Functional Reducts of Boolean Algebras
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Boolean atomless countable the denote ) ) hc en hteeycutbesrcuewith structure countable every that means which 1. 1 ∧ A A f , . . . , ( Introduction ( = x ( = ∨ ,f A, y Ba n hti vr smrhs ewe finite between isomorphism every is that , safntoa eutof reduct functional a is ) t ) 1 1 j of f , . . . , ea ler nteset the on algebra an be , satr ucinof function term a is t 1 )wee∆dntstesymmetric the denotes ∆ where ∆) t , . . . , A Ba sastructure a is n h algebra the ) h ola algebra Boolean The . k uhta every that such B A ( = B eclassify We . ,t A, ( = dcsthe educts Ba tninteresting ften Ba o detailed a for ] utomor- homogeneous 1 A O ,t A, h struc- The . ´ t , . . . , t vrfinite over s hc san is which j here ihoper- with trsare ctures f saterm a is .Homo- s. 1 ocheck to Ba t , . . . , Ba k ) We . can k ) 2 B.BODOR,K.KALINA,ANDCS.SZABO´ will show that there are exactly 13 of them up to first-order interdefinability. The example given above is denoted by (+0, 0) later in the paper. If A is a first-order structure then we will call a structure B a reduct of A if B has the same domain set as A, and all constants, relations, and functions of B are first-order definable over A. For a given first-order structure we can define a preorder on its reducts: B1 . B2 if and only if B2 is a reduct of B1. Two reducts are called first-order interdefinable if and only if B1 . B2 and B2 . B1. It is an important consequence of Theorem 3 that a reduct of an ω-categorical structure is also ω-categorical itself. The automorphism group of a reduct consists of the permutations preserving all relations of the reduct, so the automorphism group of a reduct contains the automorphism group of the original structure. In general, if B1 . B2 then Aut(B1) ≤ Aut(B2). Moreover, the group Aut(B) is closed in the topology of pointwise convergence, i.e. the subspace topology of the product topology on AA where A is equipped with the discrete topology. If the structure A is ω-categorical then there is a bijection between the closed subgroups of Sym(A) containing Aut(A) and the equivalence classes of reducts of A by first-order interdefinability. So in the case of ω-categorical structures the classification of the reducts is equivalent with the description of the closed groups containing the automorphism group. language. The classification of the reducts up to first-order interdefinability is known for a handful of ω-categorical homogeneous structures. For example the set of rationals equipped with the usual ordering has five reducts up to first- order interdefinability [8], and similarly the the random graph [16], the random tournament [2] and the random partially ordered set [14] also have five. The classification is also known for the Henson graphs [15]. The homogeneous ordered graph has more than 40 [3], and Q≤ (rationals with the usual ordering) equipped with an additional constant has 116 reducts [12]. Note that in all these classifications the languages of the structures in question do not contain any function symbols. In order to handle the reducts of these structures Bodirsky and Pinsker de- veloped some techniques using Ramsey theory and canonical functions (see [4] for instance). However, these techniques only work for homogeneous struc- tures over finite relational languages. It follows from an easy orbit counting argument that the structure Ba is not homogeneous over any finite relational −1 language: Aut(Ba) has at least 22n n-orbits, whereas the automorphism group of a homogeneous structure over a finite relational language has at most 2p(n) n-orbits for some polynomial p. Therefore the methods mentioned above cannot be applied in our case. FUNCTIONAL REDUCTS OF THE COUNTABLE ATOMLESS BOOLEAN ALGEBRA 3
In this paper we show that Ba has exactly 13 functional reducts. 11 out of these 13 reducts are reducts of Ba considered as a vector space together with the constant 1 (later denoted by (+0, 0, 1)). Apart from these there is one nontrivial reduct. This reduct can be represented as Ba with the median relation M(x, y, z)=(x ∨ y) ∧ (y ∨ z) ∧ (z ∨ x). The automorphism group of this reduct is generated by the translations tc : x 7→ x + c over Aut(Ba). We also give a list of all reducts known to us at the end of this article, the lattice of these reducts can be seen on Figure 2. This list contains 12 infinite ascending chains, and 23 sporadic reducts. Note that our list might be incomplete, but we hope that this classification of functional reducts can be used for the classification of all reducts in the future. 2. Background We can define a preorder similarly on the functional reducts of a given algebra: C1 . C2 if and only if all functions in C2 can be defined over C1 using first-order formulas. The automorphism groups of such functional reducts will also be closed. It remains true that two functional reduct is first-order interdefinable if and only if their automorphism groups are the same. But the first-order equivalence classes of functional reducts are not in a bijection with the closed groups: there may exist closed groups containing the automorphism group of the original structure which cannot be obtained as the automorphism group of a functional reduct.
