Formal language convexity in left-orderable groups

We propose a criterion for preserving the regularity of a formal language representation when passing from groups to subgroups. We use this criterion to show that the regularity of a positive cone language in a left-orderable group passes to its finite index subgroups, and to show that there exists no left order on a finitely generated acylindrically hyperbolic group such that the corresponding positive cone is represented by a quasi-geodesic regular language. We also answer one of Navas’ questions by giving an example of an infinite family of groups which admit a positive cone that is generated by exactly [Formula: see text] generators, for every [Formula: see text]. As a special case of our construction, we obtain a finitely generated positive cone for [Formula: see text].

Free division rings of fractions of crossed products of groups with Conradian left-orders

Let D be a division ring of fractions of a crossed product {F[G,\eta,\alpha]}, where F is a skew field and G is a group with Conradian left-order {\leq}. For D we introduce the notion of freeness with respect to {\leq} and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space {F((G))} of all formal power series in G over F with respect to {\leq}. From this we obtain that all division rings of fractions of {F[G,\eta,\alpha]} which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, {F[G,\eta,\alpha]} possesses a division ring of fraction which is free in this sense if and only if the rational closure of {F[G,\eta,\alpha]} in the endomorphism ring of the corresponding right F-vector space {F((G))} is a skew field.

Ordering Garside groups

We introduce a structure on a Garside group that we call Dehornoy structure and we show that an iteration of such a structure leads to a left-order on the group. We define two conditions on a Garside group [Formula: see text] and we show that if [Formula: see text] satisfies these two conditions, then [Formula: see text] has a Dehornoy structure. Then, we show that the Artin groups of type [Formula: see text] and of type [Formula: see text], [Formula: see text] satisfy these conditions, and therefore have Dehornoy structures. As indicated by the terminology, one of the orders obtained by this method on the Artin groups of type [Formula: see text] coincides with the Dehornoy order.

LEFT-ORDERABLE COMPUTABLE GROUPS

Downey and Kurtz asked whether every orderable computable group is classically isomorphic to a group with a computable ordering. By an order on a group, one might mean either a left-order or a bi-order. We answer their question for left-orderable groups by showing that there is a computable left-orderable group which is not classically isomorphic to a computable group with a computable left-order. The case of bi-orderable groups is left open.

No positive cone in a free product is regular

We show that there exists no left order on the free product of two nontrivial, finitely generated, left-orderable groups such that the corresponding positive cone is represented by a regular language. Since there are orders on free groups of rank at least two with positive cone languages that are context-free (in fact, 1-counter languages), our result provides a bound on the language complexity of positive cones in free products that is the best possible within the Chomsky hierarchy. It also provides a strengthening of a result by Cristóbal Rivas which states that the positive cone in a free product of nontrivial, finitely generated, left-orderable groups cannot be finitely generated as a semigroup. As another illustration of our method, we show that the language of all geodesics (with respect to the natural generating set) that represent positive elements in a graph product of groups defined by a graph of diameter at least 3 cannot be regular.

Left orders in Garside groups

We consider the structure group of a non-degenerate symmetric (non-trivial) set-theoretical solution of the quantum Yang–Baxter equation. This is a Bieberbach group and also a Garside group. We show this group is not bi-orderable, that is it does not admit a total order which is invariant under left and right multiplications. Regarding the existence of a left invariant total ordering, there is a great diversity. There exist structure groups with a recurrent left order and with space of left orders homeomorphic to the Cantor set, while there exist others that are even not unique product groups.

Left quotient rings of assosymmetric rings

In this paper, we prove the Common-Denominator property. If \(Q\) is Weak Fountain-Golden Left order in an assosymmetric ring \(R\), then given \(b_{1}\),..., \(b_{n} \in R\) there exist \(a\in S\), \(q_{1}\),..., \(q_{n}\in Q\) such that for every \(i = 1,...,n\), \(b_{i} = \tilde{a} q_{i}0\) and \(a \tilde{a} q_{i}=q_{i}\) and also it is shown that if \(Q\) is subring of an assosymmetric ring \(R\), (i) if $R$ is a weak Fountain-Gould left quotient ring of \(Q\), then \(R\) is a left quotient ring of \(Q\), (ii) suppose $R$ nondegenerate and coinciding with its socle, if \(Q\) is a weak Fountain-Gould left order in \(R\) then \(Q\) is a Fountain-Gould left order in \(R\), (iii) if \(R\) is also artinian then \(Q\) is a classical left order in \(R\) if and only if \(Q\) is a Fountain-Gould left order in \(R\).

Independence algebras, basis algebras and semigroups of quotients

We show that if A is a stable basis algebra satisfying the distributivity condition, then B is a reduct of an independence algebra A having the same rank. If this rank is finite, then the endomorphism monoid of B is a left order in the endomorphism monoid of A.

PALINDROMES AND ORDERINGS IN ARTIN GROUPS

The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn →Bn, [Formula: see text], defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ-1xΔ by the generalized half-twist (Garside element). More generally, the involution rev is defined for all Artin groups (equipped with Artin's presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map [Formula: see text] in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order.

COMPACTNESS OF THE SPACE OF LEFT ORDERS

A left order on a magma (e.g. semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spaces of right orders we analyze, come from knot theory and are called quandles. Our main result establishes an interesting connection between topological properties of the space of left orders on a group, and the classical algebraic result by Conrad [4] and Łoś [13] concerning the existence of left orders.