Abstract
For an algebra A, a coalgebra C and a lax entwining structure (A, C, ψ R ), in this note we provide, by means of idempotent morphisms, new criteria that allow to distinguish partial entwining structures between lax entwining structures, and refine upon those criteria given in [3]. We also introduce the notions invertible lax and invertible partial entwining structure in order to discuss differences and common properties between them and the weak entwining structures introduced in [7] and [1]. In [1] some properties of invertible weak C-Galois and invertible weak C-cleft extensions were proved. Continuing the study provided in [1], in the present work we tackle with partial and lax C-Galois and C-cleft extensions, exploring the enfeeblement of properties that arise when passing from weak to partial or lax settings.
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