Frames and rings in congruence modular varieties
The von Neumann coordinatization of frames in modular lattices is applied to algebras in congruence modular varieties and frames induced by abelian congruences. This can be seen as a `projective counterpart' to the affine coordinatization of abelian algebras and abelian congruences given by the reviewer [Geometrical methods in congruence modular algebras, Mem. Am. Math. Soc. 286 (1983; Zbl 0547.08006)] and by the reviewer and \textit{Ch. Herrmann} [Houston J. Math. 5, 503-523 (1979; Zbl 0458.08007)]. Given abelian congruences \(\alpha\) and \(\beta\) with A/\(\alpha\cong A/\beta\) the notion of ``homomorphism'' between \(\alpha\) and \(\beta\) is introduced. For \(\alpha =\beta\) this leads to a ring End(\(\alpha)\) of homomorphisms which turns out to be isomorphic to the ring given by the von Neumann coordinatization of the canonical 2-frame (M3) in Con(\(\alpha)\). Given a variety V, let FV(x,y) be the algebra freely generated by \(\{\) x,y\(\}\), let \(\tau\) be the congruence \(\theta\) (x,y) and \(\tau '=[\tau,\tau]\). Then \(\Gamma:=\tau /\tau '\) is called the `generic' congruence of V. It turns out that End(\(\Gamma)\) is isomorphic to the ring R(V), associated with a modular variety in the work of \textit{R. Freese} and \textit{R. McKenzie} [The commutator, an overview. Wadsworth, New York (1987)]. If W is a subvariety of V then R(W) is a homomorphic image of R(V). Several technical results are of independent interest. In particular, let \(\sigma\) : \(A\to B\) be a homomorphism, then a `homomorphic section' is a homomorphism \(\mu\) : \(B\to A\) with \(\sigma \mu =id\). If \(\sigma\) is surjective, then for all \(\beta\) from Con(B) \([Ker(\sigma),\mu (\beta)]=Ker(\sigma)\cap \mu (\beta)\), i.e. the commutator coincides with the meet. Applying this to free algebras and congruences given by partitions on the free generating set an easy criterion is found when the commutators of such partition congruences coincide with their meets. Moreover a number of useful congruence identities for partition congruences are derived. Finally, an equivalent definition of the commutator as a ``universal preimage'' congruence is provided.
scientific article; zbMATH DE number 4031767
scientific article