Definition 2. Let A = (A, f1,...,fn) be an algebra. Then we say that a function g : Ak → A is first-order definable over A if the relation R := {(x, g(x)) : x ∈ Ak} is definable by a first-order formula using the functions f1,...,fn. We say that a function g is a term function of the algebra A if g can be generated as a function from the functions f1,...,fn, and the projection maps k πi :(x1,...,xk) 7→ xi. Note that not every first-order definable function is a term function: if we consider any lattice as a ∧-semilattice, then the operation ∨ is first-order definable (see Lemma 11), but it is not a term function in the ∧-semilattice. Let B =(B, ∧, ∨, 0, 1, ¬) be an arbitrary Boolean algebra. Then we define the operations + and · on B as x + y := ∆(x, y) = ¬(x ∧ y) ∧ (x ∨ y), and x · y = xy = x ∧ y. It is well-known that in this case the structure B with the operations + and ·, and the constants 0 and 1 form a ring in a natural way. It is also known that this ring satisfies the identities x2 = x and 2x = 0. Using this definition the algebras B =(B, ∧, ∨, 0, 1, ¬) and B′ =(B, +, ·, 0, 1) are term equivalent meaning that a function is a term function over B if and 4 B.BODOR,K.KALINA,ANDCS.SZABO´ only if it is a term function over B′. In order to see this it is enough to show that every function of B′ is term function of B, and vice versa. The former follows directly from the definition of B′. For the other direction we can use the following definitions: x ∧ y := xy, x ∨ y := xy + x + y, ¬(x) := x +1. This observation also implies that the structures B and B′ are interdefinable. In most of our computations in this paper we will consider Ba as a ring as above. The argument above shows this does not change what is a term function of Ba or what is first-order definable over Ba. The following theorem is essential in the theory of ω-categorical structures: Theorem 3 (Engeler, Ryll-Nardzewski, Svenonius). The structure A is ω- categorical if and only if Aut(A) is oligomorphic, i.e. for all n it has finitely many n-orbits. The following two corollaries are easy consequences of Theorem 3. Corollary 4. Let A be an ω-categorical structure and f be a function or a relation on the universe of A such that every permutation of Aut(A) preserves f. Then f is first-order definable over A.
Corollary 5. Let C, C1, C2,..., Ck be some functional reducts of an algebra A. Then the following two conditions are equivalent:
• Aut(C1) ∩ Aut(C2) ∩ ... ∩ Aut(Ck) = Aut(C) • All functions of C1 ∪C2 ∪...∪Ck can be defined from C, and all functions of C can be defined from C1 ∪ C2 ... ∪ Ck. In this paper we will classify the functional reducts of the countable atom- less Boolean algebra Ba up to first-order interdefinability. Since Ba is ω- categorical, it is sufficient to describe the closed groups Aut(Ba) ≤ G ≤ Sym(Ba) which can be obtained as the automorphism group of a functional reduct. We use the following classical results in universal algebra.
Theorem 6. Let A be an algebra in which some identity f1(x1,...,xk) = f2(x1,...,xk) is satisfied for some term functions f1 and f2 of A. Let B be an A B B algebra in the variety generated by , and let fi denote the term function of A B which has the same definition as fi in . Then the identity f1 (x1,...,xk)= B B f2 (x1,...,xk) is satisfied in . The subalgebra generated by the element 0, 1 ∈ Ba is isomorphic to the two-element Boolean algebra. We denote this subalgebra by B2. FUNCTIONAL REDUCTS OF THE COUNTABLE ATOMLESS BOOLEAN ALGEBRA 5
Theorem 7. The variety generated by B2 is the class of all Boolean algebras. In particular the algebra Ba itself is contained in this variety. We refer the reader to [6] for the proofs of theorems above, as well as a more detailed discussion about universal algebra, and varieties. Theorems 6 and 7 have the following consequence for term functions of Ba.
Lemma 8. Let f1, f2 be term functions of Ba satisfying the identity f1 = f2 on B2. Then f1 = f2. Ba Proof. Since B2 is a subalgebra of Ba it follows that fi = (fi|B2 ) . Then Theorems 6 and 7 imply that the identity f1 = f2 is satisfied in Ba. Finally we provide some justification why this restricted set of reducts might be interesting for the structure Ba. Definition 9. Let A be a structure on the set A, let Aut(A) denote its auto- morphism group and let S ⊂ A be a finite subset. The group theoretic algebraic closure of S is: ACL(S)= {x ∈ A : the orbit of x under the stabilizer of S is finite} We follow the notation of [11, notation introduced before Lemma 4.1.1]. It is known that if A is ω-categorical then for finite subsets the above definition coincides with the usual model-theoretic algebraic closure [11]. We observed that every reduct of Ba that we know is similar to one of the functional reducts in the following sense: for every reduct R there is a unique functional reduct F such that the algebraic closure operator on R and on F coincides.
