Computably Enumerable Semigroups, Algebras, and Groups Bakhadyr Khoussainov The University of Auckland New Zealand Research is partially supported by Marsden Fund of New Zealand Royal Society.

Plan Equational presentations. The specification problem. A semigroup example. Algorithmically finite universal algebras. An algebra example. A group example.

Definition: A universal algebra is a tuple A=(A; f 1,f 2,…., f k, c 1,…,c r ), where A is the non-empty set (domain), each f j is a total function on A, and each c k is a constant.

Definition (terms): Variables and constants are terms. Suppose that t 1,…,t m are terms, f is a function symbol, then the expression f(t 1,…,t m ) is a term. A ground term is a term with no variables.

The set of ground terms is a universal algebra, called the term algebra. Notation: T = the term algebra Fact 1: T is finitely generated and computable. Fact 2: Every universal algebra generated by the constants is a homomorphic image of T.

An equational presentation is a finite set S of formulas of the type t=q. A quasiequational presentation is a finite set S of formulas of the type t 1 =q 1  ……  t n =q n → t=q. Here t,q are terms that may contain variables.

Definition: ( Specified Universal Algebras ) Let E(S) be the congruence relation generated by S. The universal algebra T S = T/ E(S) is called specified by S. T S is equationally presented if S is an equational presentation. T S is quasi-equationally presented if S is a quasi- equational presentation.

Examples Finitely presented groups and semigroups. Finitely presented algebras. Finitely presented semigroups with left- cancellation properties.

Properties of T S : 1.T S satisfies S and is finitely generated. 2.The equality relation E(S) of T S is computable enumerable. 3.T S is universal. 4.T S is unique.

The Specification Problem: Let A be a universal algebra. (1) Is A equationally presented? (2) Is A quasi-equationally presented? Clearly, we need to assume the following: (a) A is finitely generated. (b)The equality relation in A is c.e.

Definition (c.e. universal algebra) A c.e. universal algebra is one of the form T/E, where (1) T is the term algebra, (2) E is a c.e. congruence relation on T. Examples: finitely presented groups, semi- groups, and universal algebras.

Example: Consider the universal algebra ( ω; x+1, 2 x ). (Bergstra and Tucker): The universal algebra ( ω; x+1, 2 x ) does not have an equational presentation.

Important Observation Consider the expansion: (ω, x+1, 2 x, +, x, 0). The expanded universal algebra is now finitely presented.

Definition: (Computable Algebra) A universal algebra A=(A; f 1,f 2,…., f k, c 1,…,c r ) is computable if the set A and all functions f j are computable.

Definition: An expansion of A=(A; f 1,f 2,…., f k, c 1,…,c r ) is A’ = (A; f 1,f 2,…., f k, g 1,…,g r, c 1,…,c r ), where g 1,…,g r are new functions.

Theorem (Bergstra  Tucker, ≈1980). Every computable universal algebra can be equationally presented in an expansion.

The question of Goncharov: ( early 1980s, Goncharov (also Bersgtra and Tucker) ) Let A be a finitely generated computably enumerable universal algebra: 1.Does A have an equationally presented expansion? 2.Does A have a quasi-equationally presented expansion? Resemblance to Higman’s theorem

Theorem (Kassymov, 1988; Khoussainov, 1994) There exists a finitely generated computably enumerable universal algebra no expansion of which is equationally presented.

Theorem ( Khoussainov, 2006 ) There exists a finitely generated computably enumerable universal algebra no expansion of which is quasi-equationally presented.

Important comments The counter-examples constructed are universal algebras built specifically. The counter-examples do not belong to natural classes of structures such as the classes of groups, semi-groups, rings, etc.

Main Question: Can such examples be found among the standard algebraic structures: (1) Semigroups, (2) Algebras (these are rings that form vector spaces over fields), and (3) Groups?

Theorem A (with Hirschfeldt, 2011) There exists a finitely generated computably enumerable semigroup no expansion of which is equationally presented.

