Spectra of monoidal-lattices
We study the theory of ideals of \(m\)-lattices. Clearly, this theory extends those of the bounded distributive lattice and of the commutative rings. The main results are: (a) For those \(m\)-lattices \(L\) whose top element is also a multiplication identity (such \(m\)-lattices are also called monoidal-lattices since they are monoidal categories), we establish a separation lemma which is a common generalization of the classical separation lemmas of Krull for rings and of Stone for distributive lattices. Furthermore, we show that the properties of \(\text{Id } L\) are similar to the ideal lattices for both the distributive lattices and rings. Under this circumstance, we discuss extensively their spectra, maximal spectra and minimal spectra. Note that our results not only generalize the corresponding one for distributive lattices, but also have some ring-interpretations. (b) For general \(m\)-lattices, one can't get a similar kind of separation lemma which guarantees the existence of (\(m\)-)prime ideals. We illustrate this with an example of an \(m\)-lattice with an identity, which does not have any (\(m\)-)prime ideals. In other words, our setting to establish spectra theory is, to some extent, the most general. However, for a full discussion of it from the categorical point of view as well as the relationship with module theory over locales we refer to the author's forthcoming paper.
scientific article; zbMATH DE number 549252
scientific article