The finite lattice representation problem and intervals in subgroup lattices of finite groups William DeMeo Math 613: Group Theory 15 December 2009 Abstract A well-known result of universal algebra states: every algebraic lattice is isomorphic to the congruence lattice of an algebra. It is natural to ask whether every finite lattice occurs as the congruence lattice of a finite algebra. This fundamental question, asked over 45 years ago, is among the most elusive problems of universal algebra. Call a finite lattice L representable if it is the congruence lattice of a finite algebra. Pudlák and Tůma proved in 1980 that any finite lattice can be concretely represented as a sublattice of Eq(X), the equivalences of a finite set X. In the same year, Pálfy and Pudlák proved that the finite lattice representation problem is equivalent to a problem about lattices of subgroups of finite groups. In particular, they showed that the following statements are equivalent: (i) Any finite lattice is isomorphic to the congruence lattice of a finite algebra. (ii) Any finite lattice is isomorphic to an interval in the subgroup lattice of a finite group. We discuss this remarkable result and the key ingredients of its proof. 1 Introduction In 1963 Grätzer and Schmidt published the proof their famous result that every algebraic lattice is isomorphic to the congruence lattice of an algebra. This is an important result. For one thing, it shows that there is no lattice-theoretical condition stronger than algebraicity satisfied by all congruence lattices. A question naturally arises in the case of finite lattices. That is, given a finite lattice L, does there exist a finite algebra A such that ConA = L? This problem appears in the 1963 paper mentioned above, but it was undoubtedly asked even before the Grätzer-Schmidt result about algebraic lattices was known. Call a finite lattice representable if it is (isomorphic to) the congruence lattice of a finite algebra. The basic problem is whether every finite lattice is representable. Equivalently, given a finite lattice L, does there exist a finite algebra A such that ConA = L? In this paper, we briefly review some universal algebra and lattice theory background necessary to understand the problem at hand. Then we review some results about G-sets necessary to understand the proof of the Pálfy-Pudlák theorem. Finally, we give a fairly detailed sketch of the proof itself. 2 Algebras and their congruences In this section we present some basic universal algebra and lattice theory that is necessary to understand the finite lattice representation problem. Definition 2.1 A universal algebra A is an ordered pair A = A, F, where A is a nonempty set, called the universe of A, and F is a family of finitary operations on A. 1
2.1 Congruence relations 2 ALGEBRAS AND THEIR CONGRUENCES The arity of an operation f F is the number of operands: f is n-ary if it maps A n into A; nullary, unary, binary, and ternary operations have arities 0, 1, 2, and 3, respectively. An algebra A, F is finite if A is finite and trivial if A = 1. Given two algebras A and B, we say that B is a reduct of A if both algebras have the same universe and A can be obtained from B by simply adding more operations. Some basic examples of universal algebras are semigroups, monoids, groups, rings, modules, lattices, quasigroups, Boolean algebras, etc. As we are particularly concerned with groups and lattices, let us take a moment to define them formally, as they ought to be defined; i.e., as universal algebras. Definition 2.2 A group G is an algebra G,, 1, 1 with a binary, unary, and nullary operation satisfying G1: x (y z) (x y) z G2: x 1 1 x x G3: x x 1 x 1 x 1 for all x, y, z G. Definition 2.3 (lattice) A lattice is an algebra L = L,, with universe L, a partially ordered set, and two binary operations given by x y = g.l.b.(x, y), the meet of x and y, x y = l.u.b.