Proof. Consider the free semigroup A=({0,1} ★ ;  ). Let X be a subset of {0,1} ★. We say that a string u realizes X if u contains a substring in X. Otherwise, we say that u avoids X. Define: R(X)={u | u realizes X}. Clearly, R(X) is a subset of {0,1} ★.

Define the following relation ≈ X on {0,1} ★. u ≈ X v if either u=v or u and v both realize X. Lemma 1. The relation ≈ X is a congruence relation on the free semigroup A=({0,1} ★ ;  ). Set: A(X) = A/≈ X.

Lemma 2. (Miller) If for all k there are at most k many strings of length ≤ k+4 in X, then R(X) is co-infinite.

Lemma 3. There is a c.e. set X such that R(X) is simple. Proof. Let W 0, W 1, …. be a standard list of c.e. subsets of {0,1} ★. Put string y into X if for some i the string y is the first string of length ≥i+5 appeared W i. The set X is a desired c.e. set.

Consider the semigroup A(X) = A/≈ X. It is finitely generated, c.e., and infinite. Let h 1, …, h n be computable functions compatible with ≈ X. Consider the expansion A’(X)= ( A(X); h 1, …, h n )

Lemma 4 (Kassymov). Any c.e. universal algebra whose equality relation coincides with ≈ X is residually finite. In particular, A’(X) is residually finite. Lemma 5 (Malcev). If a universal algebra A is finitely presented and residually finite then the word problem in A is decidable. Hence, A’(X) is not equationally presented. ☐

Definition (Kassymov, Khoussainov, 1986) A finitely generated infinite c.e. universal algebra A = F/E is effectively infinite if there is an infinite c.e. sequence u 0, u 1, u 2 ….listing pair-wise distinct elements of A. If A is not effectively infinite then we call A algorithmically finite (Miasnikov).

Example. The semi-group A(X) constructed above is algorithmically finite.

Let A = T/E be an algorithmically finite universal algebra. Property 1. Each expansion of A is algorithmically finite. Property 2. Each finitely generated subalgebra is algorithmically finite.

Property 3. For every term t(x) the trace a, t(a), tt(a),… is eventually periodic. In particular, if A is a semigroup then every element of A is of finite order. Property 4. All infinite homomorphic images of A are also algorithmically finite.

Property 5. If A=T/E is residually finite then for all distinct elements x, y of A there exists a subset S(x,y) of T such that: (1) S(x,y) is E-closed and computable. (2) x belongs S(x,y). (3) y does belong to S(x,y).

Lemma. If A is residually finite, then all expansions of A are also residually finite. Proof. Let A’ be an expansion of A. Take two distinct elements x,y of A’. Select the separator set S(x,y) from Property 5. Define the following binary relation ≈ (x,y) on A’:

a ≈ (x,y) b if and only if no elements in S(x,y) and in its complement are identified by the congruence relation on A’ generated by the pair (a,b).

Properties of ≈ (x,y) : (1) ≈ (x,y) is a congruence relation on A’. (2) ≈ (x,y) is a co-c.e. relation. (3) In the quotient algebra A’/ ≈ (x,y) the images of x and y are distinct. Since A’ is algorithmically finite, A’/ ≈ (x,y) must be finite.

Theorem B. Let A be an algorithmically finite universal algebra.If A is residually finite then no expansion of A has equational presentation. Proof. If A’ is an equationally presented expansion of A, then A’ is residually finite. By Malcev’s lemma the word problem in A’ is decidable. Contradiction.

Question (Khoussainov): Are there algorithmically finite groups? Motivation of the question: Algorithmically finite groups are candidates that have no equationally presented expansions.

Theorem (Miasnikov, Osin, 2011) There exists an algorithmically finite group. Miasnikov motivates the theorem from a generic complexity view point. Algorithmically finite groups are called Dehn monsters. Miasnikov and Osin ask if there are residually finite Dehn monsters.

