Prime ideals and maximal ideals on commutative $L$-algebras
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In this paper, we focus on ideals and prime ideals of $CKL$-algebras and applications of prime ideals of $CKL$-algebras. Firstly, we prove that any self-distributive $L$-algebra is a $CKL$-algebra. Conversely, we give an example of $CKL$-algebras that is not a self-distributive $L$-algebra. Furthermore, we give a generation formula of ideals on $CKL$-algebras. Secondly, we give some equivalent descriptions of prime ideals and its properties on $CKL$-algebras. We mainly prove that maximal ideals are prime ideals on $CKL$-algebras. Next, we give a counterexample to show that commutative L-algebras may be not residuated lattices, much less $MV$-algebras. The results show that commutative L-algebras are a true promotion of $MV$-algebras. Therefore, we study some properties of commutative $L$-algebras.
Abstract
In this paper, we focus on ideals and prime ideals of $CKL$-algebras and applications of prime ideals of $CKL$-algebras. Firstly, we prove that any self-distributive $L$-algebra is a $CKL$-algebra. Conversely, we give an example of $CKL$-algebras that is not a self-distributive $L$-algebra. Furthermore, we give a generation formula of ideals on $CKL$-algebras. Secondly, we give some equivalent descriptions of prime ideals and its properties on $CKL$-algebras. We mainly prove that maximal ideals are prime ideals on $CKL$-algebras. Next, we give a counterexample to show that commutative L-algebras may be not residuated lattices, much less $MV$-algebras. The results show that commutative L-algebras are a true promotion of $MV$-algebras. Therefore, we study some properties of commutative $L$-algebras.
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