congruence_modular - MathStructures

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=====Congruence modularity=====

An algebra is \emph{congruence modular} (or CM for short) if its lattice of congruence relations is [[modular lattices|modular]].

A class of algebras is \emph{congruence modular} if each of its members is congruence modular.

Congruence modularity holds for many 'classical' varieties such as groups and rings.

A Mal'cev condition (with 4-ary terms) for congruence modularity is given by Alan Day, \emph{A characterization of modularity for congruence lattices of algebras.}, 
Canad. Math. Bull., \textbf{12}, 1969, 167-173 [[http://www.ams.org/mathscinet-getitem?mr=40:1317|MRreview]]

Another Mal'cev condition (with ternary terms) for congruence modularity is given by H.-Peter Gumm, \emph{Congruence modularity is permutability composed with distributivity}, 
Arch. Math. (Basel), \textbf{36}, 1981, 569-576 [[http://www.ams.org/mathscinet-getitem?mr=82j:08009|MRreview]]

Several further characterizations are given by Steven T. Tschantz, \emph{More conditions equivalent to congruence modularity}, 
Universal algebra and lattice theory (Charleston, S.C., 1984), 
Lecture Notes in Math.
\textbf{1149}, 1985, 270-282,  [[http://www.ams.org/mathscinet-getitem?mr=87e:08009|MRreview]]

=== Properties that imply congruence modularity ===

[[Congruence n-permutable]] for $n=2$ or $n=3$.

[[Congruence distributive]]

=== Properties implied by congruence modularity ===

Trace: • first-order_classes • directed_complete_partial_orders • ordered_semigroups • frames • representable_lattice-ordered_groups • commutative_ordered_semigroups • function_rings • boolean_groups • boolean_modules_over_a_relation_algebra • finite_ordered_semigroups_with_zero