• 한국지능시스템학회:학술대회논문집
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• 한국퍼지및지능시스템학회 1998년도 The Third Asian Fuzzy Systems Symposium
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• pp.531-536
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• 1998

It is well known that a Boolean algebra is one of the most important algebra for engineering. A fuzzy algebra, which is referred to also as a Kleen algebra, is obtained from a Boolean algebra by replacing the complementary law in the axioms of a Bloolean algebra with the Kleen's law, where the Kleen's law is a weaker condition than the complementary law. Removal of the Kleen's law from a Kleen algebra gives a De Morgan algebra. In this paper, we generate lattice structures of the above related algebraic systems having finite elements by using a computer. From the result, we could find out a hypothesis that the structure excepting for each element name between a Kleene algebra and a De Morgan algebra is the same from the lattice standpoint.

• 충청수학회지
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• 제11권1호
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• pp.13-26
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• 1998

The purpose of this note is to study the relation between Heyting algebra and t-algebra which is the dual concept of BCK-algebra. We define t-algebra with binary operation ${\rhd}$ which is a generalization of the implication in the Heyting algebra, and define a bounded ness and commutativity of it, and then characterize a Heyting algebra and a Boolean algebra as a bounded commutative t-algebra X satisfying $x=(x{\rhd}y){\rhd}x$ for all $x,y{\in}X$.

• Kyungpook Mathematical Journal
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• 제53권2호
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• pp.175-184
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• 2013

In this paper, we introduce a BN-algebra, and we prove that a BN-algebra is 0-commutative, and an algebra X is a BN-algebra if and only if it is a 0-commutative BF-algebra. And we introduce a quotient BN-algebra, and we investigate some relations between BN-algebras and several algebras.

• 호남수학학술지
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• 제29권2호
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• pp.119-192
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• 2007

A very important Hopf algebra is the graded Hopf algebra Symm of symmetric functions. It can be characterized as the unique graded positive selfdual Hopf algebra with orthonormal graded distinguished basis and just one primitive element from the distinguished basis. This result is due to Andrei Zelevinsky. A noncommutative graded Hopf algebra of this type cannot exist. But there is a most important positive graded Hopf algebra with distinguished basis that is noncommutative and that is twisted selfdual, the Malvenuto-Poirier-Reutenauer Hopf algebra of permutations. Thus the question arises whether there is a corresponding uniqueness theorem for MPR. This prepreprint records initial investigations in this direction and proves that uniquenees holds up to and including the degree 4 which has rank 24.

• Korean Journal of Mathematics
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• 제32권3호
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• pp.425-438
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• 2024

BiHom-Akivis algebras are introduced. It is shown that BiHom-Akivis algebras can be obtained either from Akivis algebras by twisting along two algebra morphisms or from a regular BiHom-algebra via the BiHom-commutator-BiHom-associator algebra. It is also proved that a BiHom-Akivis algebra associated to a regular BiHom-alternative algebra is a BiHom-Malcev algebra. Using the BiHom-Akivis algebra associated to a given regular BiHom-Leibniz algebra, a necessary and sufficient condition for BiHom-Lie admissibility of BiHom-Leibniz algebras is obtained.

• 대한수학회보
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• 제52권5호
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• pp.1587-1606
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• 2015

Comonoid, bi-algebra, and Hopf algebra structures are studied within the universal-algebraic context of entropic varieties. Attention focuses on the behavior of setlike and primitive elements. It is shown that entropic $J{\acute{o}}nsson$-Tarski varieties provide a natural universal-algebraic setting for primitive elements and group quantum couples (generalizations of the group quantum double). Here, the set of primitive elements of a Hopf algebra forms a Lie algebra, and the tensor algebra on any algebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then the underlying $J{\acute{o}}nsson$-Tarski monoid of the generating algebra is cancellative. The problem of determining when the $J{\acute{o}}nsson$-Tarski monoid forms a group is open.

• 대한수학회보
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• 제22권2호
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• pp.89-93
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• 1985

One of the most important problems in automatic continuity theory is to solve the question of continuity of an algebra homomorphism from a Banach algebra into a semisimple Banach algebra with dense range. Many results on this subject are obtained imposing some conditions on the domains or the ranges of homomorphisms. For most recent results and references in automatic continuity theory one may refer to [1], [4] and [5]. In this note we study some properties of homomorphisms from $C^{*}$-algebras into Banach algebras. It is shown that the range of an isomorphism from a $C^{*}$-algebra into a Banach algebra contains no non zero element of the radical of B. Using this result we show that the same holds for a continuous homomorphism, hence a Banach algebra which is the image of a $C^{*}$-algebra under a continuous homomorphism is necessarily semisimple. Thus if there is a homomorphism from a $C^{*}$-algebra onto a non-semisimple Banach algebra it must be discontinuous. Also it follows that every non zero homomorphism from a $C^{*}$-algebra into a radical algebra is discontinuous. Then we make a brief observation on the behavior of quasinilpotent element of noncommutative $C^{*}$-algebras in relation with continuous homomorphisms.momorphisms.

• Journal of applied mathematics & informatics
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• 제41권3호
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• pp.633-645
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• 2023

In this paper, Completely V-Regular on semiring is defined and used to derive new theorems with some of its properties. This paper also illustrates V-Regular algebra and Completely V-Regular Algebra with examples and properties. By extending completely V-Regular to fuzzy, a new concept, fuzzy V-Regular is brought out and fuzzy completely V-Regular algebra is introduced too. It is also developed by defining the ideals of Completely V -Regular Algebra and fuzzy completely V-Regular algebra. Finally, this fuzzy algebra concept is applied in image processing to detect edges. This V-Regular Algebra is novel in the research area.

• 대한수학회지
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• 제39권1호
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• pp.1-11
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• 2002

A Schur algebra was generalized to projective Schur algebra by admitting twisted group algebra. A Schur algebra is a projective Schur algebra with trivial 2-cocycle. In this paper we study situations that Schur algebra is a projective Schur algebra with nontrivial cocycle, and we find a criterion for a projective Schur algebra to be a Schur algebra.

• 대한수학회논문집
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• 제13권2호
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• pp.225-232
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• 1998

Let H be a finite dimensional Hopf algebra over a field k, and A be an H-module algebra over k which the H-action on A is D-continuous. We show that $Q_{max}(A)$, the maximal ring or quotients of A, is an H-module algebra. This is used to prove that if H is a finite dimensional semisimple Hopf algebra and A is a semiprime right(left) Goldie algebra than $A#H$ is a semiprime right(left) Goldie algebra. Assume that Asi a semiprime H-module algebra Then $A^H$ is left Artinian if and only if A is left Artinian.