Semisimples in Varieties of Commutative Integral Bounded Residuated Lattices

Abstract

In any variety of bounded integral residuated lattice-ordered commutative monoids (bounded residuated lattices for short) the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. Finally, we analyze the relationship with the property "to have radical term", especially for k-radical varieties, and for the hierarchy of varieties (WL k)k>0 defined in Cignoli and Torrens (Studia Logica 100:1107---1136, 2012 [7]).