• Journal for History of Mathematics
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• v.16 no.1
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• pp.25-44
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• 2003

In this paper, we deal with the ‘dearithmetization’ of algebra. Historically, the origin of algebra comes from arithmetic. Also school algebra is related to arithmetic in general. However, we have many difficulties in teaching school algebra, and there is many problems for students to learn algebra from elementary arithmetic knowledges. This paper supposed that the solution of these problem may be founded in the ‘dearithmetization’ of algebra. And we supposed that the ‘dearithmetization’ of algebra may be developed by three historical achievements - the completion of symbolic algebra, the principle of permanence of form, and the expansion of number concepts. In order to justify these supposition, we investigate Peacock's ideas, i.e. ‘symbolic algebra’, ‘the principle of permanence of form’, and consider how the integer is introduced in modern mathematics. And we analyze various textbooks, and investigate the ‘dearithmetization’ of school algebra which has been progressed in the three fields - symbolic algebra, the principle of permanence of form, and the expansion of number concepts.

• Korean Journal of Mathematics
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• v.18 no.1
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• pp.1-10
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• 2010

It is shown that for an almost algebra homomorphism between Banach algebras, there exists a unique algebra homomorphism near the almost algebra homomorphism. Moreover, we prove that for an almost algebra ${\ast}$-homomorphism between $C^{\ast}$-algebras, there exists a unique algebra ${\ast}$-homomorphism near the almost algebra ${\ast}$-homomorphism, and that for an almost algebra ${\ast}$-homomorphism between $JB^{\ast}$-algebras, there exists a unique algebra ${\ast}$-homomorphism near the almost algebra ${\ast}$-homomorphism.

• Journal of the Korean Institute of Intelligent Systems
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• v.13 no.2
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• pp.243-246
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• 2003

In this paper we define a weak positive implicative hyperBCK-ideal of hyperBCK-algebra. Also we investigate that every positive implicative hyperBCK-algebra is a positive implicative hyperK-algebra and then we prove that every positive implicative hyperK-algebra is a weak positive implicative hyperk-algebra.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.2
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• pp.29-39
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• 2001

For a Poisson Hopf algebra A, we find a natural Hopf structure in the Poisson enveloping algebra U(A) of A. As an application, we show that the Poisson enveloping algebra U(S(L)), where S(L) is the symmetric algebra of a Lie algebra L, has a Hopf structure such that a sub-Hopf algebra of U(S(L)) is Hopf-isomorphic to the universal enveloping algebra of L.

• Bulletin of the Korean Mathematical Society
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• v.20 no.2
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• pp.71-74
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• 1983

The problems of the continuity of homomorphisms between Banach algebras have been studied widely for the last two decades to obtain various fruitful results, yet it is far from characterizing the calss of Banach algebras for which each homomorphism from a member of the class into a Banach algebra is conitnuous. For commutative Banach algebras A and B a simple proof shows that every homomorphism .theta. from A into B is continuous provided that B is semi-simple, however, with a non semi-simple Banach algebra B examples of discontinuous homomorphisms from C(K) into B have been constructed by Dales [6] and Esterle [7]. For non commutative Banach algebras the problems of automatic continuity of homomorphisms seem to be much more difficult. Many positive results and open questions related to this subject may be found in [1], [3], [5] and [8], in particular most recent development can be found in the Lecture Note which contains [1]. It is well-known that a$^{*}$-isomorphism from a $C^{*}$-algebra into another $C^{*}$-algebra is an isometry, and an isomorphism of a Banach algebra into a $C^{*}$-algebra with self-adjoint range is continuous. But a$^{*}$-isomorphism from a $C^{*}$-algebra into an involutive Banach algebra is norm increasing [9], and one can not expect each of such isomorphisms to be continuous. In this note we discuss an isomorphism from a commutative $C^{*}$-algebra into a commutative Banach algebra with dense range via separating space. It is shown that such an isomorphism .theta. : A.rarw.B is conitnuous and maps A onto B is B is semi-simple, discontinuous if B is not semi-simple.

• Honam Mathematical Journal
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• v.40 no.2
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• pp.227-237
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• 2018

There are various papers on finding all the derivations of a non-associative algebra and an anti-symmetrized algebra. We find all the derivations of a growing algebra in the paper. The dimension of derivations of the growing algebra is one and every derivation of the growing algebra is outer. We show that there is a class of purely outer algebras in this work.

• School Mathematics
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• v.4 no.1
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• pp.29-47
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• 2002

The purpose of this study is to investigate various perspectives on the approaching to school algebra. School algebra plays an important role in school mathematics, but many students have been in difficulties for the loaming of school algebra. This problem has been continued for the long time. This paper supposed that this problem may be caused by the approaching stage to the school algebra. In order to investigate this problem, we first described the definition of algebra in chapterII. And in chapterIII, we divided the perspectives of approaching to the school algebra into the two parts. The one is the intrinsic nature of algebra, i.e. generalization, abstraction, and structural aspects. The other is the extrinsic representation of algebra, i.e. problem solving, modelling, functions. Each perspectives are investigated through various research studies. We hope that this works on each perspectives help a teacher prepare school algebra.

• Journal of Educational Research in Mathematics
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• v.13 no.3
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• pp.309-327
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• 2003

In this paper, we deal with the teaching of algebra in the elementary school mathematics, and call this algebra teaching method as ‘early algebra’. Early algebra is appeared in the 1980's with the discussion of ‘algebraic thinking’. And many studies about early algebra is in progress since 1990's. These studies aims at reducing difficulties in the teaching of algebra and the development of algebra curriculum. We investigate the background of early algebra, and justify teaching of early algebra. Also we examine the projects and studies in progress around the world. Finally through these discussions, we compare our elementary textbooks with early algebra, and verify the characters of early algebra from our arithmetic curriculum.

• Journal of applied mathematics & informatics
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• v.40 no.1_2
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• pp.317-325
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• 2022

The concept of hom-set in an implicative algebra as an implicative algebra is introduced. The idea of sub implicative algebra in hom-set, 1-commutative algebra of hom-set are investigated. Different characterization theorems are proved.

• Kyungpook Mathematical Journal
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• v.45 no.4
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• pp.481-489
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• 2005

We first show that any complete MV-algebra whose Boolean subalgebra of idempotent elements is atomic, called a complete MV-algebra with atomic center, is isomorphic to a product of unit interval MV-algebra 1's and finite linearly ordered MV-algebras of A(m)-type $(m{\in}{\mathbb{Z}}^+)$. Secondly, for a semi-simple MV-algebra A, we introduce a completion ${\delta}(A)$ of A which is a complete, MV-algebra with atomic center. Under their intrinsic topologies $(see\;{\S}3)$ A is densely embedded into ${\delta}(A)$. Moreover, ${\delta}(A)$ has the extension universal property so that complete MV-algebras with atomic centers are epireflective in semi-simple MV-algebras