3. The non-linear functional reducts In this section we will deal with the non-linear functional reducts. To do so, we determine all closed subgroups of Sym(Ba) containing Aut(Ba) which preserve some non-linear term function of Ba. We start with the following important observation. Lemma 10. The structure Ba =(B, ∧, ∨, 0, 1, ¬) is first-order interdefinable with the structures (B, ∧) and (B, ∨). Proof. We only show that Ba = (B, ∧, ∨, 0, 1, ¬) and (B, ∧) interdefinable, the other claim is analogous. For this it is enough to show that the functions ∨, ¬ and the constants 0, 1 have are all first-order definable in (B, ∧). Using the usual notation in lattice theory we know that x ≤ y if and only if x ∧ y = x, and z = x ∨ y if and only if z is the smallest element with x ≤ z 6 B.BODOR,K.KALINA,ANDCS.SZABO´ and y ≤ z. Then we have z = x ∨ y if and only if xz = x ∧ yz = y ∧ ∀w (xw = x ∧ yw = y) → zw = z. Therefore the relation ∨ is first-order definable in (B, ∧). Then we can define 0, 1 as x = 0 ⇔ ∀z(zx = x) and x = 1 ⇔ ∀z(zx = z). Finally, we can define ¬ as x = ¬(y) ⇔ (xy = 0) ∧ (x ∨ y = 1). Let f(x, y) be a binary term function of Ba. Then f(x, y) can be expressed as the sum of some monomials from the set {1,x,y,xy}. Lemma 11. If G is a closed group such that Aut(Ba) ≤ G ≤ Sym(Ba) and G preserves a non-linear binary term function f(x, y) then G = Aut(Ba). Proof. It follows from Lemma 10 and Corollary 5 that if a permutation ϕ preserves xy = x ∧ y or xy + x + y = x ∨ y then ϕ must be an automorphism. • Let f(x, y)= xy + x then f(x, f(x, y)) = x(xy + x)+ x = xy. So every permutation which preserves f must be an automorphism. The same can be said for the operation xy + y. • Let g(x, y)= xy + x + 1 then g(g(x, y),y)=(xy + x +1)y +(xy + x + 1)+1= xy + x + y. So every permutation which preserves g must be an automorphism. The same can be said for the operation xy + y + 1. • Let h(x, y) = xy + 1 then h(h(x, x), h(y,y)) = (xx + 1)(yy + 1) + 1 = xy + x + y. So every permutation which preserves h must be an automorphism. • Let k(x, y)= xy + x + y + 1 then k(k(x, x),k(y,y))=(x + 1)(y +1)+ (x +1)+(y +1)+1 = xy. So every permutation which preserves k must be an automorphism. We have checked all possible non-linear binary term functions, and thus the lemma is proved. If f is an arbitrary term function of arity k then f can be written as k εi f(x1,...,xk)= X α~ε Y xi ~ε∈{0,1}k i=1 where all coefficients α~ε are 0 or 1. If ~ε is a 0 − 1 vector of length k then k Pi=1 εi will be denoted by |~ε|. We will denote the (k − 1)-ary function f(x1,...,xj−1, xi, xj+1,...,xk) by fxixj (x1,...,xk−1). By an abuse of notation we will write fij instead of fxixj if the meaning is clear from the context. It is clear that from this definition that if a permutation ϕ preserves f, then it also preserves fij. We will need the following definition: FUNCTIONAL REDUCTS OF THE COUNTABLE ATOMLESS BOOLEAN ALGEBRA 7
Definition 12. Let c be an arbitrary element of Ba. Then the translation by c is the permutation tc(a) = a + c. The set of all translations will be denoted by T .
T = {tc : c ∈ Ba} We will now characterize the possible automorphism groups corresponding to ternary non-linear term functions. Lemma 13. Let f be a ternary non-linear term function expressed as
f(x, y, z)= xy + yz + zx + α3x + α2y + α1z + α0 where there are exactly zero or two 1s among α1,α2 and α3. Then a permu- tation ϕ preserves f if and only if ϕ can be written as ϕ = t ◦ ψ where t is a translation and ψ is an automorphism of Ba. Proof. Since f is a term function over Ba, it follows that every automorphism of Ba also preserves f. Now we show that f is also preserves by every trans- lation. Let t = tc. Then we have
f(t(x), t(y), t(z)) = f(x + c,y + c, z + c)=
= xy + yz + zx + c + α3x + α3c + α2y + α2c + α1z + α1c + α0 =
= f(x, y, z)+(α1 + α2 + α3 + 1)c = f(x, y, z)+ c = t(f(x, y, z)). For the other direction let ϕ be an arbitrary permutation which preserves f. Denote the permutation tϕ(0) ◦ ϕ byϕ ˜ (thenϕ ˜(x) = ϕ(x)+ ϕ(0)). We will show that this permutation is an automorphism:ϕ ˜ ∈ Aut(Ba). Since ϕ˜ preserves f andϕ ˜(0) = 0, the permutationϕ ˜ must preserve the function g(x, y) = f(x, y, 0). This g is a binary non-linear term function, soϕ ˜ is an −1 automorphism by Lemma 11. The decomposition of ϕ as tϕ(0) ◦ ϕ˜ = ϕ is of the required form. Lemma 14. Let f be a ternary non-linear term function expressed as
f(x, y, z)= α7xyz + α6xy + α5yz + α4zx + α3x + α2y + α1z + α0 where at least one of the coefficients α7,α6,α5 and α4 is equal to 1. Let ϕ be a permutation which preserves f. Then one of the following two possibilities holds for ϕ: • ϕ is an automorphism: ϕ ∈ Aut(Ba). • ϕ can be obtained as the composition of an automorphism and a non- identical translation. 8 B.BODOR,K.KALINA,ANDCS.SZABO´
Proof. Since the permutation ϕ preserves f, it also preserves the function fxy, fyz and fzx. Therefore if at least one of the functions fxy, fyz and fzx are non-linear then it follows from Lemma 11 that f ∈ Aut(Ba), and therefore the statement of the lemma holds.
• If α7 = 1 and α6 = α5 then fyz is a binary non-linear term function. Since there are two identical values among α4,α5 and α6 in the case of α7 = 1 the statement of the Lemma will hold. • If α7 = 0 and exactly one of α4,α5,α6 is 1: by symmetry we can assume that α4 = 1, then fyz is non-linear. • If α7 = 0 and exactly one of α4,α5,α6 is 0: by symmetry we can assume that α4 = 0, then fxz is non-linear. • If α7 = 0 and α4 = α5 = α6 = 1 then f can be written as
f(x, y, z)= xy + yz + zx + α3x + α2y + α1z + α0.
