• 대한수학회논문집
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• 제35권2호
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• pp.347-358
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• 2020

In this paper, we simplify the natural partial ordering ≼ on the semigroup 𝒪([n]) under composition of all order-preserving maps on [n] = {1, …, n}, and describe its maximal elements. Also, we show that the poset (𝒪([n]), ≼) is weakly graded and determine when (𝒪([n]), ≼) has a structure of (i + 1)-avoidance.

• Kyungpook Mathematical Journal
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• 제54권2호
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• pp.197-201
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• 2014

In this paper, we study some properties of finite or infinite poset P determined by properties of the ideal based zero-divisor graph properties $G_J(P)$, for an ideal J of P.

• 호남수학학술지
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• 제40권1호
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• pp.75-82
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• 2018

Let P be a commutator poset with Z(P) its set of zero-divisors. The annihilator graph of P, denoted by AG(P), is the (undirected) graph with all elements of $Z(P){\setminus}\{0\}$ as vertices, and distinct vertices x, y are adjacent if and only if $ann(xy)\;{\neq}\;(x)\;{\cup}\;ann(y)$. In this paper, we study basic properties of AG(P).

• 대한수학회논문집
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• 제22권4호
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• pp.503-508
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• 2007

Using the notion of posets, a method to make BCK-algebras is considered. We show that if a poset has the least element, then the induced BCK-algebra is bounded.

• 대한수학회보
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• 제53권5호
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• pp.1281-1289
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• 2016

Richard Stanley suggested the problem of finding combinatorial proofs of formulas for counting regions of certain hyperplane arrangements defined by hyperplanes of the form $x_i=0$, $x_i=x_j$, and $x_i=2x_j$ that were found using the finite field method. We give such proofs, using embroidered permutations and linear extensions of posets.

• Kyungpook Mathematical Journal
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• 제56권3호
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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.

• International Journal of Contents
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• 제7권4호
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• pp.56-63
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• 2011

Quantum mechanics is a branch of physics for a mathematical description of the particle wave, and it is applied to information technology such as quantum computer, quantum information, quantum network and quantum cryptography, etc. In 1936, Garrett Birkhoff and John von Neumann introduced the logic of quantum mechanics (quantum logic) in order to investigate projections on a Hilbert space. As another type of quantum logic, orthomodular implication algebra was introduced by Chajda et al. This algebra has the logical implication as a binary operation. In pure mathematics, there are many algebras such as Hilbert algebras, implicative models, implication algebras and dual BCK-algebras (DBCK-algebras), which have the logical implication as a binary operation. In this paper, we introduce the definitions and some properties of those algebras and clarify the relations between those algebras. Also, we define the implicative poset which is a generalization of orthomodular implication algebras and DBCK-algebras, and research properties of this algebraic structure.

• 대한수학회보
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• 제48권2호
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• pp.315-324
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• 2011

In this paper, we study the effects of a deformation mapping on the resulting deformation d/BCK-algebra obtained via such a deformation mapping. Besides providing a method of constructing d-algebras from BCK-algebras, it also highlights the special properties of the standard BCK-algebras of posets as opposed to the properties of the class of divisible d/BCK-algebras which appear to be of interest and which form a new class of d/BCK-algebras insofar as its not having been identified before.

• 대한수학회보
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• 제32권2호
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• pp.171-177
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• 1995

Let P be a finite poset and let $\mid$P$\mid$ be the number of vertices in pp. A subposet of P is a subset of P with the induced order. A chain C in P is a subposet of P which is a linear order. The length of the chain C is $\mid$C$\mid$ - 1. A linear extension of a poset P is a linear order $L = x_1, x_2, \ldots, x_n$ of the elements of P such that $x_i < x_j$ is P implies i < j. Let L(P) be the set of all linear extensions of pp. E. Szpilrajn [5] showed that L(P) is not empty.

• 호남수학학술지
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• 제33권2호
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• pp.279-300
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• 2011

For a subset system M on any poset, M-Scott notions, such as M-way below relation,M-continuity,M-Scott convergence (of nets and filters respectively) and M-Scott topology are proposed Any approximating auxiliary relation on a poset can be represented by an M-way below relation such that this poset is M-continuous. It is shown that a poset is M-continuous iff the M-Scott topology is completely distributive. The topology induced by the M-Scott convergence coincides with the M-Scott topology. If the M-way below relation satisfies the property of interpolation then a poset is M-continuous if and only if the M-Scott convergence coincides with the M-Scott topological convergence. Also, M-continuity is characterized by a certain Galois connection.