# Logics for Classes of Boolean Monoids

DOI: 10.1023/B:JLLI.0000028336.64373.f6

- Cite this article as:
- Allwein, G., Demir, H. & Pike, L. Journal of Logic, Language and Information (2004) 13: 241. doi:10.1023/B:JLLI.0000028336.64373.f6

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## Abstract

This paper presents the algebraic and Kripke modelsoundness and completeness ofa logic over Boolean monoids. An additional axiom added to thelogic will cause the resulting monoid models to be representable as monoidsof relations. A star operator, interpreted as reflexive, transitiveclosure, is conservatively added to the logic. The star operator isa *relative modal operator*, i.e., one that is defined in terms ofanother modal operator. A further example, relative possibility,of this type of operator is given. A separate axiom,antilogism, added to the logic causes the Kripke models to support acollection of abstract topological uniformities which become concretewhen the Kripke models are dual to monoids of relations. The machineryfor the star operator is shownto be a recasting of Scott-Montague neighborhood models. An interpretationof the Kripke frames and properties thereof is presented in terms ofcertain CMOS transister networks and some circuit transformation equivalences.The worlds of the Kripke frame are wires and the Kripke relation is a specializedCMOS pass transistor network.