# The properties of \(\models \)-filters of a topological system

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## Abstract

The aim of this paper is to build relationships between point logics and logical algebras. Firstly, by modifying Vickers’s Scott open filters, the notion of the \(\models \)-filters of a topological system is introduced. It is proved that the \(\models \)-filters are lattice filters, but the converse is not true. Secondly, the concrete forms of infimum, supremum and implication of the set of all \(\models \)-filters are obtained. It is shown that the set of all \(\models \)-filters of a topological system is a (co)frame and a completely distributive lattice. Finally, we prove that the set of all maximal \(\models \)-filters are endowed with two topologies forming a \(T_{2}\) space and a \(T_{1}\) space.

## Keywords

Topological systems BL-algebra Maximal \(\models \)-filters## Notes

### Acknowledgements

This research is supported by a Grant of National Natural Science Foundation of China (11531009).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of authors.

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