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Logic Journal of IGPL 1993 1(1):91-98; doi:10.1093/jigpal/1.1.91
© 1993 by Oxford University Press
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Original Articles

Undecidable Varieties of Semilattice—ordered Semigroups, of Boolean Algebras with Operators, and logics extending Lambek Calculus

Á. KURUCZ, I. NÉMETI, I. SAIN and A. SIMON

Mathematical Institute of the Hungarian Academy of Sciences Budapest, PO Box 127, H-1364, Hungary E-mail: kurucz{at}rmk530.rmki.kfki.bu, {h1469nem,h1468sain} @ella.hu, andras@rmk30.rmki.kfki.hu

We prove that the equational theory of a semigroups becomes undecidable if we add a semilattice structure with a ‘touch of symmetric difference’. As a corollary we obtain that the variety of all Boolean algebras with an associative binary operator has a ‘hereditarily’ undecidable equational theory. Our results have implications in logic, e.g. they imply undecidability of modal logics extending the Lambek Calculus and undecidability of Arrow Logics with an associative arrow modality.


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