semi-continuity of betweenness, and a type of internal continuity of the betweenness. They obtain results connecting the convexity and the topology in compact connected Hausdorff spaces which are aposyndetic or hereditary unicoherent. In particular, they study how the span and the convex hull interact with the topological closure and interior operators.
Reviewer: Alejandro Illanes (Ciudad de MÃ©xico)