Some abstract maximal ideal-like spaces
The motivation of for this paper is to find a common pattern for some facts which occur in a much similar way in both cases of distributive lattices and of commutative rings. Namely, the existence of a certain spatial frame of ideals (the o-ideals for lattices [\textit{W. H. Cornish}, Rev. Roum. Math. Pur. Appl. 22, 1059-1067 (1977; Zbl 0382.06011); the authors, J. Pure Appl. Algebra 45, 213-223 (1987; Zbl 0646.06009)], and the virginal ideals for rings [\textit{F. Borceux}, \textit{H. Simmons} and \textit{G. van den Bossche}, Proc. Lond. Math. Soc., III. Ser. 48, 230-246 (1984; Zbl 0536.18003)]) which is in particular cases isomorphic to the frame of open sets of the maximal ideal space. This, in turn, ensures interesting properties for sheaf representations. Notice that in the first case the lattice of all ideals is distributive, while in the second it is not. The abstract setting that is found to encompass both cases above is that of integral complete lattice-ordered groupoids [\textit{G. Birkhoff}, Lattice theory, 3rd ed. (1967; Zbl 0153.025)], which are algebraic lattices. For brevity, the structures studied are called multiplicative ideal structures (abbreviated mi-structures). The paper has three sections. The first one contains some preliminary material and the description of the frame of radical elements in a multiplicative ideal structure, which are an obvious generalization of the radical ideals in a ring. Notions and results related to some of those in this paragraph appear in slightly different contexts in papers of \textit{K. Keimel} [Acta Math. Acad. Sci. Hung. 23, 51-69 (1972; Zbl 0265.06016)] and \textit{J. Martinez} [Ordered algebraic structures, Lect. Notes Pure Appl. Math. 99, 125-137 (1985; Zbl 0576.06015)]. In the second section the notion of virginal element in a commutative mi-structure is introduced and the relationship between the frames of virginal elements and radical elements, respectively, is established. This permits the use of some general results from the earlier paper of the authors [loc. cit.]. The third section contains the study of normal mi-structures, particular cases of which are the ideal lattices and the lattice of L- ideals of an F-ring. From the viewpoint of this paper, the significant feature of these structures is the role played by the virginal elements and by the morphisms between the frames of radical and virginal elements. Two consequences of these facts are: any algebraic structure representable by the global sections of a sheaf based on the spectrum of a normal mi-structure, has also a representing sheaf based on the compact Hausdorff space of maximal elements; a section representation result for any universal algebra whose congruence lattice is a normal mi-structure in which the virginal elements permute. (From the text.)
integral complete lattice-ordered groupoids
scientific article; zbMATH DE number 4122043
scientific article