The main goal of this paper is to introduce analogously modal operators on $L$-algebras and study their properties. To begin with, we introduce the notion of modal operators on $L$-algebras and investigate some important properties of this operator. In order for the kernel of modal operator to be ideal, we investigate what conditions are required. Relations between modal operator and endomorphism of $L$-algebras are investigated. Also, we define the concept of positive $L$-algebra and some characterizations of positive $L$-algebra are established. Finally, we introduce a map $k_{a}$ and show that $k_{a}$ is a modal operator and we prove that the set of all $k_{a}$ on a positive $L$-algebra makes a dual BCK-algebra.

It is known that even duals of a Banach algebra A with one of Arens products are Banach algebras, these products are natu-ral multiplications extending the one on A. But the essence of A ∗,is completely di, erent. By de, ning new products, we investigate some algebraic and spectral properties of odd duals of A. We will show re-lations between these products and Arens products, weak or weak-star continuity, commutativity and unit elements of these algebras. We also determine the spectrum and multiplier algebra for A ∗, , and we calculate the quasi-inverses, spectrum and spectral radius for elements of these kinds of algebras.

We show that the radical bundle of an algebra bundle is a characteristic ideal bundle. Further we prove an algebra bundle is semisimple if and only if its derivation algebra bundle is either semisimple or zero.

We introduce and investigate the concept of flexibility in a groupoid, and extend it to RM-algebras, and present several examples and these properties. We observe that condition flexibility is strong, and most known algebras of logic will be trivial with this condition. It is proved that every KL-algebra and BE-algebra are flexible, and every flexible CI-algebra is a BE- algebra. We present that the flexible groups are trivial. Moreover, we apply two conditions (I) and (H) in a groupoid and prove that in a RM-algebra, they will be equivalent.