The completion of a poset P by ideals (in the sense of Frink) is characterized abstractly as the ... more The completion of a poset P by ideals (in the sense of Frink) is characterized abstractly as the smallest algebraic complete lattice containing P as a set of compact elements, while the completion by directed ideals is the smallest algebraic poset with the corresponding propert y It is known that both types of ideal completions may also be interpreted as certain topological completions. We show that a suitable intermediate completion by so-called Cauchy ideals may be regarded as a uniform completion of P. For lattices, the Cauchy ideal completion coincides with the Frink ideal completion (and with the completion by directed ideals, provided a least element exists). In the theory of partially ordered sets (posets), various types of ideals have been introduced for concrete constructions of certain universal completions. Prominent examples are AP, the Alexandroff completion by order ideals (lower sets), MP, the Dedekind-MacNeille completion by normal ideals (cuts),
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
Proceedings of the American Mathematical Society, 1995
In this paper we deal with the (o)-convergence and the order topology in the hilbertian logic L (... more In this paper we deal with the (o)-convergence and the order topology in the hilbertian logic L ( H ) \mathcal {L}(H) of closed subspaces of a separable Hilbert space H. We compare the order topology on L ( H ) \mathcal {L}(H) with some other topologies. The main result is a theorem which asserts that the weak convergence of a sequence of unit vectors to zero in H is equivalent to the convergence of the sequence of one-dimensional subspaces generated by these vectors to the zero subspace in the order topology on L ( H ) \mathcal {L}(H) .
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
The completion of a poset P by ideals (in the sense of Frink) is characterized abstractly as the ... more The completion of a poset P by ideals (in the sense of Frink) is characterized abstractly as the smallest algebraic complete lattice containing P as a set of compact elements, while the completion by directed ideals is the smallest algebraic poset with the corresponding propert y It is known that both types of ideal completions may also be interpreted as certain topological completions. We show that a suitable intermediate completion by so-called Cauchy ideals may be regarded as a uniform completion of P. For lattices, the Cauchy ideal completion coincides with the Frink ideal completion (and with the completion by directed ideals, provided a least element exists). In the theory of partially ordered sets (posets), various types of ideals have been introduced for concrete constructions of certain universal completions. Prominent examples are AP, the Alexandroff completion by order ideals (lower sets), MP, the Dedekind-MacNeille completion by normal ideals (cuts),
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
Proceedings of the American Mathematical Society, 1995
In this paper we deal with the (o)-convergence and the order topology in the hilbertian logic L (... more In this paper we deal with the (o)-convergence and the order topology in the hilbertian logic L ( H ) \mathcal {L}(H) of closed subspaces of a separable Hilbert space H. We compare the order topology on L ( H ) \mathcal {L}(H) with some other topologies. The main result is a theorem which asserts that the weak convergence of a sequence of unit vectors to zero in H is equivalent to the convergence of the sequence of one-dimensional subspaces generated by these vectors to the zero subspace in the order topology on L ( H ) \mathcal {L}(H) .
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
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