• Communications of the Korean Mathematical Society
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• v.20 no.2
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• pp.221-229
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• 2005

In this paper we consider the decompositions of subdirect sums and direct sums in bounded BCK-algebras. The main results are as follows. Given a bounded BCK-algebra X, if X can be decomposed as the subdirect sum $\bar{\bigoplus}_{i{\in}I}A_i$ of a nonzero ideal family $\{A_i\;{\mid}\;i{\in}I\}$ of X, then I is finite, every $A_i$ is bounded, and X is embeddable in the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$ ; if X is with condition (S), then it can be decomposed as the subdirect sum $\bar{\bigoplus}_{i{\in}I}A_i$ if and only if it can be decomposed as the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$ ; if X can be decomposed as the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$, then it is isomorphic to the direct product $\prod_{i{\in}I}A_i$.

• 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 $x,y{\in}X$.

• Journal of applied mathematics & informatics
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• v.5 no.1
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• pp.91-98
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• 1998

In [6] Y. B. Jun et al. fuzzified the concept of BCK-filters in BCK-algebrs and investigated its properties. In this paper we investigate further properties of fuzzy BCK-filters.

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.27-36
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• 2006

In this paper, we establish a generalization of fundamental homomorphism theorem in BCK-algebras by using fuzzy BCK-filters. We prove that if ${\mu}$. (resp. v) is a fuzzy BCK-filter of abounded BCK-algebra X (resp. Y), then $\frac{X{\times}Y}{{\mu}{\times}v}{\approxeq}X/{\mu}{\times}Y/v;\;and\;if\;{\mu}$ and F is a BCK-filter in a bounded BCK-algebra X such that $F/{\mu}$ is a BCK-filter of $X/{\mu}$, then $\frac{X/{\mu}}{F/{\mu}}{\approxeq}X/F$.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.499-518
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• 1995

In this paper we consider more systematically the centralizer C(S) of the set $S = {f_a $\mid$ f_a : X \to X ; x \longmapsto x * a, a \in X}$ with respect to the semigroup End(X) of all endomorphisms of an implicative BCK-algebra X with the condition (S). We obtain a series of interesting results. The main results are stated as follows : (1) C(S) with repect to a binary operation * defined in a certain way forms a bounded implicative BCK-algebra with the condition (S). (2) X can be imbedded in C(S) such that X is an ideal of C(S)/ (3) If X is not bounded, it can be imbedded in a bounded subalgebra T of C(S) such that X is a maximal ideal of T. (4) If $X (\neq {0})$ is semisimple, C(S) is BCK-isomorphic to $\prod_{i \in I}{A_i}$ in which ${A_i}_{i \in I}$ is simple ideal family of X.

• Communications of the Korean Mathematical Society
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• v.22 no.4
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• pp.503-508
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• 2007

Using the notion of posets, a method to make BCK-algebras is considered. We show that if a poset has the least element, then the induced BCK-algebra is bounded.

• Journal for History of Mathematics
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• v.13 no.1
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• pp.125-136
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• 2000

The notion of MV-algebra was introduced by C.C. Chang in 1958 to provide an algebraic proof of the completeness of Lukasiewicz axioms for infinite valued logic. These algebras appear in the literature under different names: Bricks, Wajsberg algebra, CN-algebra, bounded commutative BCK-algebras, etc. The purpose of this paper is to give a topological lattice completion of semisimple MV-algebras. To this end, we characterize the complete atomic center MV-algebras and semisimple algebras as subalgebras of a cube. Then we define the $\delta$-completion of semisimple MV-algebra and construct the $\delta$-completion. We also study some important properties and extension properties of $\delta$-completion.