• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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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.

• Journal of the Korean Mathematical Society
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• v.39 no.1
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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.

• Journal of the Korean Mathematical Society
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• v.40 no.2
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• pp.273-286
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• 2003

Let $F^{\alpha}$G be a twisted group algebra. A subalgebra of $F^{\alpha}$G generated by all class sums of partition P of G is called a projective class algebra in $F^{alpha}$G associated with partition P. In this paper we study various partitions of G determined by actions of certain operator groups on G and construct projective class algebras depending on the actions. With regard to projective class algebras, we investigate structures of associated skew group algebras and fixed group algebras.

• Korean Journal of Mathematics
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• v.7 no.1
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• pp.103-110
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• 1999

For a group action ${\alpha}$ on a Kac algebra $\mathbb{K}$ with the crossed product Kac algebra $\mathbb{K}{\rtimes}_{\alpha}G$, we will show that ${\pi}_{\alpha}(\mathbb{K})$ is a sub-Kac algebra of $\mathbb{K}{\rtimes}_{\alpha}G$. We will also investigate the intrinsic group $G(\mathbb{K})$ of $\mathbb{K}$ and get a group action ${\beta}$ on a symmetric Kac algebra $\mathbb{K}_s(G(\mathbb{K})$ with the crossed product sub-Kac algebra $\mathbb{K}_s(G(\mathbb{K}){\rtimes}_{\beta}G$ of $\mathbb{K}{\rtimes}_{\alpha}G$.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.803-814
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• 2001

A projective Schur algebra associated with a partition of finite group G can be constructed explicitly by defining linear transformations of G. We will consider various linear transformations and count the number of equivalent classes in a finite group. Then we construct projective Schur algebra dimension is determined by the number of classes.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.453-467
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• 2005

Let K be an algebraic number field. If k is the maximal cyclotomic subextension in K then the Schur K-group S(K) is obtained from the Schur k-group S(k) by scalar extension. In the paper we study projective Schur group PS(K) which is a generalization of Schur group, and prove that a projective Schur K-algebra is obtained by scalar extension of a projective Schur k-algebra where k is the maximal radical extension in K with mild condition.

• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.1105-1115
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• 2022

The main objective of this paper is to introduce the notion of AMR-algebra and its generalization, and to compare them with other algebras such as BCK, BCI, BCH, · · ·, etc. We show moreover that the K-part of AMR-algebra is an abelian group, and the weak AMR-algebra is also an abelian group and generalizes many known algebras like BCI, BCH, and G.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.453-459
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• 2004

A left-invariant flat Riemannian connection on a Lie group makes its Lie algebra a left symmetric algebra compatible with an inner product. The left symmetric algebra is decomposed into trivial ideal and a subalgebra of e(l). Using this result, the Lie group is embedded isomorphically into the direct product of O(l) $\times$ $R^{k}$ for some nonnegative integers l and k.

• Honam Mathematical Journal
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• v.26 no.4
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• pp.471-481
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• 2004

In this paper we introduce the notion of a (pre-)Coxeter algebra and show that a Coxeter algebra is equivalent to an abelian group all of whose elements have order 2, i.e., a Boolean group. Moreover, we prove that the class of Coxeter algebras and the class of B-algebras of odd order are Smarandache disjoint. Finally, we show that the class of pre-Coxeter algebras and the class of BCK-algebras are Smarandache disjoint.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1275-1283
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• 2010

If the Wiener-Hopf $C^*$-algebra W(G,M) for a discrete group G with a semigroup M has the uniqueness property, then the structure of it is to some extent independent of the choice of isometries on a Hilbert space. In this paper we show that if the Wiener-Hopf $C^*$-algebra W(G,M) of a partially ordered group G with the positive cone M has the uniqueness property, then (G,M) is weakly unperforated. We also prove that the Wiener-Hopf $C^*$-algebra W($\mathbb{Z}$, M) of subsemigroup generating the integer group $\mathbb{Z}$ is isomorphic to the Toeplitz algebra, but W($\mathbb{Z}$, M) does not have the uniqueness property except the case M = $\mathbb{N}$.