Sheaves and duality

It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalization of this fact and prove a converse of the generalization....

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Bibliographic Details
Published in:Journal of pure and applied algebra 2018-08, Vol.222 (8), p.2164-2180
Main Authors: Gehrke, Mai, v. Gool, Samuel J.
Format: Article
Language:English
Subjects:
General Topology
Mathematics
Rings and Algebras
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Summary:It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalization of this fact and prove a converse of the generalization. To be precise, we exhibit a one-to-one correspondence (up to isomorphism) between soft sheaf representations of universal algebras over stably compact spaces and frame homomorphisms from the dual frames of such spaces into subframes of pairwise commuting congruences of the congruence lattices of the universal algebras. For distributive-lattice-ordered algebras this allows us to dualize such sheaf representations.
ISSN:0022-4049
1873-1376
DOI:10.1016/j.jpaa.2017.09.004