In  (see also  ), J. Howie shows that the classes of locally indicable groups and right orderable groups are quasivarieties and he also asserts that the class of locally indicable groups is a subclass of the class of right orderable groups. He posed the question of whether or not these two classes coincide. It turns out that the answer is No, indeed, the class of locally indicable groups is a proper subclass of the class of right orderable groups. The first examples of this were given by George Bergman in . There he shows that the universal covering group of SL(2, R) is right-orderable but not locally indicable. It has been shown that the braid groups are right-orderable but that braid groups on more than 4 strings are not locally indicable (see  ).
Howie does not find an explicit set of quasi-identities determining the class of right orderable groups but instead deduces the result rather quickly from the fact that the class is closed under ultraproducts. Clearly, a similar argument shows that the class of orderable groups is a quasi-variety. It would be of interest to find explicit quasi-identities determining the class right orderable group. In this paper, we find explicit sets of universal sentences determining these quasi-varieties. Furthermore, we show that each of the quasi-varieties of orderable groups and right-orderable groups has undecidable theory. Since the paper of Howie, there has been a significnat amount of work on orderable groups (See    and the references there).
We note the classic result that free groups are orderable. It follows that every universally free group is orderable since orderability is captured by universal sentences, We remark that, in the finitely generated case, universally free groups are precisely, in the terminology of Sela, the non-abelian limit groups.
2. Preliminaries from Group Theory
Definition 2.1. Let G be a group. G is locally indicable provided every finitely generated subgroup H, ; admits a surjective homomorphism onto the infinite cyclic group.
Definition 2.2. Let G be a group. G is right-orderable provided it admits a total order ≤ satisfying whenever . G is orderable provided it admits a total order ≤ satisfying both
Clearly, both right-orderability and orderability are inherited by subgroups.
If G is a group and H and K are (not necessarily distinct) subgroups in G, then shall be the subgroup of G generated by all commutators as h and k vary independently over H and K respectively. The lower central series of G is defined recursively by and, if and has already been defined, then .
Theorem 2.3. (Iwasawa , B. H. Neumann  ) Suppose G is a group such that
and, for each n, , is torsion free. Then G is orderable.
Since, by a classical result of Magnus , the hypotheses of Theorem 2.3 are satisfied by free groups, we have the immediate.
Corollary 1. Every free group is orderable.
3. Preliminaries from Logic
A standard reference for the material in this section is the book of Fine, Gaglione, Mysanikov, Spellman and Rosneberger . [Let L0 be the first-order language with equality containing a binary operation symbol • (suppressed notationally in favor of juxtaposition), a unary operation symbol−1 and a constant symbol 1. Thus, an L0-structure is a set G provided with a binary operation , a unary operation and a distinguished element .
A group is then an L0-structure which is a model of the axioms
Suppose is the set of variables of L0. The set of terms of L0 is defined recursively as follows:
Every variable is a term; moreover, the constant symbol 1 is a term. If t is a term already defined, then is a term. If is an ordered pair of terms already defined, then is a term. Modulo the group axioms, every term is equal to a word on the variables and their formal inverses. Here, 1 is identified with the empty word.
An identity or law of L0 is a universal sentence of the form where is a tuple of distinct variables and and are terms of L0 involving at most the variables in . Thus, for example, the group axioms are identities of L0.
A quasi-identity of L0 is a universal sentence of the form
where is a tuple of distinct variables and and are terms of L0 involving at most the variables in . Note that the identity is equivalent to the quasi-identity so that identities may be considered as special cases of quasi-identities. Note also that, modulo the group axioms, the quasi-identity
is equivalent to one of the form
where the and are words in at most the variables in and their formal inverses. A quasi-variety of groups is the model class of a set of quasi-identities of L0 together with the group axioms. Following Chang and Keisler  let us call a class of L0-structures elementary provided it is the model class of at least one set of sentences of L0. A theorem of Mal’cev  asserts that a nonempty elementary class of groups is a quasi-variety of groups if and only if it is closed under taking subgroups and (unrestricted) direct products.
Two (not necessarily distinct) L0-structures G and H are elementarily equivalent, in symbols , provided they satisfy precisely the same sentences of L0. (In particular, if , then H is a group if and only if G is a group.) The next theorem gives an algebraic characterization of elementary equivalence. It was initially proven by Keisler assuming the Generalized Continuum Hypothesis and subsequently proven by Shelah without need of that assumption.
Theorem 3.1. (Keisler-Shelah  ) Two L0-structures are elementarily equivalent if and only if they have isomorphic ultrapowers.
(For a discussion of ultraproducts, see, for example, [CK]).
Theorem 3.2. (Chang and Keisler  ) A class of L0-structures is an elementary class if and only if it is closed under both elementary equivalence and ultraproducts.
To show undecidbability, we will need the following result.
