Let
S
S
be a regular semigroup and let
ρ
\rho
be a congruence relation on
S
S
. The kernel of
ρ
\rho
, in notation
ker
ρ
\ker \rho
, is the union of the idempotent
ρ
\rho
-classes. The trace of
ρ
\rho
, in notation
tr
ρ
\operatorname {tr}\,\rho
, is the restriction of
ρ
\rho
to the set of idempotents of
S
S
. The pair
(
ker
ρ
,
tr
ρ
)
(\ker \rho ,\operatorname {tr}\,\rho )
is said to be the congruence pair associated with
ρ
\rho
. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple
(
(
ρ
∨
L
)
/
L
,
ker
ρ
,
(
ρ
∨
R
)
/
R
)
((\rho \vee \mathcal {L})/\mathcal {L},\ker \rho ,(\rho \vee \mathcal {R})/\mathcal {R})
is said to be the congruence triple associated with
ρ
\rho
. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of
S
S
. For congruence relations
ρ
\rho
and
θ
\theta
, put
ρ
T
l
θ
[
ρ
T
r
θ
,
ρ
T
θ
]
\rho {T_l}\theta \;[\rho {T_r}\theta ,\rho T\theta ]
if and only if
ρ
∨
L
=
θ
∨
L
[
ρ
∨
R
=
θ
∨
R
,
tr
ρ
=
tr
θ
]
\rho \vee \mathcal {L} = \theta \vee \mathcal {L}\;[\rho \vee \mathcal {R} = \theta \vee \mathcal {R},\operatorname {tr}\rho = \operatorname {tr}\theta ]
. Then
T
l
,
T
r
{T_l},{T_r}
and
T
T
are complete congruences on the congruence lattice of
S
S
and
T
=
T
l
∩
T
r
T = {T_l} \cap {T_r}
.