• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.777-788
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• 2018

A simple poset is a poset whose linear discrepancy increases if any relation of the poset is removed. In this paper, we investigate more important properties of simple posets such as its width and height which help to construct concrete simple poset of linear discrepancy l. The simplicity of a poset is similar to the ld-irreducibility of a poset. Hence, we investigate which posets are both simple and ld-irreducible. Using these properties, we characterize n-posets of linear discrepancy n - k for k = 2, 3, and, lastly, we also characterize a poset of linear discrepancy 3 with simple posets and ld-irreducible posets.

• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.917-931
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• 2013

It is known that being hierarchical is a necessary and sufficient condition for a poset to admit MacWilliams identity. In this paper, we completely characterize the structures of posets which have an association scheme structure whose relations are indexed by the poset distance between the points in the space. We also derive an explicit formula for the eigenmatrices of association schemes induced by such posets. By using the result of Delsarte which generalizes the MacWilliams identity for linear codes, we give a new proof of the MacWilliams identity for hierarchical linear poset codes.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.4
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• pp.195-213
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• 2023

Discrete volumes of poset polytopes for a variant of up-down posets introduced in [4] were studied. We obtained the generating functions for the discrete volumes of poset polytopes for a variant of up-down poset using the characteristic matrices.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.11 no.1
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• pp.59-64
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• 2011

In 2001, the notion of a fuzzy poset defined on a set X via a triplet (L, G, I) of functions with domain X ${\times}$ X and range [0, 1] satisfying a special condition L+G+I = 1 is introduced by J. Negger and Hee Sik Kim, where L is the 'less than' function, G is the 'greater than' function, and I is the 'incomparable to' function. Using this approach, we are able to define a special class of fuzzy posets, and define the 'skeleton' of a fuzzy poset in view of major relation. In this sense, we define the linear discrepancy of a fuzzy poset of size n as the minimum value of all maximum of I(x, y)${\mid}$f(x)-f(y)${\mid}$ for f ${\in}$ F and x, y ${\in}$ X with I(x, y) > $\frac{1}{2}$, where F is the set of all injective order-preserving maps from the fuzzy poset to the set of positive integers. We first show that the definition is well-defined. Then, it is shown that the optimality appears at the same injective order-preserving maps in both cases of a fuzzy poset and its skeleton if the linear discrepancy of a skeleton of a fuzzy poset is 1.

• Communications of the Korean Mathematical Society
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• v.26 no.4
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• pp.543-549
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• 2011

In 1945, the notion of free lattice over a poset was introduced by R. Dilworth (Trans. Am. Math. Soc. 57 (1945), 123{154). In this note, a construction of the free algebra over a poset in varieties finitely generated is shown. Finally, this result is applied to different classes of algebras.

• Korean Journal of Mathematics
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• v.17 no.2
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• pp.169-179
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• 2009

Let P be a (n, n − 1, j)-poset, which is a partially ordered set of cardinality n with n − 1 maximal elements and $j(1{\leq}j{\leq}n-1)$ minimal elements, and $P^*$ the dual poset of P. In this paper, we obtain two types of incidence codes of nonempty proper subset S of P and $P^*$, respectively, by using Bogart's method [1] (see Theorem 3.3).

• Korean Journal of Mathematics
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• v.16 no.3
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• pp.425-437
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• 2008

Let P be a poset and G = G(P) be the incomparability graph of P. Stanley [7] defined the chromatic symmetric function $X_{G(P)}$ which generalizes the chromatic polynomial ${\chi}_G$ of G, and showed all coefficients are nonnegative in the e-expansion of $X_{G(P)}$ for a poset P of rank 1. In this paper, we construct a sign reversing involution on the set of special rim hook P-tableaux with some conditions. It gives a combinatorial proof for (3+1)-free conjecture of a poset P of rank 1.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.727-735
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• 2016

In this paper, we study and establish some interesting results of ideals in a poset. It is shown that for a nonzero ideal I of a poset P, there are at most two strongly prime ideals of P that are minimal over I. Also, we study the notion of primal ideals in a poset and the relationship among the primal ideals and strongly prime ideals is considered.

• Korean Journal of Mathematics
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• v.29 no.2
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• pp.305-320
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• 2021

In this paper, the relations between L-fuzzy semi-prime (respectively, L-fuzzy prime) ideals of a poset and L-fuzzy semi-prime (respectively, L-fuzzy prime) ideals of the lattice of all ideals of a poset are established. A result analogous to Separation Theorem is obtained using L-fuzzy semi-prime ideals.

• Annals of Fuzzy Mathematics and Informatics
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• v.16 no.3
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• pp.285-299
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• 2018

Many generalizations of ideals of a lattice to an arbitrary poset have been studied by different scholars. In this paper, we introduce several L-fuzzy ideals of a poset which generalize the notion of an L-fuzzy ideal of a lattice and give several characterizations of them.