Theorem 3.3. Let be an elementary class of groups. If contains a finitely presented group with unsolvable word problem, then the theory of is undecidable.
Proof. Suppose G is a group in the class and suppose that G has finite presentation
Suppose further that G has unsolvable word problem. For each word on the distinct variables and their formal inverses let be the sentence
If there were a recursive algorithm to decide whether or not each is true in every group in , then we would have an algorithm to solve the word problem for G. The contradiction shows the theory of is undecidable. +
Finally in this section, we explicitly mention the positive solution to the Tarski question.
Theorem 3.4. (Kharlampovich and Myasnikov , Sela  ) Every non-abelian free group is elementarily equivalent to the free group .
4. The Main Results
In this section, we show that the class of orderable groups forms a quasi-variety and further the theory of the orderable groups is undecidable.
Let G be a group and S be a subsemigroup of G. We call S normal in G provided it is invariant under conjugation by arbitrary elements in G. If n is a positive integer and we let be the least normal subsemigroup of G containing as a subset and the least subsemigroup of G containing as a subset.
Let be the ring of integers, be the class of posiitve integers and be its group of units.
Theorem 4.1. (See Passman  ) (a) A necessary and sufficient condition for a group G to be left orderable is that, for every finite subset , the intersection of the 2n semigroups is empty as varies over .
(b) A necessary and sufficient condition for a group G to be orderable is that, for every finite subset , the intersection of the 2n normal subsemigroups is empty as varies over .
Theorem 4.2. The class of orderable groups is elementary.
Proof. For each , and each let be a word of positive length at most on the free semigroup generators (regarded as compound symbols; so, no formal cancellation is permitted) , . In view of Lorenzen’s Theorem (see  ), the class of orderable groups is axiomatized by the group axioms together with the sentences
as n varies over , and the vary over and as the vary over all possible choices. (Note: so ). +
Corollary 2. Any elementary free group and more generally any universally free group is orderable.
Theorem 4.3. The class of left-orderable groups is elementary.
Proof. To prove this we shall utilize the characterization of elementary classes given in Theorem 3.2. Suppose is a family of left-orderable groups indexed by a nonempty set I. For each , let be a left order on . Let D be an ultrafilter on I and let G be the ultraproduct of the family with respect to the ultrafilter D on I.
Then is well-defined on G by insisting and it is easily seen to be a left order on G. Thus, the class of left-orderable groups is closed under taking ultraproducts. Now suppose H is a left-orderable group and . By the Keisler-Shelah characterization of elementary equivalence (  ), H and K have isomorphic ultrapowers and . The fact that the class of left-orderable groups is closed under taking ultraproducts implies that admits a left order. Then the isomorphic group admits a left order. But K embeds in and the left order is inherited by subgroups. Thus, H left-orderable and implies K left-orderable. Hence, the class of left-orderable groups is closed under elementary equivalence as well as ultraproducts. Therefore the class of left-orderable groups is elementary. +
Theorem 4.4. The class of orderable groups is a quasi-variety of groups with an undecidable theory.
Proof. Using Mal’cev’s characterization of quasi-varieties it suffices to show that the class of orderable groups is closed under taking subgroups and unrestricted direct products. We have already noted that order is inherited by subgroups. (Alternatively, since the class has a set of universal axioms it is closed under taking substructures).
Suppose is an order on the group and is an order on the group . Then the lexicographic order on (i.e., if , then provided either or and ) makes into an ordered group. Corollary 2 of Chapter 7, Section 47, p. 292 of Grätzer  asserts that if an elementary class is closed under the direct product of two structures, then it is closed under arbitrary direct products of nonvoid families of structures in .
Alternatively, we could argue as follows. We may well-order the index set of any nonvoid family of orderable groups. There is no loss of generality in taking the index set to be an ordinal . Suppose is a left order on for all ordinals . Then the lexicographical order on
(i.e., if , then provided where is the least ordinal such that ) makes G into an ordered group. It follows that the class of orderable groups is a quasi-variety of groups.
Theorem 4.5. The class of left-orderable groups is a quasi-variety of groups.
We must now show undecidability.
Theorem 4.6. The theory of orderable groups is undecidable.
Proof. By a result of Bludov and Glass , there is a finitely presented orderable group with unsolvable word problem. The result then follows from the proof of Theorem 2.3. +
Exactly the same proof shows.
Theorem 4.7. The theory of left-orderable groups is undecidable.
Remark 4.8. It would be of interest to find explicit quasi-identities axiomatizing the class of left-orderable groups.
 Rhemtulla, A. and Rolfsen, D. (2002) Local Indicability in Ordered Groups: Braids and Elementary Amenable Groups. Proceedings of the American Mathematical Society, 30, 2569-2577.
 Bludov, V.V. and Glass, A.M.W. (2012) A Finitely Presented Orderable Group with Insoluble Word Problem. Bulletin of the London Mathematical Society, 44, 85-98.