In his study of spectra of f -rings via pointfree topology, Banaschewski [6] considers lattices of `-ideals, radical `-ideals, and saturated `-ideals of a given f -ring A. In each case he shows that the lattice of each of these kinds of ideals is a coherent frame. This means that it is compact, generated by its compact elements, and the meet of any two compact elements is compact. This will form the basis of our main goal to show that the lattice-ordered rings studied in [6] are coherent frames. We conclude the dissertation by revisiting the d-elements of Mart́ınez and Zenk [30], and characterise them analogously to d-ideals in commutative rings. We extend these characterisations to algebraic frames with FIP. Of necessity, this will require that we reappraise a great deal of Banaschewski’s work on pointfree spectra, and that of Mart́ınez and Zenk on algebraic frames.

Introduction to category theory in automata and systems

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Frames of ideals of commutative f-rings

References

Frames of ideals of commutative f-rings

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