The case when there are exactly zero or two 1s are among α1,α2 and α3 is handled in Lemma 13. In the case when there are exactly one or three 1s among α1,α2,α3 then f(a, a, a)=(fyz)xy(a)=0or f(a, a, a)=1 for every a ∈ Ba. In this case g(x, y) = f(x, y, f(x, x, x)) is a binary non-linear term function, and then we can apply Lemma 11 again. Theorem 15. Let M denote the ternary function median: M(x, y, z)= xy + yz +zx. This is the same as the usual lattice-theoretic (lower or upper) median (x ∨ y) ∧ (y ∨ z) ∧ (z ∨ x). Let Aut(M) denote the group of all permutations preserving M. Then Aut(M)= T ⋊ Aut(Ba) Proof. By Lemma 13 every permutation of Aut(M) can be written as ϕ = t◦ψ (where t ∈ T and ψ ∈ Aut(Ba)). Moreover, every permutation in Aut(Ba) and T preserves the median so Aut(M)= hAut(Ba), T i. The intersection of the groups Aut(Ba) and T is the trivial group because every element of Aut(Ba) preserves 0 and the identity is the only 0-preserving translation. Finally, the group T is a normal subgroup of Aut(M) = hAut(Ba), T i. For this it is enough to show that the group T is closed under conjugation of elements of Aut(Ba). Let tc be an arbitrary element of T and ϕ be an arbitrary element of Aut(Ba). Then
−1 −1 −1 −1 (ϕ ◦ tc ◦ ϕ)(x)= ϕ (ϕ(x)+ c)=(ϕ ◦ ϕ)(x)+ ϕ (c)= tϕ−1(c)(x) −1 so ϕ ◦ tc ◦ ϕ ∈ T . FUNCTIONAL REDUCTS OF THE COUNTABLE ATOMLESS BOOLEAN ALGEBRA 9
The case of the non-linear term functions with more than three variables can be reduced to the previous cases by the following lemma. Lemma 16. Let f be a non-linear term function of arity k ≥ 4. Then there exist indices 1 ≤ i < j ≤ k such that the function g = fij is non-linear. Proof. We will prove the statement by checking the following three cases: • The first case is when there is an ~ε such that 2 ≤ |~ε| ≤ k − 2 and α~ε = 1 holds. We have two indices i and j such that 1 ≤ i < j ≤ k and k εi ~εi = ~εj = 0. Then for this ~ε the term Qi=1 xi is with coefficient 1 in the functions f and fij, as it is not altered by collapsing xi and xj and this term cannot show up by collapsing those two variables, either. So fij is non-linear. • If the only non-linear monomial is the product of all the variables then every pair of indices i and j is suitable. • The remaining case is when all non-linear monomials have a degree at least (k − 1) and there is a monomial with a degree of exactly (k − 1). Let ~ε denote a vector corresponding to such monomial. Let i and j denote two indices such that ~εi = ~εj = 1. Then if we replace xj with xi the monomial of f corresponding to ~ε will become a monomial of fij with a degree of (k − 2). This monomial of fij cannot be formed from any other monomial of f, so it will not be cancelled out. Lemma 17. Let f be a non-linear term function. Then Aut(Ba) ≤ Aut(f) ≤ Aut(M). Proof. First we show that there exists a non-linear term function g such that the arity of g is at most 3, and Aut(Ba) ≤ Aut(f) ≤ Aut(g). Let k be the arity of f. If k ≤ 3, then there is nothing to prove. If k ≥ 4, then by Lemma 16 we know that fij is non-linear for some indices 1 ≤ i < j ≤ k. Then Aut(Ba) ≤ Aut(f) ≤ Aut(fij). By iterating this construction we can find a non-linear function g of arity at most 3 such that Aut(Ba) ≤ Aut(f) ≤ Aut(g). By Lemma 14 we have Aut(g) ⊂ hAut(Ba), T i. The right handside is equal to Aut(M) by Lemma 15. Therefore Aut(Ba) ≤ Aut(f) ≤ Aut(g) ≤ Aut(M). Lemma 18. Let f be an arbitrary non-linear term function. Then Aut(f)= Aut(Ba) or Aut(f) = Aut(M). Proof. Let Aut(f) be denoted by G. The group G is uniquely determined by the subgroup T ∩ G because Aut(M) = T ⋊ Aut(Ba) and Aut(Ba) ≤ G ≤ Aut(M). Let c and d be two elements of Ba such that they are on the same 10 B.BODOR,K.KALINA,ANDCS.SZABO´ orbit of Aut(Ba). We will show that tc ∈ G implies td ∈ G. Let ϕ ∈ Aut(Ba) an automorphism such that ϕ(d)= c then −1 −1 −1 −1 (ϕ ◦ tc ◦ ϕ)(x)= ϕ (ϕ(x)+ c)=(ϕ ◦ ϕ)(x)+ ϕ (c)= td(x) The automorphism group Aut(Ba) has three orbits on Ba: {0}, {1} and Ba \ {0, 1} so one of the following four possibilities must hold (using that t0 = id ∈ T ∩ G):
• T ∩ G = {tc|c ∈{0}} • T ∩ G = {tc|c ∈{0, 1}} • T ∩ G = {tc|c ∈ Ba \{1}} • T ∩ G = {tc|c ∈ Ba} The first case corresponds to the case G = Aut(Ba) and the fourth corresponds to the case G = Aut(M). We will rule out the second and third possibility. The third possibility can be ruled out by noticing that the set {tc|c ∈ Ba \ {1}} is not a subgroup. Let a ∈ Ba \{0, 1} be an arbitrary element then the translations ta and ta+1 must be in {tc|c ∈ Ba \{1}} = T ∩ G so their composition (ta+1 ◦ ta)(x) = x + a + a +1= t1(x) must also be contained in T ∩ G. The second case gives us a closed group Aut(Ba) ≤ G ≤ Aut(M). We will show that this group cannot be obtained as the automorphism group of a functional reduct. Let f be a term function which is preserved by the translation t1. Then the following two equations hold:
f(x1 +1, x2 +1,...,xk +1)= f(x1, x2,...,xk)+1, and clearly
f(x1 +0, x2 +0,...,xk +0)= f(x1, x2,...,xk)+0.
