Notion of prime congruences

Question

We have the idea of a prime ideal in a commutative ring $R$ but in universal algebra, we generalize the notion of ideal to that of a congruence. I’ve thought over the question of what a prime congruence would look like and all I could say is that maximal (proper) congruences should be prime and the trivial congruence shouldn’t be. Given the importance of prime ideals, I’d imagine someone’s put thought into how this would generalize though I haven’t been able to find anything.

I ask this question because I’m thinking about that the spectrum of a (commutative?) algebraic structure would look like in general. Any pointers would be helpful.

(Should I put the AG tag here?)

Answer 1

What's a prime congruence?

The lattice of ideals of a commutative ring has three well-studied operations:

1. Sum: $I+J$
2. Intersection: $I\cap J$
3. Product: $I\cdot J$.

Primeness is defined with the third operation: $P$ is prime if $I\cdot J\subseteq P$ implies $I\subseteq P$ or $J\subseteq P$.

For general algebraic structures, we have three corresponding operations on congruences:

1. Join: $\alpha\vee \beta$
2. Meet or Intersection: $\alpha\wedge \beta = \alpha\cap \beta$
3. Commutator: $[\alpha,\beta]$.

General primeness is defined in terms of the commutator operation: congruence $\pi$ is prime if $[\alpha,\beta]\subseteq \pi$ implies $\alpha\subseteq \pi$ or $\beta\subseteq \pi$.

You can find this definition (for congruence modular varieties) on page 64 of

Freese, Ralph; McKenzie, Ralph,
Commutator theory for congruence modular varieties.
London Mathematical Society Lecture Note Series, 125. Cambridge University Press, Cambridge, 1987.

Remark: for noncommutative rings, $[I,J] = IJ+JI$. For groups or Lie algebras, $[I,J]$ is the usual commutator of normal subgroups or Lie ideals.

Since you are thinking about the spectrum, let me point you to a couple of papers:

Agliano, Paolo,
The prime spectrum of a universal algebra.
Riv. Mat. Pura Appl. No. 5 (1989), 97-105.

Georgescu, George; Mureşan, Claudia,
Going up and lying over in congruence-modular algebras.
Math. Slovaca 69 (2019), no. 2, 275-296.

Answer 2

Primeness is not a statement purely in terms of congruences; the naive translation also gives special treatment to the binary operation $\cdot$ and the nullary operation $0$. Given an algebra $\mathfrak{A}$, a binary term $*$, and a nullary term $c$ in the relevant language, we can define a $(*,c)$-prime congruence on $\mathfrak{A}$ as one satisfying $a*b\approx c$ (or maybe "$a*b\approx c*c$" is more natural?) iff $a\approx c$ or $b\approx c$; however, it's not clear how to find the "right" $*$ and $c$ in a given algebra. Moreover, and more importantly I think, there's no particular reason to expect this notion to be important in general algebras.

That said, there is at least one notion of "prime congruence" in the literature. In the book Kearnes/Kiss, The shape of congruence lattices, the authors introduce the following notion: a congruence $\theta$ on an algebra ${\bf A}$ is prime iff $\theta$ is meet-irreducible in $\mathrm{Con}({\bf A})$ and additionally $\theta\blacktriangleleft 1$. This latter condition is a bit of a definition chase, and the relevant pages are 27/28, 126, and 147. This notion does generalize that of prime ideals from ring theory (in the sense that in a commutative unital ring $R$, an ideal $I$ is prime iff the corresponding congruence $\approx_I$ is Kearnes/Kiss-prime). However, I think the technical detail needed to make sense of the definition indicates just how special the setting of rings is; I suspect that we should not in general expect KK-primeness to have a simple description in general (even natural) classes of algebras rather than rings.