• Title, Summary, Keyword: Algebra

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Generation of Finite Fuzzy Algebra and Finite De Morgan Algebra Using a Computer

  • Tastumi, Hisayuki;Araki, Tomoyuki;Mukaidono, Masao;Tokumasu, Shinji
    • Proceedings of the Korean Institute of Intelligent Systems Conference
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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.

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HEYTING ALGEBRA AND t-ALGEBRA

  • Yon, Yong Ho;Choi, Eun Ai
    • Journal of the Chungcheong Mathematical Society
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    • v.11 no.1
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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 $x,y{\in}X$.

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On BN-algebras

  • Kim, Chang Bum;Kim, Hee Sik
    • Kyungpook Mathematical Journal
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    • v.53 no.2
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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.

ALMOST HOMOMORPHISMS BETWEEN BANACH ALGEBRAS

  • Lee, Sung Jin;Park, Choonkil
    • 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.

ON SOME SCHUR ALGEBRAS

  • Choi, Eun-Mi;Lee, Hei-Sook
    • 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.

ON HOPF ALGEBRAS IN ENTROPIC JÓNSSON-TARSKI VARIETIES

  • ROMANOWSKA, ANNA B.;SMITH, JONATHAN D.H.
    • 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.

TOWARDS UNIQUENESS OF MPR, THE MALVENUTO-POITIER-REUTENAUER HOPF ALGEBRA OF PERMUTATIONS

  • Hazewinkel, Michiel
    • Honam Mathematical Journal
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    • v.29 no.2
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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.

A study on elementary school algebra -focusing on 'early algebra'- (초기대수'를 중심으로 한 초등대수 고찰)

  • 김성준
    • 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.

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BCK-ALGEBRAS WITH SUPREMUM

  • Jun, Young-Bae;Lee, Kyoung-Ja;Park, Chul-Hwan
    • The Pure and Applied Mathematics
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    • v.16 no.1
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    • pp.1-11
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    • 2009
  • The notion of a BCK-algebra with supremum (briefly, sBCK-algebra) is introduced, and several examples are given. Related properties are investigated. We show that every sBCK-algebra with an additional condition has the condition (S). The notion of a dry ideal of an sBCK-algebra is introduced. Conditions for an sBCK-algerba to be an spBCK-algebra are provided. We show that every sBCK-algebra satisfying additional condition is a semi-Brouwerian algebra.

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AN ALGEBRA WITH RIGHT IDENTITIES AND ITS ANTISYMMETRIZED ALGEBRA

  • Choi, Seul-Hee
    • Honam Mathematical Journal
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    • v.30 no.2
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    • pp.273-281
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    • 2008
  • We define the Lie-admissible algebra NW$({\mathbb{F}}[e^{A[s]},x_1,{\cdots},x_n])$ in this work. We show that the algebra and its antisymmetrized (i.e., Lie) algebra are simple. We also find all the derivations of the algebra NW$(F[e^{{\pm}x^r},x])$ and its antisymmetrized algebra W$(F[e^{{\pm}x^r},x])$ in the paper.