In particular for all x1, x2,...,xk,y ∈{0, 1} we have
f(x1 + x, x2 + x,...,xk + y)= f(x1, x2,...,xk)+ y. This means that the identity
f(x1 + x, x2 + x,...,xk + y)= f(x1, x2,...,xk)+ y is satisfied in the subalgebra B2. Therefore by Lemma 8 this identity also holds in Ba. Setting y = c ∈ Ba we obtain that
f(x1 + c, x2 + c,...,xk + c)= f(x1, x2,...,xk)+ c holds for all x1,...,xk ∈ Ba. Therefore tc preserves f for every c ∈ Ba. FUNCTIONAL REDUCTS OF THE COUNTABLE ATOMLESS BOOLEAN ALGEBRA 11
Assume G = Aut(f) for some term function f. Then the argument above shows that t1 ∈ Aut(f) implies that tc ∈ Aut(f) for arbitrary c ∈ Ba. This excludes the second case. Lemma 18 can be formulated in terms of first-order definability as follows. Lemma 19. Let f be an arbitrary non-linear term function. Then the func- tion f is first-order interdefinable with either the operation ∧ or the median operation M. Proof. By Lemma 18 we know that either Aut(f) = Aut(Ba) or Aut(f) = Aut(M). By Lemma 10 we have Aut(Ba) = Aut(∧). Then the statement of the lemma follows directly from Corollary 4.
4. The linear functional reducts In this section we will classify the linear functional reducts up to first-order interdefinability. All linear term functions can be written as l(x1, x2,...,xk)= x1 + x2 + ...+ xk + α where α is either 0 or 1. Consider the following eight term functions: (1) 0 (2) 1 (3) x (4) ¬(x)= x +1 (5) +0(x, y)= x + y (6) +1(x, y)= x + y +1 (7) Σ(x, y, z)= x + y + z (8) Σ1(x, y, z)= x + y + z +1 We will refer to this functions as canonical linear functions. We will need the corresponding automorphism groups: Group 1: Aut(0)
Aut(0) is the stabilizer of 0 in the whole symmetric group Sym(Ba). This is a maximal proper subgroup of Sym(Ba) because for every permutation ϕ∈ / Aut(0) the subgroup hAut(0),ϕi is the whole Sym(Ba).
Group 2: Aut(1)
Similarly Aut(1) is the stabilizer of 1 in the whole symmetric group Sym(Ba). This is also a maximal proper subgroup. 12 B.BODOR,K.KALINA,ANDCS.SZABO´
Group 3: Aut(∅)
The group Aut(∅) = Aut(x) is the whole symmetric group.
Group 4: Aut(¬)
For ¬(x)= x + 1 let I denote an arbitrary maximal ideal of Ba. Define the Ba following two groups: Let Sym{I} ( ) be defined as an extension of the action of Sym (I) to the whole Ba. For ϕ ∈ Sym (I) define the action as ϕ(x)= ϕ(x) if x ∈ I and ϕ(x)= ϕ(x + 1) + 1 if x∈ / I. I I Let Z2 denote the group consisting of the following permutations: ϕ ∈ Z2 Ba I if and only if for every x ∈ ϕ(x) = x or ϕ(x) = x + 1. This Z2 group is always the same regardless of the choice of the maximal ideal I because I I we do not refer to in its definition. The notation is justified because Z2 is isomorphic to a direct power of Z2 where the direct factors are indexed with the elements of I. I Ba We will show that Aut(¬)= Z2 ⋊ Sym{I} ( ). I I The group Z2 is a normal subgroup in Aut(¬). Let ϕ ∈ Z2 and ψ ∈ Aut(¬) −1 I be two permutations, we need that the permutation ψ ϕψ is also in Z2 . This holds if and only if for every x ∈ Ba the image ψ−1ϕψ(x) is either x or x + 1. So we have the following two cases: • If ϕ(ψ(x)) = ψ(x) then ψ−1ϕψ(x)= x. • If ϕ(ψ(x)) = ψ(x) + 1 then ψ−1ϕψ(x)= ψ−1(¬ψ(x)) = ¬ψ−1(ψ(x)) = x + 1. I Ba The intersection of the groups Z2 and Sym{I} ( ) is trivial because there are I I no permutations in Z2 such that the image of any element of is another element of I. We will show that every ϕ permutation in Aut(¬) can be written as the Ba I composition of a permutation from Sym{I} ( ) and a permutation from Z2 . Let ϕ denote an arbitrary permutation from Aut(¬), first we will define a permutationϕ ˜: • If x ∈ I then letϕ ˜(x) be the element from {ϕ(x),ϕ(x)+1} the one which is in I. • If x∈ / I then letϕ ˜(x) be from {ϕ(x),ϕ(x)+1} the one which is not in I. Ba −1 Thisϕ ˜ will be an element of Sym{I} ( ) andϕ ˜ ◦ ϕ will be an element of I Z2 . The composition of these two permutations is ϕ.
Group 5: Aut(+0) FUNCTIONAL REDUCTS OF THE COUNTABLE ATOMLESS BOOLEAN ALGEBRA 13
In case of the operation +0(x, y) = x + y the functional reduct is a vector ∞ space F2 . It has the automorphism group Aut(+0) = GL(∞, 2).
Group 6: Aut(+1)
In case of the operation +1(x, y)= x + y + 1 the functional reduct is also a vector space, but in this case the zero element of the vector space is 1, and the addition is the operation +1(x, y)= x + y + 1. This vector space is isomorphic to the vector space in Group 5, the τ1 translation is an isomorphism:
τ1(+1(x, y)) = x + y +1+1= x +1+ y +1=+0(τ1(x), τ1(y)) and −1 −1 −1 τ1 (+0(x, y)) = x + y +1=+1(τ1 (x), τ1 (y)) 1 We will denote the automorphism group Aut(+1) by GL (∞, 2). Since the two vector spaces are isomorphic, their automorphism groups are also isomorphic: 1 ∼ Aut(+1)=GL (∞, 2) = GL(∞, 2) = Aut(+0).
