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In this paper, we investigate the properties of A-nuclei and [Formula: see text]-A-nuclei of a quasigroup including connections between components of [Formula: see text]-A-nuclei and local identity elements. To make the study easier, we give the explicit structure of [Formula: see text]-A-nuclei of a loop, a unipotent left loop and a unipotent right loop in terms of translation maps. We also find the left, right and middle A-nuclei of a quasigroup which is an isostrophic image of a loop in terms of translation maps.

In this paper, we investigate the properties of [Formula: see text]-A-nuclei of a quasigroup including relations between them and relations between their respective component sets, where [Formula: see text]. We also find connections between components of [Formula: see text]-A-nuclei of a quasigroup and components of [Formula: see text]-A-nuclei of the isostrophic images. Further, we investigate the properties of various inverse quasigroups using the derived connections. These properties will not only make the study of [Formula: see text]-A-nuclei of a quasigroup simple but also reduce the time required in the computation of [Formula: see text]-A-nuclei of a quasigroup for different values of [Formula: see text].