• 충청수학회지
• /
• 제29권4호
• /
• pp.613-629
• /
• 2016

In this paper, we introduced the notion of multiplier of Boolean algebras and discuss related properties between multipliers and special mappings, like dual closures, homomorphisms on B. We introduce the notions of xed set $Fix_f(X)$ and normal ideal and obtain interconnection between multipliers and $Fix_f(B)$. Also, we introduce the special multiplier ${\alpha}_p$a nd study some properties. Finally, we show that if B is a Boolean algebra, then the set of all multipliers of B is also a Boolean algebra.

• Korean Journal of Mathematics
• /
• 제12권1호
• /
• pp.49-54
• /
• 2004

In this paper we introduce the notion of quotient Boolean algebra and study the relation between the ideals of Boolean algebra ${\wp}(S)$ and the ideals of quotient Boolean algebra ${\wp}(S)/I$.

• 대한수학회지
• /
• 제54권3호
• /
• pp.847-865
• /
• 2017

For a finite or an infinite set X, let $2^X$ be the power set of X. A class of simple graph, called strong Boolean graph, is defined on the vertex set $2^X{\setminus}\{X,{\emptyset}\}$, with M adjacent to N if $M{\cap}N={\emptyset}$. In this paper, we characterize the annihilating-ideal graphs $\mathbb{AG}(R)$ that are blow-ups of strong Boolean graphs, complemented graphs and preatomic graphs respectively. In particular, for a commutative ring R such that AG(R) has a maximum clique S with $3{\leq}{\mid}V(S){\mid}{\leq}{\infty}$, we prove that $\mathbb{AG}(R)$ is a blow-up of a strong Boolean graph if and only if it is a complemented graph, if and only if R is a reduced ring. If assume further that R is decomposable, then we prove that $\mathbb{AG}(R)$ is a blow-up of a strong Boolean graph if and only if it is a blow-up of a pre-atomic graph. We also study the clique number and chromatic number of the graph $\mathbb{AG}(R)$.

• 한국IT서비스학회지
• /
• 제7권4호
• /
• pp.221-230
• /
• 2008

Green's relations are five equivalence relations that characterize the elements of a semigroup in terms of the principal ideals. The J relation is one of Green's relations. Although there are known algorithms that can compute Green relations, they are not useful for finding all J relations in the semigroup of all $n{\times}n$ Boolean matrices. Its computation requires multiplication of three Boolean matrices for each of all possible triples of $n{\times}n$ Boolean matrices. The size of the semigroup of all $n{\times}n$ Boolean matrices grows exponentially as n increases. It is easy to see that it involves exponential time complexity. The computation of J relations over the $5{\times}5$ Boolean matrix is left an unsolved problem. The paper shows theorems that can reduce the computation time, discusses an algorithm for efficient J relation computation whose design reflects those theorems and gives its execution results.

• 한국IT서비스학회지
• /
• 제9권4호
• /
• pp.219-229
• /
• 2010

The Green's equivalence relations have played a fundamental role in the development of semigroup theory. They are concerned with mutual divisibility of various kinds, and all of them reduce to the universal equivalence in a group. Boolean matrices have been successfully used in various areas, and many researches have been performed on them. Studying Green's relations on a monoid of boolean matrices will reveal important characteristics about boolean matrices, which may be useful in diverse applications. Although there are known algorithms that can compute Green relations, most of them are concerned with finding one equivalence class in a specific Green's relation and only a few algorithms have been appeared quite recently to deal with the problem of finding the whole D or J equivalence relations on the monoid of all $n{\times}n$ Boolean matrices. However, their results are far from satisfaction since their computational complexity is exponential-their computation requires multiplication of three Boolean matrices for each of all possible triples of $n{\times}n$ Boolean matrices and the size of the monoid of all $n{\times}n$ Boolean matrices grows exponentially as n increases. As an effort to reduce the execution time, this paper shows an isomorphism between the R relation and L relation on the monoid of all $n{\times}n$ Boolean matrices in terms of transposition. introduces theorems based on it discusses an improved algorithm for the J relation computation whose design reflects those theorems and gives its execution results.

• Kyungpook Mathematical Journal
• /
• 제43권4호
• /
• pp.539-545
• /
• 2003

We present a set of equations that axiomatizes the class of all lattice implication algebras. The construction is different from the one given in [7], and the proof is direct: i.e., it does not rely on results from outside the realm of the lattice implication algebras, such as the theory of BCK-algebras. Then we show that the lattice H implication algebras are nothing more than the familiar Boolean algebras. Finally we obtain some negative results for the embedding of lattice implication algebras into Boolean algebras.

• Journal of applied mathematics & informatics
• /
• 제18권1_2호
• /
• pp.465-486
• /
• 2005

We study the relationship between intuitionistic fuzzy ideals and intuitionistic fuzzy congruences on a distributive lattice. And we prove that the lattice of intuitionistic fuzzy ideals is isomorphic to the lattice of intuitionistic fuzzy congruences on a generalized Boolean algebra.

• 대한수학회지
• /
• 제56권2호
• /
• pp.539-558
• /
• 2019

An ideal of a commutative ring is called a d-ideal if it contains the annihilator of the annihilator of each of its elements. Denote by DId(A) the lattice of d-ideals of a ring A. We prove that, as in the case of f-rings, DId(A) is an algebraic frame. Call a ring homomorphism "compatible" if it maps equally annihilated elements in its domain to equally annihilated elements in the codomain. Denote by $SdRng_c$ the category whose objects are rings in which the sum of two d-ideals is a d-ideal, and whose morphisms are compatible ring homomorphisms. We show that $DId:\;SdRng_c{\rightarrow}CohFrm$ is a functor (CohFrm is the category of coherent frames with coherent maps), and we construct a natural transformation $RId{\rightarrow}DId$, in a most natural way, where RId is the functor that sends a ring to its frame of radical ideals. We prove that a ring A is a Baer ring if and only if it belongs to the category $SdRng_c$ and DId(A) is isomorphic to the frame of ideals of the Boolean algebra of idempotents of A. We end by showing that the category $SdRng_c$ has finite products.

• 호남수학학술지
• /
• 제37권2호
• /
• pp.187-205
• /
• 2015

We discuss the relationship between interval-valued fuzzy ideals and interval-valued fuzzy congruence on a distributive lattice L and show that for a generalized Boolean algebra the lattice of interval-valued fuzzy ideals is isomorphic to the lattice of interval-valued fuzzy congruences. Finally we consider the products of interval-valued fuzzy ideals and obtain a necessary and sufficient condition for an interval-valued fuzzy ideal on the direct sum of lattices to be representable as a direct sum of interval-valued fuzzy ideals on each lattice.

• 정보관리학회지
• /
• 제21권2호
• /
• pp.235-248
• /
• 2004

웹 문서 정색을 위해 키워드와 불리언 연산식을 사용하는 것에 비해 자연어 질의 문장을 입력하는 방법은 검색 시스템 사용자에게 훨씬 이상적인 인터페이스이다. 본 논문은 사용자가 입력하는 자연어 질의 문장을 구문 분석하고 그 구문 구조에 기반하여 검색어를 확장하는 다중 검색 기법을 제안한다. 구문 트리를 순회하여 구조적으로 연관된 복합 명사를 조합하거나 분할하는 과정을 거치고, 이형 표기 및 축약 표기 용어들에 대해 확장 다중 검색함으로써 웹 정보 검색 시스템의 재현율과 정확도를 높일 수 있다.