Group 7: Aut(Σ)
In case of the operation Σ(x, y, z)= x+y+z the functional reduct is an affine space. Let x, y, z and v be four pairwise different elements from Ba. Then Σ(x, y, z)= v holds if and only if these four elements form a two dimensional affine subspace. If Σ(x, y, z)= v ⇔ x+y +z = v then {x,y,z,v} is a translate of the linear subspace {0,y + x, z + x, v + x} by x. The operation Σ is preserved by all translations T and all linear transforma- tions GL(∞, 2). The group of translations T is a normal subgroup in Aut(Σ) and Aut(Σ) is generated by the subgroups T and GL(∞, 2). So Aut(Σ) is a semidirect product T ⋊ Aut(+0).
Group 8: Aut(Σ1)
In case of operation Σ1(x, y, z)= x + y + z + 1 we can define both ¬(x) = Σ(x, x, x) and Σ(x, y, z) = Σ1(x, y, ¬(z)). We can also define Σ1(x, y, z) using ¬(x) and Σ(x, y, z): Σ1(x, y, z) = Σ(x, y, ¬(z)). By Corollary 5 Aut(Σ1) = Aut(Σ) ∩ Aut(¬). T is a subgroup of Aut(Σ1) because all translations preserve Σ1. Using the fact that T ⊳ Aut(Σ) the group T will be a normal subgroup in Aut(Σ1), too. The group GL(∞, 2)∩Aut(Σ1) consists of the complement-preserving linear permutations. These are exactly those linear permutations which fix the 1. We can conclude that 14 B.BODOR,K.KALINA,ANDCS.SZABO´
Aut(Σ1) = Aut(Σ) ∩ Aut(¬)=(T ⋊ Aut(+0)) ∩ Aut(¬)=
= T ⋊ (Aut(+0) ∩ Aut(¬)) = T ⋊ (Aut(+0) ∩ Aut(+1))
Lemma 20. For every linear term function l, there uniquely exists a canonical linear function f such that l and f are first-order interdefinable.
Proof. Let l denote l(x1, x2,...,xk) = x1 + x2 + ... + xk + α. First we will prove the existence of such f, then prove the uniqueness. If the arity of l is at most one then l is a canonical linear function so we can choose itself as f. Let k denote the arity of l. If k is even and at least two, then f(x, y)= x+y+ α will be first-order interdefinable with l. The function f can be defined from l because f(x, y)= l(x,y,y,...,y). For the other direction define the following series of functions: let l2 be l2 = f(x1, x2) and recursively lj(x1, x2,...,lj) = f(lj−1(x1, x2,...,xj−1), xj) if j > 2. Then l = lk so l can be defined from f. If k is odd and at least 3 then f(x, y, z) = x + y + z + α will be first- order interdefinable with l. The function f can be defined from l because f(x, y, z) = l(x,y,z,z,...,z). Define the following series of functions: let l1 be l1 = f(x1, x2, x3) and recursively
lj(x1, x2,...,x2j+1)=
= f(lj−1(x1, x2,...,x2j−1), f(x2j, x2j, x2j), f(x2j+1, x2j+1, x2j+1)).
Then l = l(k−1)/2 so l can be defined from f. We will show the uniqueness of the previous f. If for a given l there are two different canonical linear functions f1, f2 such that l is first-order interdefinable with both, then f1 will be interdefinable with f2. So their automorphism group will be the same. Using Corollary 4 it is enough to show that the automorphism groups of two different canonical linear functions are different. We have described these automorphism groups, and they are pairwise distinct. Our goal is to show that the automorphism groups of the functional reducts of Ba ordered by inclusion form the lattice on Figure 1. First we will prove that the (supposed) coatoms of the lattice form an an- tichain. Next we will describe the possible intersections of the coatoms. Finally we determine the places of the two possible automorphism groups of the non- linear functional reducts. There are three groups which can be obtained as the intersection of some coatoms and have not been characterized yet: Group 9: Aut(0, 1) FUNCTIONAL REDUCTS OF THE COUNTABLE ATOMLESS BOOLEAN ALGEBRA 15
∅
0 Σ 1 ¬
0, 1
+0, 0 ¬, 0, 1 +1, 1 Σ1
+0, 0, 1 M
·, +0, 0, 1
Figure 1. The lattice of the functional reducts, ordered by the inclusion of their automorphism groups.
The group Aut(0, 1) is the pointwise stabilizer of {0, 1}. It is a proper maximal subgroup of Aut(0) and Aut(1).
Group 10: Aut(¬, 0, 1)
The group Aut(¬, 0, 1) is the intersection of the groups Aut(0), Aut(1) and Aut(¬). Like the group Aut(¬) can be obtained as a semidirect product I Ba Z2 ⋊ Sym{I} ( ) the group Aut(¬, 0, 1) can be decomposed as Aut(¬, 0, 1) = I I Z Sym I (Ba) . Here Z and Sym I (Ba) denote the stabilizer 2 0 ⋊ { } 0 2 0 { } 0 I Ba of the 0 in the groups Z2 and Sym{I} ( ), respectively.
Group 11: Aut(+0, 0, 1) 16 B.BODOR,K.KALINA,ANDCS.SZABO´
The group Aut(+0, 0, 1) was briefly discussed in the section about Aut(Σ1). It is the group of linear permutations which preserve 1 and the complementa- tion.
