Is there a Jacobson congruence?
Let me try to discourage everyone from defining
the Jacobson congruence of an algebraic structure
to be the intersection of all its maximal congruences.
Even for rings, this definition is wrong.
The Jacobson radical of a ring $R$ is the intersection
of maximal left ideals of $R$.
This definition is introduced
to describe the ideal of those elements
that annihilate every simple left $R$-module.
That is, if $R$ is a ring, then the
set of elements that act trivially (i.e., like $0$)
on every simple
$R$-module equals the intersection of the maximal
left ideals of $R$, and this is a 2-sided ideal
of $R$ called the Jacobson radical (usually written $J(R)$).
If instead you form the intersection of all maximal 2-sided ideals
of $R$, then you get the Brown-McCoy radical.
Therefore, it might be reasonable to call the intersection
of all maximal congruences of an algebraic structure
the Brown-McCoy congruence of the structure.
It is reasonable to consider the Frattini subgroup of a group to be the analog of the Jacobson radical of a ring.
That is, if $G$ is a group, then the
set of elements that act trivially (i.e., like $1$)
on every simple
$G$-set
equals the intersection of the maximal
subgroups of $G$, and this is a normal subgroup
of $G$ called the Frattini subgroup (usually written $\Phi(G)$).
The universal algebraic
formulation of the Frattini congruence was
first introduced and applied in the paper
E. W. Kiss, S. M. Vovsi,
Critical algebras and the Frattini congruence
Algebra Universalis 34 (1995), 336-344.
Let me define the Frattini congruence of
an algebra $\mathbf{A}$.
For a congruence $\theta$ of $\mathbf{A}$
and a subalgebra $\mathbf{S}$ of $\mathbf{A}$,
define the saturation
of $\mathbf{S}$ by $\theta$ (usually written $\mathbf{S}^{\theta}$)
to be the least subalgebra of $\mathbf{A}$ that contains
$\mathbf{S}$ and is a union of $\theta$-classes.
Equivalently, if $\nu\colon \mathbf{A}\to \mathbf{A}/\theta$
is the natural map, then the saturation of $\mathbf{S}$
by $\theta$ is $\mathbf{S}^{\theta} = \nu^{-1}(\nu(\mathbf{S}))$.
Let's say that $\mathbf{S}$ is saturated by
$\theta$ if $\mathbf{S}^{\theta}=\mathbf{S}$.
The Frattini congruence of $\mathbf{A}$ is the largest congruence
$\Phi(\mathbf{A})$ of $\mathbf{A}$ that saturates all maximal
proper subalgebras of $\mathbf{A}$.
When $\mathbf{A}$ is finitely generated
the Frattini congruence can be characterized
as the 'congruence of nongenerators'
in the following sense: $\Phi(\mathbf{A})$ is the largest congruence on $\mathbf{A}$ with the property that if $\mathbf{S}$ is any subalgebra
of $\mathbf{A}$ and $\mathbf{S}^{\Phi(\mathbf{A})} = \mathbf{A}$,
then $\mathbf{S}=\mathbf{A}$. (This says that $\mathbf{A}$ cannot
be obtained from any proper subalgebra through saturation
by $\Phi(\mathbf{A})$, and that $\Phi(\mathbf{A})$ is the largest congruence of $\mathbf{A}$ with this property.)
