Congruence modularity implies the Arguesian law for single algebras with a difference term
A general commutator theory which was developed for algebras belonging to a congruence modular variety is used to improve known results or give simpler proofs to such results and to (partially) solve some problems. Commutator theory is extended to a somewhat larger setting. Generalizing a result of Freese and Jónsson, it is proved that the congruence lattice of a congruence modular algebra having a difference term is Arguesian. A difference term on an algebra \({\mathbb A}=(A,F)\) is a ternary term \(d\) on \({\mathbb A}\) satisfying \(d(x,y,y)=x\) for all \(x,y\in A\) and \(d(x,x,y) [\alpha,\alpha] y\) for all \(\alpha\in\text{Con }{\mathbb A}\) and all \((x,y)\in\alpha\). Here \([\alpha,\beta]\) denotes the commutator of \(\alpha,\beta\in\text{Con }{\mathbb A}\).
scientific article; zbMATH DE number 1346324
scientific article