# Boolean algebras (with operators)

I am looking for an introductory, recent discussion (in the last 30 years or so) of the link between Boolean algebras with operators and modal logic.

My interest in the topic is in understanding the ramifications of the fact that prime ideals (filters) of a Boolean algebra correspond to homomorphisms and the relationship between partial orders in a distributive lattice and homomorphisms. It has been claimed (Fox, C. and Lappin, S., 2005, Foundations of Intensional Semantics, Malden, MA: Blackwell.) there are serious ramifications of these facts for natural language semantic theories that employ modal logic and I want to assess the arguments.

Can you recommend any relatively introductory books or articles (written in the last 30 years or so) of the link between Boolean algebras with operators and modal logic?

I would also like to know which of the following texts would be better to read to have an introduction to Boolean algebra for someone with limited time: Halmos $\textit{Lectures on Boolean Algebra}$ or Halmos and Givant $\textit{Introduction to Boolean Algebras}$.

I am conscious of the fact that Halmos and Givant (p.x) write their introduction is "a substantially revised version of the second author’s $\textit{Lectures on Boolean Algebras}$", that "tries to steer a middle course" between "elementary texts" and "advanced treatises". But I fear that it may be too long for me to read, and am wondering how much I will lose out by reading the earlier book by Halmos (and not the later co-authored book).

Bearing in mind my interests, which should I read?

• Have you heard about Modal Logic, by Blackburn, de Rijke and Venema (2002)? I think this use to be a great reference... – amrsa Mar 14 '17 at 14:54
• I've dipped into it, but not much. I'll go back to it. – user65526 Mar 14 '17 at 15:03
• In the Blackburn, de Rijke, Venema book what you're asking for is chapter 5. – Apostolos Mar 14 '17 at 21:50