(x, y), the join of x and y. A bounded lattice is one with a least element and a greatest element, often denoted 0 L and 1 L (or sometimes and ), respectively. Some examples of lattices are: subsets of a set, closed subsets of a topology, subgroups of a group, normal subgroups of a group, ideals of a ring, submodules of a module, invariant subspaces of an operator, etc. A lattice L L,, is called distributive if and only if, for all x, y, z L, x (y z) = (x y) (x z) A lattice L is called modular if and only if, for all x, y, z L, 2.1 Congruence relations x (y (x z)) = (x y) (x z) If f : A B is a mapping of one set to another, the relation θ on A defined by x θ y f(x) = f(y) is an equivalence relation. If A and B happen to be algebras, and f Hom(A, B) is an algebra homomorphism, then the relation just defined is θ = ker f = {(x, y) A 2 f(x) = f(y)}, and is called a congruence relation on A. It is useful to have some equivalent ways to define and think about congruence relations. In what follows, A = A, F is an algebra. A relation on A is a congruence relation if and only if it is an equivalence relation which is a subalgebra of the product algebra A 2. 2
2.2 Subgroup lattices 3 GROUPS AND G-SETS An equivalence relation θ Eq(A) is a congruence on A if and only if θ admits F ; that is, for n-ary f F, and elements a i, b i A, In other words, f respects θ for all f F. if (a i, b i ) θ, then (f(a 1,..., a n ), f(b 1,..., b n )) θ. We denote the set of all equivalence relations on a set A by Eq(A). Note that it doesn t make sense to talk about the set of all congruences relations on a set A. Congruence relations require some algebraic structure (given by F ). If A = A, F is an algebra, then we denote the set of all congruences on A by Con(A). (Thus, Con(A) doesn t make sense and, for that matter, neither does Eq(A), since equivalences are simply the partitions of the set A, and have nothing to do with any algebraic structure on A.) Note that Eq(A) P(A A) and we can partially order the set Eq(A) by set inclusion on A A. The result is a bounded lattice: Eq(A) = Eq(A), = Eq(A),, where meet is set intersection, and join is the equivalence generated by set union; respectively, α β = α β and α β = {θ Eq(A) α θ, β θ}. The greatest equivalence is the all relation, = A A, and the least equivalence is the diagonal relation: = {(x, y) A A x = y} The set Con(A) of all congruences of an algebra A is actually a sublattice of Eq(A). As usual, we signify this additional algebraic structure with bold font: i.e., ConA = Con(A),,. Consider, for example, a group G. In this case ConG is precisely the lattice of normal subgroups. This is clear since, as indicated by one of the above equivalent definitions, congruences are simply kernels of homomorphisms. For the same reason, the congruence lattice of a ring is the lattice of ideals. For a module, M, the lattice of submodules is ConM. It turns out that, for most classical algebras, ConA is modular. For lattices, as well as the algebras appearing in logic, ConA is distributive. 2.2 Subgroup lattices Another important example is the lattice of subgroups of a group G, which we denote by Sub[G] = Sub[G], = Sub[G],,, Of course, the universe, Sub[G], is the set of subgroups of G. The meet is set intersection, and the join is the subgroup generated by set union; that is, for subgroups H, K Sub[G], H K = H K and H K = {J Sub[G] H K J}. For example, the lattice Sub[D 4 ] of subgroups of the dihedral group D 4 is shown in Figure 2.2 3 Groups and G-sets Let G = G,, 1, 1 G be a group, A a set. Definition 3.1 A G-set is a unary algebra A = A, G, where G = {ḡ : g G}, 1 G = id A, and g 1 g 2 = g 1 g 2, for all g i G. 3
3.1 Intervals in subgroup lattices 3 GROUPS AND G-SETS Figure 1: The lattice of subgroups of the dihedral group D 4, represented as groups of rotations and reflections of a plane figure. For any a A, the stabilizer of a is the set Some basic facts about the G-set A, G are the following: Each ḡ G is a permutation of A. If [a] is the subalgebra generated by a A, then The stabilizer Stab(a) is a subgroup of G. Stab(a) = {g G ḡ(a) = a}. [a] = {ḡ(a) g G} = the orbit of a in A. If A = A, G has only one orbit, we say G acts transitively on A, or A is a transitive G-set. 3.1 Intervals in subgroup lattices If G is a group and H Sub[G] is a subgroup, define [H, G] = {K Sub[G] H K}, Call [H, G] an (upper) interval in the lattice Sub[G]. An important lemma about transitive G-sets is the following Lemma 3.1 If A = A, G is a transitive G-set, then for any a A, ConA = [Stab(a), G] This lemma plays a crucial role in the Pálfy-Pudlák result, which we discuss in the next section. 4
