A study on Boolean like algebras

The idea of Boolean ring has been generalized in a variety of ways by several Mathematicians. Von-Neumann regular rings, p-rings, p^k−rings, Boolean–like rings , Associative rings and P₁ & P₂−rings are a few interesting ring theoretic generalizations of Boolean rings. Boolean-like rings developed by A.L. Foster [3] preserve and confirm to the formal properties of the Boolean ring. V. Swaminathan [4] studied the structure of Boolean-like rings and established new results regarding Boolean-like rings besides extending many of the results true for Boolean rings to Boolean-like rings. The concept of Boolean- like algebra is introduced. It is proved that a Boolean-like algebra is a Boolean algebra if and only if aΛa = a for all a. Some results pertaining to the mutual implications of Boolean-like algebra and Boolean-like ring are presented in this paper. As in the case of Boolean algebra and Boolean ring, it is proved that the Boolean-like algebra and Boolean-like ring are equivalent abstract structures.