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We investigate the relation between bifix codes and interval exchange transformations. We prove that the class of natural codings of regular interval echange transformations is closed under maximal bifix decoding.
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We prove new results concerning the relation between bifix codes, episturmian words and subgroups offree groups. We study bifix codes in factorial sets of words. We generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set $F$ of words ($F$-maximal bifix codes). In the case of bifix codes contained in Sturmian sets of words, we obtain several new results. Let $F$ be a Sturmian set of words, defined as the set of factors of a strict episturmian word. Our results express the fact that an $F$-maximal bifix code of degree $d$ behaves just as the set of words of $F$ of length $d$. An $F$-maximal bifix code of degree $d$ in a Sturmian set of words on an alphabet with $k$ letters has $(k-1)d+1$ elements. This generalizes the fact that a Sturmian set contains $(k-1)d+1$ words of length $d$. Moreover, given an infinite word $x$, if there is a finite maximal bifix code $X$ of degree $d$ such that $x$ has at most $d$ factors of length $d$ in $X$, then $x$ is ultimately periodic. Our main result states that any $F$-maximal bifix code of degree $d$ on the alphabet $A$ is the basis of a subgroup of index $d$ of the free group on~$A$.
We introduce a class of sets of words which is a natural common generalization of Sturmian sets and of interval exchange sets. This class of sets consists of the uniformly recurrent tree sets, where the tree sets are defined by a condition on the possible extensions of bispecial factors. We prove that this class is closed under maximal bifix decoding. The proof uses the fact that the class is also closed under decoding with respect to return words.
We introduce a definition of admissibility for subintervals in interval exchange transformations. Using this notion, we prove a property of the natural codings of interval exchange transformations, namely that any derived set of a regular interval exchange set is a regular interval exchange set with the same number of intervals. Derivation is taken here with respect to return words. We characterize the admissible intervals using a branching version of the Rauzy induction. We also study the case of regular interval exchange transformations defined over a quadratic field and show that the set of factors of such a transformation is primitive morphic. The proof uses an extension of a result of Boshernitzan and Carroll.
An interval $k$-graph is the intersection graph of a family $mathcal{I}$ of intervals of the real line partitioned into at most $k$ classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we discuss the interval $k$-graphs that are the incomparability graphs of orders; i.e., cocomparability interval $k$-graphs or interval $k$-orders. Interval $2$-orders have been characterized in many ways, but we show that analogous characterizations do not carry over to interval $k$-orders, for $k > 2$. We describe the structure of interval $k$-orders, for any $k$, characterize the interval $3$-orders (cocomparability interval $3$-graphs) via one forbidden suborder (subgraph), and state a conjecture for interval $k$-orders (any $k$) that would characterize them via two forbidden suborders.
We prove that the order of an ordered group is an interval order if and only if it is a semiorder. Next, we prove that every semiorder is isomorphic to a collection $mathcal J$ of intervals of some totally ordered abelian group, these intervals being of the form $[x, x+ alpha[$ for some positive $alpha$. We describe ordered groups such that the ordering is a semiorder and we introduce threshold groups generalizing totally ordered groups. We show that the free group on finitely many generators and the Thompson group $mathbb F$ can be equipped with a compatible semiorder which is not a weak order. On another hand, a group introduced by Clifford cannot.
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