Prime spectra in modular varieties
Let \(V\) be a congruence modular variety. A congruence \(c\) in \(V\)-algebra \(A\) is prime if \(c\supseteq [a,b]\), \(a,c\in\text{Con}(A)\) implies either \(c\supseteq a\), or \(c\supseteq b\). Denote by \(\text{Spec}(A)\) the set of all prime congruences in \(A\) provided with Zariski topology. It is shown that each irreducible closed subset in \(\text{Spec}(A)\) has a unique generic point. A congruence \(a\) in \(A\) is radical if \(a\) is the intersection of all prime congruences \(b\) in \(A\) such that \(b\supseteq a\). If \(\text{Con}(A)\) satisfies the ascending chain condition on radical congruences, then there exists a commutative ring \(R\) with unit such that \(\text{Spec}(A)\simeq\text{Spec}(R)\) as topological spaces. There is given a characterization of topological spaces \(X\) which are isomorphic to \(\text{Spec}(L)\) for some distributive lattice \(L\). There is also found a Mal'tsev characterization of varieties of algebras in which no nontrivial algebras has a one-element subalgebra. In these varieties the author finds a topological characterization of the spaces \(\text{Spec}(A)\).
scientific article; zbMATH DE number 488268
scientific article