We show that for any class of Boolean algebras with an associative operator, if it contains the complex algebra of ( P ( N ) , ∪ ) (P(N), ∪ ) left parenthesis script upper P left parenthesis double struck upper N right parenthesis comma union right parenthesis , its equational theory is undecidable. Equivalently, any associative normal modal logic valid over the frame ( P ( N ) , ∪ ) (P(N), ∪ ) left parenthesis script upper P left parenthesis double struck upper N right parenthesis comma union right parenthesis is undecidable. This settles a long-open question on the decidability of hyperboolean modal logic [16], and addresses several related problems.
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