If S is an inverse semigroup
and 𝜃 is the relation on the lattice Λ(S) of congruences on S defined by saying that
two congruences ρ1,ρ2 are 𝜃-equivalent if and only if they induce the same partition
of the idempotents then 𝜃 is a congruence on Λ(S) and each 𝜃-class is a complete
modular sublattice of Λ(S). If X is a partially ordered set then JX denotes the
inverse semigroup of one-to-one partial transformations of X which are order
isomorphisms of ideals of X onto ideals of X, while if X is a semilattice, TX
denotes the inverse subsemigroup of JX consisting of those elements α whose
domain Δ(α) and range ∇(α) are principal ideals. It is shown that any inverse
semigroup is isomorphic to an inverse subsemigroup of JX for some semilattice
X.