5 PROOF OF THE MAIN THEOREM 4 The theorem of Pálfy-Pudlák In [2], Pálfy and Pudlák prove Theorem 4.1 The following statements are equivalent: (i) Any finite lattice is isomorphic to the congruence lattice of a finite algebra. (ii) Any finite lattice is isomorphic to an interval in the subgroup lattice of a finite group. Below, we will refer to this theorem as the main theorem or the Pálfy-Pudlák theorem. One should be careful to note that this theorem is an equivalence of two statements concerning all finite lattices. The following interpretation 1 is clearly false: In AU 11, Pálfy and Pudlák proved that...a finite lattice is representable if and only if it occurs as an interval in the subgroup lattice of a finite group. Nonetheless, the claim made in this quote is interesting, and we might ask whether it is, in fact, true that a finite lattice is representable if and only if it occurs as an interval in the subgroup lattice of a finite group. Indeed, one direction is true and quite easy to prove. Namely, if a lattice occurs as an interval in the subgroup lattice of a finite group, then it is representable. We give a proof of this fact below. 4.1 The seminal lemma of tame congruence theory Besides the fundamental lemma about G-sets stated above, the most important lemma appearing in the Pálfy-Pudlák paper is the following Lemma 4.1 Let A = A, F be a unary algebra with F a monoid, and e 2 = e F. Define B = B, G with B = e(a) and G = {ef B : f F }. Then is a lattice epimorphism. Con(A) θ θ (B B) Con(B) This lemma spawned a whole body of research in the early 80 s, and this area is now called tame congruence theory. It is fully described in [1], and is said to be largely responsible for most of the progress made in lattice theory and universal algebra in recent years. 5 Proof of the Main Theorem Pálfyand Pudlák use lemma 4.1 (among other things) to arrive at the following Theorem 5.1 Let L = L,, be a finite lattice and A = A, F a finite unary algebra of minimal cardinality for which Con(A) = L. Then we have 1. if L satisfies (A) below, every idempotent mapping in F is either the identity or a constant mapping; 2. if L satisfies (A) and (B) below, every element of F is either a permutation or a constant mapping; 3. if L satisfies (A), (B) and (C) below, then A has no proper subalgebra. 1 This quote appears in a MathSciNet review, not of the article [2] (fortunately), but of a related article. The details are unimportant. It is only quoted here to caution against misinterpreting the theorem. 5
REFERENCES The conditions are (A) L is simple. (B) For any nonzero element x L there exist two elements y l, y 2 L such that x y l = x y 2 = 1 L and y l y 2 = 0 L. (C) L 2 and, for any x L not an atom and not 0 L, there exist at least 4 atoms less than x. Given this theorem, it is not hard to arrive at the following conclusion: If L is a finite lattice satisfying conditions (A), (B), (C) of the theorem, and if A = A, F be a finite unary algebra of minimal cardinality such that ConA = L, then A must be a transitive G-set. Then the basic lemma about transitive G-sets can be applied to show that ConA = [Stab(a), G] for any a A. It follows that L is isomorphic to the subgroup lattice interval [Stab(a), G]. Next, Pálfy and Pudlák proved that any finite lattice can be embedded as a sublattice of a lattice L which satisfies conditions (A), (B), (C) of the theorem. Therefore, if we assume part (i) of the main theorem i.e. that every finite lattice is representable then it follows that L L = ConA = [Stab(a), G] where A is a transitive G-set and a A. This shows that (i) implies (ii), which is the hard direction of the Pálfy-Pudlák theorem. The fact that (ii) implies (i) in is actually quite easy to prove. Suppose L = [H, G] for some finite group G = G,, 1, 1. Let A = {gh g G} be the set of cosets of H. For each g G, define the group action ḡ : A A by ḡ(a) = ga. Then A = A, {ḡ : g G} is clearly a transitive G-set. Once again, we apply the basic G-set lemma to see that ConA = [Stab(a), G]. Letting a = H, the universe of Stab(a) is then since e H. It follows that L = [H, G] = A, as desired. Stab(a) = {g G ḡ(a) = a} = {g G gh = H} = H, 6 Conclusion Given a finite lattice L, does there exist a finite algebra A such that L = ConA? It is generally believed the answer is no. About 30 years ago Pálfy and Pudlák translated this problem into one for the group theorists, yet the answer is not forthcoming. References [1] D. Hobby and R. McKenzie. The Structure of Finite Algebras. American Mathematical Society, 1988. [2] Peter Pálfy and Pavel Pudlák. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis, 11:22 27, 1980. 6