Remarks about a closure algebra in which closed elements are open

Abstract

1. C-5 algebras. A C-5 algebra is defined as follows: DEFINITION 1.1. An ordered quadruple a= (A, *, -, C) is said to be a C-5 algebra if the following conditions are satisfied: (i)(A, *,-) is a Boolean algebra. (We shall use + as the Boolean operation of addition; < as the Boolean relation of less than or equal to; 0 and 1 as the zero and unit elements of the Boolean algebra.) (ii) C is a unary function from A to A. (iii) x < Cx for all x in A. (iv) CCx= Cx for all x in A. (v) C(x+y) = Cx+ Cy for all x and y in A. (vi) CO = 0. (vii) -C-Cx=Cx for all x in A. REMARK. The algebra a of 1.1 is said to be a closure algebra if it satisfies (i)-(vi).