Lemma 21. The automorphism groups of the canonical linear functions are ordered by inclusion as the [Aut(+0, 0, 1), Aut(∅)] interval on Figure 1. Proof. We will show that the groups Aut(0), Aut(1), Aut(Σ) and Aut(¬) form an antichain. • Let a, b ∈ Ba \{0, 1} be arbitrary elements such that a =6 ¬b. Then the 4-cycle (a, a+1, b, 1) is in Aut (0). We will denote this permutation by ϕ. Then Aut(0) * Aut(1) because ϕ(1) = a =6 1. Aut(0) * Aut(¬) because ϕ(¬a) = b =6 a = ¬ϕ(a). And Aut(0) * Aut(Σ) because ϕ(Σ(a, a +1, 1)) = 0 =6 a + b +1=Σ(ϕ(a),ϕ(a + 1),ϕ(1)). • Similarly Aut(1) * Aut(0),Aut(1) * Aut(¬) and Aut(1) * Aut(Σ). • Let a, b ∈ Ba \{0, 1} be arbitrary elements such that a =6 ¬b. Then the 6-cycle (a, b, 0, a +1, b +1, 1) is in Aut(¬). We will denote this permutation by ϕ. Aut(¬) * Aut(0) and Aut(¬) * Aut(1) because ϕ does not fix 0 and 1. Finally Aut(¬) * Aut(Σ) because ϕ(Σ(a, b, 0)) = a + b =6 a + b +1=Σ(ϕ(a),ϕ(b),ϕ(0)) • Let τa(x)= x + a be an arbitrary translation. Then τa preserves Σ but if a =6 0 then τa does not preserve 0 or 1. So Aut(Σ) * Aut(0) and Aut(Σ) * Aut(1). Let ϕ be a permutation from GL(∞, 2) such that ϕ does not fix 1. Then ϕ does not preserve ¬ so Aut(Σ) * Aut(¬) because Aut(+0) = GL(∞, 2) ⊂ Aut(Σ). We will determine the possible intersections of Aut(0), Aut(1), Aut(Σ) and Aut(¬). The intersections of every possible pair: • Aut(0) ∩ Aut(1) = Aut(0, 1) by the definition of stabilizers. • Aut(0) ∩ Aut(Σ) = Aut(+0) = GL(∞, 2) because +0(x, y)=Σ(x, y, 0), 0=+0(x, x) and Σ(x, y, z)=+0(x, +0(y, x)), using Corollary 5 • Aut(0) ∩ Aut(¬) = Aut(¬, 0, 1) 1 • Aut(1)∩Aut(Σ) = Aut(+1)=GL (∞, 2) because +1(x, y)=Σ(x, y, 1), 1=+1(x, x) and Σ(x, y, z)=+1(x, +1(y, x)) using Corollary 5 • Aut(0) ∩ Aut(¬) = Aut(¬, 0, 1) • Aut(Σ) ∩ Aut(¬) = Aut(Σ1) because Σ(x, y, z) = Σ1(x, y, Σ1(z,z,z)), ¬x = Σ1(x, x, x) and Σ1(x, y, z)=Σ(x, y, ¬z) using Corollary 5. The possible triple intersections:
• Aut(0) ∩ Aut(1) ∩ Aut(Σ) = Aut(+0, 1) because 0 = +0(x, x), 1 = 1, Σ(x, y, z)=+0(x, +0(y, x)) and +0(x, y) = Σ(x, y, 0), 1 = 1 using Corollary 5. FUNCTIONAL REDUCTS OF THE COUNTABLE ATOMLESS BOOLEAN ALGEBRA 17
• Aut(0) ∩ Aut(1) ∩ Aut(¬) = Aut(¬, 0, 1) • Aut(0)∩Aut(Σ)∩Aut(¬) = Aut(+0, 1) because 0 = +0(x, x), Σ(x, y, z)= +0(x, +0(y, x)), ¬x =+0(x, 1) and +0(x, y)=Σ(x, y, 0), ¬0 = 1 using Corollary 5. • Aut(1) ∩ Aut(Σ) ∩ Aut(¬) = Aut(+0, 1) because 1 = 1, Σ(x, y, z) = +0(x, +0(y, x)), ¬x =+0(x, 1) and +0(x, y)=Σ(x, y, ¬1), 1 = 1 using Corollary 5. The intersection of all four elements of the antichain:
• Aut(0)∩Aut(1)∩Aut(Σ)∩Aut(¬) = Aut(+0, 1) because 0 = +0(x, x), 1 = 1, Σ(x, y, z)=+0(x, +0(y, x)), ¬x = +0(x, 1) and +0(x, y) = Σ(x, y, 0), ¬0 = 1 using Corollary 5. So we can conclude that the automorphism group of any canonical linear func- tion can be obtained as the intersection of some of the groups Aut(0), Aut(1), Aut(Σ) and Aut(¬). The group Aut(x) = Sym(Ba) corresponds to the empty inter- section. There are three other groups which can be obtained as the intersection of some coatoms: Ba • The group Aut(0, 1) = Sym(0,1)( ). • The group Aut(¬, 0, 1). • The group Aut(+0, 1). These are the groups of number 9, 10 and 11.
5. The functional reducts of the countable atomless Boolean algebra In this section we will finish the classification of the functional reducts. In the previous section we finished the case of the linear functional reducts, the remaining case is the non-linear one. Theorem 22. The possible automorphism groups of functional reducts are ordered by inclusion as in Figure 1. Proof. If f is a non-linear term function then Aut(f) = Aut(Ba) or Aut(f)= Aut(M) by Lemma 18. The group Aut(Ba) is the minimal element of the lat- tice. We need to find the place of Aut(M). Aut(M) * Aut(0) and Aut(M) * Aut(1) because T ≤ Aut(M). We will show that Aut(M) ≤ Aut(Σ1). By The- orem 15 it is enough to show that translations preserve Σ1. Let τa(x)= x + a be arbitrary translation then
τa(Σ1(x, y, z)) = x + y + z +1+ a = x + a + y + a + z + a +1=
= Σ1(τa(x), τa(y), τa(z)) 18 B.BODOR,K.KALINA,ANDCS.SZABO´ so translations preserve Σ1. In order to finish our proof it is enough to show that Aut(Ba) =6 Aut(+0, 0, 1) and Aut(M) =6 Aut(Σ1). Let a, b ∈ Ba two elements such that 0
Definition 24. Let Wn be an n-codimensional subspace of Ba containing 1. Then hn ∈ Sym(Ba) is defined by hn(v) = v +1 for v ∈ Wn, and hn(v) = v otherwise. We define Hn to be the closure of the group hAut(+0, 0, 1), hni.
