Remarks about a closure algebra in which closed elements are open
Jean E. Rubin · Proceedings of the American Mathematical Society · 1956
The closure algebra1 which is considered here is one in which the closure operation C satisfies the condition :(1) -C -Cx = Cx for all x in the domain of C.A closure algebra which satisfies (1) will be called a C-5 algebra.2In §1, it is shown that condition (1) may be expressed in many diverse forms.For example, in a closure algebra, (1) is equivalent to: C(x-Cy) = CxCy for all x and y in the domain of C.It is also shown that the closure operator in a C-5 algebra is completely distributive over addition.In §2, MacNeille's extension3 of a C-5 algebra is defined and it is shown that the extension is a C-5Received by the editors January 18, 1955. 1 For a definition of a closure algebra and a discussion of its properties see Mc-Kinsey and Tarski [3]. 1 The name "C-5 algebra" is used because the algebra is an algebraic model (characteristic matrix) for Lewis' system of modal logic S5.A closure algebra is an algebraic model for Lewis' system S4.For a definition of S4 and S5 see Lewis and Langford [2], Appendix II. 3 For a definition of MacNeille's extension of a partially ordered set see MacNeille [4, p. 443 ff.].