Congruence Modularity Implies the Arguesian Law for Single Algebras with a Difference Term

Recently, a generalization of commutator theory has been developed for algebraic systems belonging to a congruence modular variety. This general commutator theory is used here both to provide a very simple proof of a classical result by R. Freese and B. Jónsson and to solve a problem raised by S. Tschantz. Some other known results are improved or are given a simple proof. Moreover, we pursue the goal of extending commutator theory to a somewhat larger setting; thus we give a more general version of the result by Freese and Jónsson (partially solving a problem we raised in P. Lipparini, 1994, Trans. Amer. Math. Soc.346, 177–202).