AcademicArticleSCO_85053634797

abstract

• © 2018, Springer Nature B.V. We are concerned with the boolean or more generally with the complemented properties of idioms (complete upper-continuous modular lattices). Simmons (Cantor¿Bendixson, socle, and atomicity. http://www.cs.man.ac.uk/~hsimmons/00-IDSandMODS/002-Atom.pdf, 2014) introduces a device which captures in some informal speaking how far the idiom is from being complemented, this device is the Cantor-Bendixson derivative. There exists another device that captures some boolean properties, the so-called Boyle-derivative, this derivative is an operator on the assembly (the frame of nuclei) of the idiom. The Boyle-derivative has its origins in module theory. In this investigation we produce an idiomatic analysis of the boolean properties of any idiom using the Boyle-derivative and we give conditions on a nucleus j such that [j, tp] is a complete boolean algebra. We also explore some properties of nuclei j such that Aj is a complemented idiom.