Department of Mathematics
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The concept of varieties defined by forbidden algebras (for example, the familiar modularity is equivalent to the non-occurrence of the 5-element pentagon as a sublattice) is kind of unique to lattice theory. In this talk, we try to extend this phenomenon to other algebraic theories, in particular, to various generalizations of lattices - generalizations obtained by syntactic modifications of lattice identities including e.g regular identities, external identities, normal identities and hyper-identities. We prove these new theorems using several known results on subdirectly irreducible algebras in regular varieties of lattices as well as the fact that forbidden lattices are indeed finite subdirectly irreducible sublattices of free lattices.