Ideal lattice isomorphisms of semigroups

A lattice isomorphism $\psi$ of a semigroup $S$ upon a semigroup $T$ is called an ideal lattice isomorphism if it induces a bijection of the set of ideals of $S$ onto the corresponding set of $T$. Left and right ideal lattice isomorphisms are defined in a similar way. The order on idempotents and the property of being a subgroup are proved to retain under lattice isomorphisms of these kinds. The property of a semigroup of being decomposable in a semilattice of Archimedean semigroups is retained as well. Mappings that induce ideal lattice isomorphisms of idempotent semigroups are described. In particular, each left ideal or right ideal lattice isomorphism of an idempotent semigroup is induced by an isomorphism.