Published online by Cambridge University Press: 20 January 2009
Let be an arbitrary semigroup. A congruence γ on
is a group congruence if
/γ is a group. The set of group congruences on
is non-empty since
×
is a group congruence. The lattice of congruences on a semigroup
will be denoted by
(
) and the set of group congruences on
will be denoted by
(
). If
(
) is a lattice then it is modular and γ ∨ ρ = γ ο ρ = ρ ο γ for all γ, ρ ε
(
). The main result is that γ ν ρ = γ ο ρ ο γ for any γ ε
(
) and ρ ε
(
) (whence every element of the set
(
) is dually right modular in
(
). This result has appeared, for particular classes of semigroups, many times in the literature. Also γ ν ρ = γ ο ρ ο γ = ρ ο γ ο ρ for all γ, ρ ε
(
) which is similar to the well known result for the join of congruences on a group. Furthermore, if γ ∩ ρ ε
(
) then γ ν ρ = γ ο ρ = ρ ο γ.