Construction of quotient BCI(BCK)-algebra via a fuzzy ideal |
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Authors: | Yong Lin Liu Jie Meng |
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Institution: | 1. Department of Mathematics, Nanping Teachers College, 353000, Nanping, Fujian, P.R. China 2. Department of Mathematics, NorthWest University, 710069, Xian, P.R. China
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Abstract: | The present paper gives a new construction of a quotient BCI(BCK)-algebraX/μ by a fuzzy ideal μ inX and establishes the Fuzzy Homomorphism Fundamental Theorem. We show that if μ is a fuzzy ideal (closed fuzzy ideal) ofX, thenX/μ is a commutative (resp. positive implicative, implicative) BCK (BCI)-algebra if and only if μ is a fuzzy commutative (resp positive implicative, implicative) ideal ofX. Moreover we prove that a fuzzy ideal of a BCI-algebra is closed if and only if it is a fuzzy subalgebra ofX. We show that if the period of every element in a BCI-algebraX is finite, then any fuzzy ideal ofX is closed. Especially, in a well (resp, finite, associative, quasi-associative, simple) BCI-algebra, any fuzzy ideal must be closed. |
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