Vol. 41, No. 1, 1972

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Inverse semigroups of partial transformations and 𝜃-classes

Norman R. Reilly

Vol. 41 (1972), No. 1, 215–235

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 ρ12 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.

Mathematical Subject Classification 2000
Primary: 20M20
Received: 3 June 1970
Revised: 15 September 1971
Published: 1 April 1972
Norman R. Reilly