Split Hausdorff internal topologies on posets. (English) Zbl 1435.06004
Summary: In this paper, the concepts of weak quasi-hypercontinuous posets and weak generalized finitely regular relations are introduced. The main results are: (1) when a binary relation \(\rho: X \rightharpoonup Y\) satisfies a certain condition, \(\rho\) is weak generalized finitely regular if and only if \(( \varphi_\rho (X, Y), \subseteq )\) is a weak quasi-hypercontinuous poset if and only if the interval topology on \(( \varphi_\rho (X, Y), \subseteq )\) is split \(T_2\); (2) the relation \(\nleq\) on a poset \(P\) is weak generalized finitely regular if and only if \(P\) is a weak quasi-hypercontinuous poset if and only if the interval topology on \(P\) is split \(T_2\).
MSC:
| 06B35 | Continuous lattices and posets, applications |
| 54H10 | Topological representations of algebraic systems |
| 06A06 | Partial orders, general |
Keywords:
split \(T_2\); \(T_2\) property; weak generalized finitely regular; weak quasi-hypercontinuous posetReferences:
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