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September 9, 2019

Semisimplicity and the Excluded Middle (continued)

I will present some recent joint research with Tomáš Lávička concerning the logical interpretation of semisimplicity. We shall see that, under some mild assumptions, a quasivariety (or a logic) is semisimple if and only if it enjoys a certain general form of the law of the excluded middle. We then apply this general result to provide a simple proof of the result of Kowalski that a variety of FLew-algebras (i.e. bounded commutative integral residuated lattices) is semisimple if and only if it validates the equation 1 = x v ~(x^n) for some n. In addition to being brief and purely syntactic, our proof also has the virtue of immediately extending to varieties of bounded commutative residuated lattices without any additional work.