Abstract
In this paper, we study those rings whose semiring of ideals can be given the structure of a Gödel algebra. Such rings are called Gödel rings. We investigate such structures both from an algebraic and a topological point of view.
Our main result states that every Gödel ring R is a subdirect product of prime Gödel rings R i , and the Gödel algebra Id(R) associated to R is subdirectly embeddable as an algebraic lattice into \({{\prod_{i}}Id(R_{i})}\), where each Id(R i ) is the algebraic lattice of ideals of R i that can be equipped with the structure of a Gödel algebra. We see that the mapping associating to each Gödel ring its Gödel algebra of ideals is functorial from the category of Gödel rings with epimorphisms into the full subcategory of frames whose objects are Gödel algebras and whose morphisms are complete epimorphisms.
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Presented by C. Tsinakis.
Marchioni acknowledges partial support from the Spanish projects MULOG2 (TIN2007- 68005-C04), Agreement Technologies (CONSOLIDER CSD2007-0022, INGENIO 2010), the Generalitat de Catalunya grant 2009-SGR-1434, and Juan de la Cierva Program of the Spanish MICINN, as well as the ESF Eurocores-LogICCC/MICINN project (FFI2008-03126-E/FILO). The authors would also like to thank the anonymous referee for several suggestions that have led to a substantial improvement of this work.
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Belluce, L.P., Di Nola, A. & Marchioni, E. Rings and Gödel algebras. Algebra Univers. 64, 103–116 (2010). https://doi.org/10.1007/s00012-010-0092-1
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DOI: https://doi.org/10.1007/s00012-010-0092-1