Since Aut(+0, 0, 1) acts transitively on subspaces of codimension n, we see that the group Hn does not depend on our particular choice of Wn, only on the FUNCTIONAL REDUCTS OF THE COUNTABLE ATOMLESS BOOLEAN ALGEBRA 19 codimension n. By Proposition 6 in [5] we know that the groups H1,H2,... form an infinite ascending chain. Now we are ready to give our list of reducts. We only label those reducts here which are not functional reducts. We will describe the reduct, its auto- morphism group, or both, without any proofs. There are 12 individual reducts which are not functional reducts. There are indexed by lowercase letters (a)- (l). The reducts in the 12 infinite ascending chains are labelled by a lowercase letter (m)-(x), and a natural number indicating their place in the chain. Two of the reducts in these chains are also functional reducts. The automorphism group of a reduct labelled x will be denoted by Gx.
(a) Let f denote the permutation f(0) = 0, f(1) = 1 and f(x) = x +1 for all other elements. Then Ga = hAut(Ba), fi. The group Aut(Ba) is a subgroup of index 2 in Ga. (b) Let g denote the permutation g(0) = 1,g(1) = 0 and g(x)= x for all other elements. Then Gb = hAut(Ba),gi. The group Aut(Ba) is a subgroup of index 2 in Gb. (c) Let h denote the permutation h(x) = x + 1. Then Gc = hAut(Ba), hi. The group Aut(Ba) is a subgroup of index 2 in Gc. (d) Let f,g,h denote permutations as in the definitions of the reducts (a), (b) and (c). Note that fg = h. Then Gd = hAut(Ba),f,gi. The group Aut(Ba) is a subgroup of index 4 in Gd. (e) This reduct can be characterized by the the independence relation (Defini- tion 23). The automorphism group Ge consists of the permutations of the form g1g2 where g1 ∈ Aut(Ba) and g2(x) ∈{x, x +1} for every x ∈ Ba. (f) The group Gf can be obtained as Gf = Ge ∩ Aut(0, 1). (g) The group Gg can be obtained as Gg = Cl hAut(M),Gei. (h) The group Gh can be obtained as Gh = Cl hAut(+0, 0, 1),Gei. (i) The group Gi can be obtained as Gi = Cl hAut(+0, 0, 1),Gf i. (j) The group Gj can be obtained as Gj = Cl hAut(Σ1),Gei. (k) The group Gk is the group of permutations fixing the set {0, 1} setwise. (l) The group Gl is the subgroup of Aut(¬) fixing the set {0, 1} setwise.
(m) We put Gmi := Hi ∩Aut(0) (see Definition 24). Then Gm1 = H1 ∩Aut(0) = Aut(+0, 0, 1).
(n) Let f be the permutation defined in the definition of Ga. Then Gni =
hGmi , fi.
(o) Let g be the permutation defined in the definition of Gb. Then Goi =
hGmi ,gi.
(p) Let h be the permutation defined in the definition of Gc. Then Gpi =
hGmi , hi = Hi. 20 B.BODOR,K.KALINA,ANDCS.SZABO´
(q) Let f,g be the permutation defined in the definition of Ga and Gb. Then
Gqi = hGmi ,f,gi.
(r) The group Gri can be obtained as Gri = Cl hAut(M),Gmi i. Then Gr1 = Aut(Σ1).
(s) The group Gsi can be obtained as Gsi = Gmi ∩ Ge.
(t) Let f be the permutation defined in the definition of Ga. Then Gti =
hGsi , fi.
(u) Let g be the permutation defined in the definition of Gb. Then Gui =
hGsi ,gi.
(v) Let h be the permutation defined in the definition of Gc. Then Gvi =
hGsi , hi. (w) Let f,g be the permutation defined in the definition of Ga and Gb. Then
Gwi = hGsi ,f,gi.
(x) The group Gxi can be obtained as Gxi = Cl hAut(M),Gsi i. We hope that the above list is close to complete, and that this classification of functional reducts will help for classifying all reducts. FUNCTIONAL REDUCTS OF THE COUNTABLE ATOMLESS BOOLEAN ALGEBRA 21
∅
0 1 k ¬ Σ 0,1 l
j ¬, 0, 1
g h
+0, 0 +1, 1 e
i
f
q3 r3
n3 o3 p3 w3 x3
m3 t3 u3 v3
s3 q2 r2
n2 o2 p2 w2 x2
m2 t2 u2 v2
s2 q1 Σ1
n1 o1 p1 w1 x1
+0, 0, 1 M t1 u1 v1 d
s1 a b c
·, +0, 0, 1
Figure 2. The lattice of all reducts known to us, ordered by the inclusion of their automorphism groups. 22 B.BODOR,K.KALINA,ANDCS.SZABO´
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(1,2,3) Eotv¨ os¨ Lorand´ University, Department of Algebra and Number The- ory, Hungary, 1117 Budapest, Pazm´ any´ Peter´ set´ any´ 1/c E-mail address: [email protected]
E-mail address: [email protected]
E-mail address: [email protected]