Vol. 23, No. 2, 1967

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Congruences on regular semigroups

Norman R. Reilly and Herman Edward Scheiblich

Vol. 23 (1967), No. 2, 349–360

For any regular semigroup S the relation 𝜃 is defined on the lattice, Λ(S), of congruences on S by: (ρ,τ) 𝜃 if and only if ρ and τ induce the same partition of the idempotents of S. Then 𝜃 is an equivalence relation on Λ(S) such that each equivalence class is a complete modular sublattice of Λ(S). If S is an inverse semigroup then 𝜃 is a congruence on Λ(S), Λ(S)∕𝜃 is complete and the natural homomorphism of Λ(S) onto Λ(S)∕𝜃 is a complete lattice homomorphism. Any congruence on an inverse semigroup S can be characterized in terms of its kernel, namely, the set of congruence classes containing the idempotents of S. In particular, any congruence on S induces a partition of the set ES of idempotents of S satisfying certain normality conditions. In this note, those partitions of ES which are induced by congruences on S and the largest and smallest congruences on S correspond ing so such a partition of ES are characterized.

Mathematical Subject Classification
Primary: 20.92
Received: 13 June 1966
Revised: 8 November 1966
Published: 1 November 1967
Norman R. Reilly
Herman Edward Scheiblich