We prove that the equation «1 theory of semigroup* becomes undecidable if we add a semilattice structure with a 'touch of symmetric difference'. At 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 undeddabUi ty of modal logics extending the Lambek Calculus and undeddabiBty of Arrow Logics with an associative arrow modality. Below we prove that semilattice-ordered semigroups with a 'touch of symmetric difference' have a 'hereditarily' undecidable equational theory. There were results and methods available showing that many discriminator varieties having an associative binary operation are undecidable. See e.g. AndrekaGivant-Nemeti [2] for a relatively large variety of such methods and references. For completeness we note that Tarski around 1950 and Maddux around 1977 already had such kinds of results. One of the main aims of our method herein is to elaborate a general method which works for varieties which are very far from discriminator varieties. By 'very far' we mean that we do not use discriminator algebras at any point of our proof. E.g. one of the proofs in Kurucz-Nemeti-Sain-Gyuris [10] proves that a certain variety, say, V is undecidable by proving that for every algebra A € V one can define a new structure B (with both universe and operations different from those of A) being a subdirect product of discriminator algebras, then one can use the discriminator property of B. In contrast, the present proof method does not use discriminator algebras even in this indirect way.