Abstract
By an “effective” subdirect decomposition we mean a subdirect decomposition for which an effective construction of the corresponding family of factor congruences is given. In this paper we construct several commuting pairs of factor congruences which decompose any regular semigroup into a pullback product. For regular and completely regular semigroups whose idempotents form subsemigroups belonging to certain varieties of bands, we give precise structural descriptions of the components in these pullback products.
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