Let K be a finite field. Consider the algebra F=K of polynomials in non-commuting variables. We can represent F as the direct sum ∑Fn∑Fn where F n is the vector space spanned over monomials of degree n.

Let H be a set of homogeneous polynomials, I be the ideal generated by H. Theorem (Golod Shafarevich). Let r n be the number of polynomials in H of degree n, and ε be such that 0< ε < m/2 and r n ≤ ε 2 (m-2 ε) n-2. Then the algebra A=F / I is infinite dimensional.

Let H be a subset of {x,y} ★ constructed in Lemma 3 above. Consider the ideal I=. Theorem C. The algebra A=F/I satisfies the following properties: (1) A is effectively infinite. (2) All expansions of A are residually finite. (3) A has no equationally presented expansions.

Proof. It is clear that the algebra is infinite. Write any polynomials f of F in the form a+ h, where a is a sum of monomials not from KH and h is a sum of monomials in KH. So: f=(a a n )+(h h k ). Identify this sum with the set {a 1,.... a n,h 1,....+h k }, and call the set {a 1,.... a n } the true representative of a.

Since H is simple but not hypersimple, H has a strong array of finite sets for the complement of H. The identification of polynomials with finite subsets implies that the algebra A is effectively infinite.

Claim 1. The collection of all true representatives is an immune set. Claim 2. For all distinct elements x, y of A there exists a subset S(x,y) of F such that: (1) S(x,y) is I-closed and computable. (2) x belongs S(x,y). (3) y does belong to S(x,y).

Claim 3. A ll expansions of A are residually finite. Proof. Take two distinct elements x,y of an expansion A’. Select the separator set S(x,y) from Claim 2. Define the following binary relation ≈ (x,y) :

f ≈ (x,y) g if and only if no elements in S(x,y) and in its complement are identified by the congruence relation on A’ generated by the pair (f,g).

Properties of ≈ (x,y) : (1) ≈ (x,y) is a congruence relation on A’. (2) ≈ (x,y) is a co-c.e. relation. (3) In the quotient algebra A’/ ≈ (x,y) the images of x and y are distinct.

Claim 4. The collection of all true representatives that belong to distinct ≈ (x,y) -equivalence classes is a c.e. set. Thus, the A’/ ≈ (x,y) must be finite. Hence, A’ is c.e., infinite, residually finite. Therefore, the algebra can not be equationally specified by Theorem B.

Theorem C. There exists an algorithmically finite and residually finite algebra. Proof. Consider F=K. Construct a set H of homogeneous polynomials by stages as follows. Let W 0, W 1, …. be a list of c.e. subsets of F.

For each i, let f and g the first polynomials occurring in W i such that: (1) f=f 1 +f 2, g=g 1 + g 2, f 1 = g 1, and (2) the degrees of homogeneous polynomials occurring in both f 2 and g 2 are greater than i+64. Put all homogeneous polynomials occurring in both f 2 and g 2 into H.

For each n >2, in H there are at most 2n homogeneous polynomials of degree n. Hence, for a small ε we have 0< ε < m/2 and r n ≤ ε 2 (m-2 ε) n-2 for all n. By Golod Shafarevich theorem, we have that the algebra A=F / I is infinite. By construction, it is algorithmically finite.

It is well-known that Golod-Shafarevich algebras are residually finite (Golod). So, A is algorithmically finite and residually finite. By Theorem B no expansion of A is equationally presented.

Theorem D. There exists an algorithmically finite and residually finite group G. Proof. Consider the algebra A constructed above. The semigroup G=G(A) generated by the elements (1+x)/I and (1+y)/I of the algebra A forms a group under the product operation. The group G is the one desired.

As a corollary we obtain the following theorem. Theorem E. There exists a group that has no equationally presented expansion. Open Question: Are there semigroups, algebras and groups whose all expansions are not quasi- equationally presented?