Abstract
In the present paper, we given sorne equivalent conditions of (*)-ideals and positive implicative ideals in BCI-algebras and we clarify the relation of the two classes of ideals. Also, we obtain further properties of these ideals. Finally, we discuss a quotient algebra of a BCI-algebra by a closed (*)-ideal and a quotient algebra by positive implicative ideal.
Highlights
A nonempty subset J of a BCI-algebra X is called an ideal of X if (i)OEJ, (ii) x E I whenever X*Y E J and y E J
Theorem 2 : If J is an ideal of a BCI-algebra X, and L (X) ~ J, J is a (*)-ideal of X
Theorem 16 : If J is an ideal of a BCI-algebra X, the following are equivalent : (8) J is positive implicative, (9) (x*y) *Y E J implies X*Y E J, for all x,y E X, (10) (x *Y)* z E J implies (x * z) *(y* z) E J for ali x,y,z E J
Summary
Theorem 9 : An ideal J of a BCI-algebra X is a closed (*)-ideal if and only if O* x E J for all x E X. Proof : Let J be a closed (*)-ideal and x E X. Theorem 10 : A nonempty subset J of a BCI-algebra X is a closed (*)-ideal of X if and only if (i) O E J, and (v) for all x,y,z E X, X*Y E J and y E J imply x * z E J.
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