Mono-unary algebras are uniquely determined by their lattices of fuzzy weak subalgebras

In this paper we give an abstract characterization of the lattices that are isomorphic to the lattice of fuzzy weak subalgebras of some partial algebra, and we show that from this lattice we can extract more information about the algebra than from its lattice of weak subalgebras. We use it to prove that the directed graph associated to a unary partial algebra is always uniquely determined (up to isomorphisms) by the algebra's lattice of fuzzy weak subalgebras, and in particular that if two mono-unary partial algebras (i.e., two partial algebras over a signature containing only one operation symbol, which is moreover unary) have their lattices of fuzzy weak subalgebras isomorphic, then they are isomorphic.