• Korean Journal of Mathematics
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• v.29 no.1
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• pp.81-89
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• 2021

In this paper, we introduce the notions of a distance function, Alexandrov topology and ⊖-upper (⊕-lower) approximation operator based on complete co-residuated lattices. Under various relations, we define (⊕, ⊖)-fuzzy rough set on complete co-residuated lattices. Moreover, we study their properties and give their examples.

• Korean Journal of Mathematics
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• v.23 no.4
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• pp.557-569
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• 2015

We dene a fuzzy lattice as a fuzzy relation, develop some basic properties of the fuzzy lattice, show that the operations of join and meet in fuzzy lattices are isotone and associative, characterize a fuzzy lattice by its level set, and show that the direct product of two fuzzy lattices is a fuzzy lattice.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.373-381
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• 2010

In this paper we determine all elements in the root lattice of symmetrizable generalized Kac-Moody algebras whose reflections preserve the root systems. Also we discuss elements in the root lattices whose reflection preserve the root lattices.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.15-36
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• 1999

In this paper, we determine the dimension of the rectangular product of certain finite lattices. In face, if L1 and a L2 be finite lattices which satisfy the some conditions, then we have dim (L1$\square$L2) = dim(L1) + dim(L2) - 1.

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.285-292
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• 1995

Some nonpath-connected orthomodular lattices are given : Every infinite direct product of othomodular lattices containing infinitely many non-Boolean factors is a nonpath-connected orthomodular lattice. The orthomodular lattice of all closed subspaces of an infinite dimensional Hilbert space is a nonpath-connected orthomodular lattice.

• East Asian mathematical journal
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• v.26 no.5
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• pp.649-654
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• 2010

A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find some finiteness theorems which ensure 2-universality of Hermitian lattices over several imaginary quadratic number fields.

• Honam Mathematical Journal
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• v.40 no.3
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• pp.401-431
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• 2018

In this paper, some kinds of (skew) filters are defined and are studied in residuated skew lattices. Some relations are got between these filters and quotient algebras constructed via these filters. The Green filter is defined which establishes a connection between residuated lattices and residuated skew lattices. It is investigated that relationships between Green filter and other types of filters in residuated skew lattices and the relationship between residuated skew lattice and other skew structures are studied. It is proved that for a residuated skew lattice, skew Hilbert algebra and skew G-algebra are equivalent too.

• Journal of the Korean Mathematical Society
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• v.59 no.6
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• pp.1047-1065
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• 2022

For two dimensional lattices, a Gaussian basis achieves all two successive minima. For dimension larger than two, constructing a pairwise Gaussian basis is useful to compute short vectors of the lattice. For three dimensional lattices, Semaev showed that one can convert a pairwise Gaussian basis to a basis achieving all three successive minima by one simple reduction. A pairwise Gaussian basis can be obtained from a given basis by executing Gauss algorithm for each pair of basis vectors repeatedly until it returns a pairwise Gaussian basis. In this article, we prove a necessary and sufficient condition for a pairwise Gaussian basis to achieve the first k successive minima for three dimensional lattices for each k ∈ {1, 2, 3} by modifying Semaev's condition. Our condition directly checks whether a pairwise Gaussian basis contains the first k shortest independent vectors for three dimensional lattices. LLL is the most basic lattice basis reduction algorithm and we study how to use LLL to compute a pairwise Gaussian basis. For δ ≥ 0.9, we prove that LLL(δ) with an additional simple reduction turns any basis for a three dimensional lattice into a pairwise SV-reduced basis. By using this, we convert an LLL reduced basis to a pairwise Gaussian basis in a few simple reductions. Our result suggests that the LLL algorithm is quite effective to compute a basis with all three successive minima for three dimensional lattices.

• Journal of the Korean Mathematical Society
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• v.38 no.5
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• pp.937-954
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• 2001

This paper is a natural continuation of [2], [3], [4] and [5]. Localization techniques for modular lattices are developed. These techniques are applied to study liftings of linear order types from quotient lattices and to find Г-dense sets in certain lattices without Г-deviation in the sense of [4], where Г is a set of indecomposable linear order types.

• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.227-237
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• 2007

We define weak distributive $n$-semilattices and $n$-lattices, using variants of the absorption law and those of the distributive law. From a weak distributive $n$-semilattice, we construct direct system of subalgebras which are weak distributive $n$-lattices and show that its direct limit is a reflection of the category $wDn$-SLatt of the weak distributive $n$-semilattices.