• East Asian mathematical journal
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• v.18 no.2
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• pp.205-209
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• 2002

In the paper the following propositions are proved. 1) If ($Q,{\cdot},e$) is a B-algebra, then there exists a group($Q,A,^{-1}$, 1) such that the following equalities hold e=1 and ${\cdot}=^{-1}A$, where $^{-1}A(x,y)=z{\Longleftrightarrow^{def}}A(z,y)=x$; and 2) If ($Q,A,^{-1}$, e) is a group, then ($Q,^{-1}A$, e) is a B-algebra.

• Korean Journal of Mathematics
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• v.12 no.1
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• pp.23-32
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• 2004

Let (B, $m_B$, ${\kappa}$) be a maximal commutative ${\kappa}$-subalgebra of a matrix algebra $M_n(\kappa)$. We will construct a maximal commutative ${\kappa}$-subalgebra (R, $m$, ${\kappa}$) of $M_n+3(\kappa)$ from the algebra B such that the algebra R has dimension greater than the dimension of B by 3. Moreover, we will show a $C_i$-construction doesn't imply a $C^3_2$-construction for $i=1,2$.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.499-505
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• 1997

This paper is a continuation of [3]. We prove that if A is quasi-associative (resp. an implicative) ideal of a BCI-algebra X then the k-nil radical of A is a quasi-associative (resp. an implicative) ideal of X. We also construct the quotient algebra $X/[Z;k]$ of a BCI-algebra X by the k-nhil radical [A;k], and show that if A and B are closed ideals of BCI-algebras X and Y respectively, then

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1333-1354
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• 2019

In authors' paper in 2007, it was shown that the BSE-extension of $C^1_0(R)$, the algebra of continuously differentiable functions f on the real number space R such that f and df /dx vanish at infinity, is the Lipschitz algebra $Lip_1(R)$. This paper extends this result to the case of $C^n_0(R^d)$ and $C^{n-1,1}_b(R^d)$, where n and d represent arbitrary natural numbers. Here $C^n_0(R^d)$ is the space of all n-times continuously differentiable functions f on $R^d$ whose k-times derivatives are vanishing at infinity for k = 0, ${\cdots}$, n, and $C^{n-1,1}_b(R^d)$ is the space of all (n - 1)-times continuously differentiable functions on $R^d$ whose k-times derivatives are bounded for k = 0, ${\cdots}$, n - 1, and (n - 1)-times derivatives are Lipschitz. As a byproduct of our investigation we obtain an important result that $C^{n-1,1}_b(R^d)$ has a predual.

• 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.

• Korean Journal of Mathematics
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• v.22 no.1
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• pp.91-121
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• 2014

Let X and Y be vector spaces. It is shown that a mapping $f:X{\rightarrow}Y$ satisfies the functional equation ${\ddag}$ $$2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^jx_j}{2d})-2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^{j+1}x_j}{2d})=2\sum_{j=2}^{2d}(-1)^jf(x_j)$$ if and only if the mapping $f:X{\rightarrow}Y$ is additive, and prove the Cauchy-Rassias stability of the functional equation (${\ddag}$) in Banach modules over a unital $C^*$-algebra, and in Poisson Banach modules over a unital Poisson $C^*$-algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras. As an application, we show that every almost homomorphism $h:\mathcal{A}{\rightarrow}\mathcal{B}$ of $\mathcal{A}$ into $\mathcal{B}$ is a homomorphism when $h(d^nuy)=h(d^nu)h(y)$ or $h(d^nu{\circ}y)=h(d^nu){\circ}h(y)$ for all unitaries $u{\in}\mathcal{A}$, all $y{\in}\mathcal{A}$, and n = 0, 1, 2, ${\cdots}$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras, and of Lie $JC^*$-algebra derivations in Lie $JC^*$-algebras.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.357-365
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• 2013

Fix $n-2$ elements $h_1,{\cdots},h_{n-2}$ of the quotient field B of the polynomial algebra $\mathbb{C}[x_1,x_2,{\cdots},x_n]$. It is proved that B is a Poisson algebra with Poisson bracket defined by $\{f,g\}=det(Jac(f,g,h_1,{\cdots},h_{n-2})$ for any $f,g{\in}B$, where det(Jac) is the determinant of a Jacobian matrix.

• Journal of the Korean Mathematical Society
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• v.43 no.2
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• pp.323-356
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• 2006

Let X and Y be vector spaces. It is shown that a mapping $f\;:\;X{\rightarrow}Y$ satisfies the functional equation $$mn_{mn-2}C_{k-2}f(\frac {x_1+...+x_{mn}} {mn})$$ $(\ddagger)\;+mn_{mn-2}C_{k-1}\;\sum\limits_{i=1}^n\;f(\frac {x_{mi-m+1}+...+x_{mi}} {m}) =k\;{\sum\limits_{1{\leq}i_1<... if and only if the mapping $f : X{\rightarrow}Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation $(\ddagger)$ in Banach modules over a unital $C^*-algebra$. Let A and B be unital $C^*-algebra$ or Lie $JC^*-algebra$. As an application, we show that every almost homomorphism h : $A{\rightarrow}B$ of A into B is a homomorphism when $h(2^d{\mu}y) = h(2^d{\mu})h(y)\;or\;h(2^d{\mu}\;o\;y)=h(2^d{\mu})\;o\;h(y)$ for all unitaries ${\mu}{\in}A,\;all\;y{\in}A$, and d = 0,1,2,..., and that every almost linear almost multiplicative mapping $h:\;A{\rightarrow}B$ is a homomorphism when h(2x)=2h(x) for all $x{\in}A$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*-algebras$ or in Lie $JC^*-algebras$, and of Lie $JC^*-algebra$ derivations in Lie $JC^*-algebras$.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1727-1748
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• 2018

In this paper, we characterize the graded post-Lie algebra structures on the W-algebra W(2, 2). Furthermore, as applications, the homogeneous Rota-Baxter operators on W(2, 2) and solutions of the formal classical Yang-Baxter equation on $W(2,2){\ltimes}_{ad^*} W(2,2)^*$ are studied.

• Bulletin of the Korean Mathematical Society
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• v.38 no.3
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• pp.527-541
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• 2001

We find the necessary and sufficient conditions for the smash product algebra structure and the crossed coproduct coalgebra structure with th dual cocycle $\alpha$ to afford a Hopf algebra (A equation,※See Full-text). If B and H are finite algebra and Hopf algebra, respectively, then the linear dual (※See Full-text) is also a Hopf algebra. We show that the weak coaction admissible mapping system characterizes the new Hopf algebras (※See Full-text).