Natural dualities for varieties of distributive lattices with a quantifier
Subdirectly irreducible non-idempotent groupoids satisfying and are studied.
We consider algebras determined by all normal identities of -algebras, i.e. algebras of many-valued logics. For such algebras, we present a representation based on a normalization of a sectionally involutioned lattice, i.e. a -lattice, and another one based on a normalization of a lattice-ordered group.