Abstract
Nuclei and prenuclei have proved popular for providing quotients in frame theory; moreover the collection of all nuclei is itself a frame with useful functorial properties. Another natural approach to quotients in the frame setting, much used by algebraists, uses congruences as a tool. In partial frames, nuclei no longer suffice for constructing quotients, but congruences do, and it is to these that we turn in this paper. Partial frames are meet-semilattices in which not all subsets need have joins; a selection function, \(\mathcal {S}\), specifies, for all meet-semilattices, certain subsets under consideration; an \(\mathcal {S}\)-frame then must have joins of all such subsets and binary meet must distribute over these. Examples of these are \(\sigma \)-frames, \(\kappa \)-frames and frames themselves. The first part of this paper investigates the structure and functorial properties of the congruence frame of a partial frame; the second constructs the least dense quotient, which we call the Madden quotient, in three different ways. We include some functoriality properties in the subcategory of partial frames with skeletal maps.
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A. Schauerte acknowledges support from the NRF.
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Frith, J.L., Schauerte, A. The congruence frame and the Madden quotient for partial frames. Algebra Univers. 79, 73 (2018). https://doi.org/10.1007/s00012-018-0554-4
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DOI: https://doi.org/10.1007/s00012-018-0554-4