# Classifications of finite simple objects

I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "classification" in the informal sense of the term, but also answers that take into account a more sophisticated viewpoint (as in this mo question) are welcome, as well as answers that consider reasonable weaker notions of classification (as e.g. this mo question, that asks about a classification of finite simple groups up to finitely many exceptions).

The (apperarently incomplete) case of finite commutative rings has already been discussed here. The finite p-groups have been considered here.

Also answers/remarks involving the classification of "finite simple objects" of some category (or higher category) are considered in topic (provided that a reasonable definition of "finite" and "simple" is suggested in that context).

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Did you mean finite or are you also interested in finite dimensional objects? Some of the answers are over non-finite fields. – Yiftach Barnea Apr 3 '11 at 18:05
I think you might enjoy reading en.wikipedia.org/wiki/ADE_classification – Igor Pak Apr 3 '11 at 20:17

The classification that has been an inspiration for over a century is the Cartan-Killing classification of finite-dimensional simple complex Lie algebras. This precedes Wedderburn theory. This is surely (?) the most significant classification.

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The classification isn't known (to me anyway), but there is an interesting notion of 'simplicity' for finite graphs:

A graph homomorphism is a map from vertices to vertices such that adjacent vertices remain adjacent. Two graphs are homomorphism-equivalent if there are homomorphisms in both directions. A graph is said to be a 'core' if all endomorphisms are automorphisms. Every finite graph $G$ maps onto a core, and this core is unique up to isomorphism; moreover, the core can be obtained as an induced subgraph that is the image of an idempotent endomorphism of $G$. So some problems in finite graph theory reduce to problems about cores in the same way that some problems in finite group theory reduce to problems about simple groups.

So what are the core graphs? It is easy to see, for instance, that all complete graphs are cores, and that the only bipartite cores are the complete graphs on less than 3 vertices.

I know about this from Peter Cameron, who has done a significant amount of work on the subject.

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For any field $K$, finite-dimensional simple $K$-algebras (simple in the sense of having no proper non-zero two sided ideals) are famously classified by a theorem of Wedderburn. More generally, simple Artinian rings are classified by the Artin-Wedderburn theorem. Some say that the latter was the beginning of ring theory as a grown up discipline.

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Central simple algebras over local and global fields are classified up to Morita equivalence by class field theory.

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Somewhat related to Igor Pak's comment is the classification of the finite irreducible Coxeter groups. Of course they are not "simple" as groups, but the irreducibility seems the natural replacement for simplicity; here "irreducible" means that the Coxeter diagram is connected, or equivalently, that the Coxeter system does not split as the direct product of two Coxeter systems.

The outcome is the famous list $A_n$, $B_n = C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$, $G_2$, $H_3$, $H_4$, $I_{(n)}$, where the last three items are maybe less well known people only familiar with Lie groups and Lie algebras and/or algebraic groups since they don't survive there.

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Kurokawa's Zeta functions of categories contains the following definitions.

Definition 1: In a category $C$ with a zero object, a simple object is an object $X$ such that, for every object $Y \in C$, $\text{Hom}(X, Y)$ consists only of monomorphisms and zero-morphisms. (I don't know enough to say whether this is equivalent to the nLab definition.)

Definition 2: A non-zero object of $C$ is finite if $\text{Hom}(A, A)$ is finite.

So there are some easy examples: if $R$ is a commutative ring, then the finite simple objects of $R\text{-Mod}$ are precisely the simple modules $R/m$ where $m$ is a maximal ideal with finite residue field. In particular if $R = \mathbb{Z}$ then the finite simple objects are the modules $\mathbb{Z}/p\mathbb{Z}$.

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This more of a generalization than a direct answer, but it may prove interesting and/or useful to the original poster.

Universal Algebra has the student look at the lattice of congruences of various algebraic structures. Simple algebras are then algebras with a lattice of two congruences, i.e. no nontrivial congruences, which for groups corresponds to no nontrivial normal subgroups. A related concept is that of subdirectly irreducible algebra. Here the congruence lattice has a unique nontrivial smallest congruence, that is a congruence which is contained in any other congruence on the algebra (except the trivial one induced by an isomorphism). Any simple algebra is subdirectly irreducible. The utility of the latter concept is that any algebra has a representation as a subdirect product (subalgebra of a direct product) of subdirectly irreducible algebras. So when one looks at classes of algebras (of a single similarity type) which are closed under taking direct products and subalgebras (and often isomorphic images of such), one finds the subdirectly irreducible algebras as natural building blocks to form the class.

I recall that semilattices and Boolean algebras had nice classifications of finite subdirectly irreducible algebras. I am confident the general algebra literature contains more.