Extending congruences on semigroups

Abstract

The two main results are: (1) Let S be a semigroup which satisfies the relation a b c d = a c b d abcd = acbd , let A be a subsemigroup of Reg S which is a band of groups and let [ φ ] [\varphi ] be a congruence on A. Then [ φ ] [\varphi ] can be extended to a congruence on S. (2) Let S be a compact topological semigroup which satisfies the relation a b c d = a c b d abcd = acbd , let A be a closed subsemigroup of Reg S and let [ φ ] [\varphi ] be a closed congruence on A such that dim φ ( A ) | H = 0 \dim \,\varphi (A)|\mathcal {H} = 0 . Then [ φ ] [\varphi ] can be extended to a closed congruence on S.