On infinite variants of De Morgan law in locale theory

Abstract

A locale, being a complete Heyting algebra, satisfies De Morgan law $(a\vee b{)}^{\ast }=a^{\ast }\wedge b^{\ast }$ for pseudocomplements. The dual De Morgan law $(a\wedge b{)}^{\ast }=a^{\ast }\vee b^{\ast }$ (here referred to as the second De Morgan law) is equivalent to, among other conditions, $(a\vee b{)}^{\ast \ast }=a^{\ast \ast }\vee b^{\ast \ast }$, and characterizes the class of extremally disconnected locales. This paper presents a study of the subclasses of extremally disconnected locales determined by the infinite versions of the second De Morgan law and its equivalents.