Related papers: On Unique Factorization of Non-periodic Words
While modal extensions of decidable fragments of first-order logic are usually undecidable, their monodic counterparts, in which formulas in the scope of modal operators have at most one free variable, are typically decidable. This only…
By strengthening known results about primitivity-blocking words in free groups, we prove that for any nontrivial element w of a free group of finite rank, there are words that cannot be subwords of any cyclically reduced automorphic image…
Given a finite alphabet $\Sigma$ and a right-infinite word $w$ over the alphabet $\Sigma$, we construct a topological space ${\rm Rec}(w)$ consisting of all right-infinite recurrent words whose factors are all factors of $w$, where we work…
This note constructs a finitely generated group $W$ whose word-growth is exponential, but for which the infimum of the growth rates over all finite generating sets is 1 -- in other words, of non-uniformly exponential growth. This answers a…
The category of all idempotent generated semigroups with a prescribed structure $\mathcal{E}$ of their idempotents $E$ (called the biordered set) has an initial object called the free idempotent generated semigroup over $\mathcal{E}$,…
We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w-values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G) is locally finite. If G is…
We show that a free-by-cyclic group with a polynomially growing monodromy is subgroup separable exactly when it is virtually $F_n \times \mathbb{Z}$. We also prove that random deficiency 1 groups are not subgroup separable with positive…
A class of acyclic digraphs $\mathscr{C}$ is linearly unavoidable if there exists a constant $c$ such that every digraph $D\in \mathscr{C}$ is contained in all tournaments of order $c\cdot |V(D)|$. The class of all acyclic digraphs is not…
We find all words $W(x,y,z)$ in the free group $F(x,y,z)$, such that for every group $G$ and an element $c\in G$ the algebraic system $(G,*_{W,c})$ with the binary operation $*_{W,c}$ given by $a*_{W,c}b=W(a,b,c)$ for $a,b\in G$ is a…
The number of frequencies of factors of length $n+1$ in a recurrent aperiodic infinite word does not exceed $3\Delta \C(n)$, where $\Delta \C (n)$ is the first difference of factor complexity, as shown by Boshernitzan. Pelantov\'a together…
We show that for any continuous monotonic fixed-point free automorphism $f$ on a $\sigma$-compact subgroup $G\subset \mathbb R$ there exists a binary operation $+_f$ such that $\langle G, +_f\rangle$ is a topological group topologically…
Fix a word $w$ in a free group $F$ on $r$ generators. A $w$-random permutation in the symmetric group $S_N$ is obtained by sampling $r$ independent uniformly random permutations $\sigma_{1},\ldots,\sigma_{r}\in S_{N}$ and evaluating…
This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly…
We construct a (bi)cyclic sieving phenomenon on the union of dominant maximal weights for level $\ell$ highest weight modules over an affine Kac-Moody algebra with exactly one highest weight being taken for each equivalence class, in a way…
Recently, Kitaev and Remmel [Classifying descents according to parity, Annals of Combinatorics, to appear 2007] refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers. Results in that paper…
We show that for any finite group $G$ and for any $d$ there exists a word $w\in F_{d}$ such that a $d$-tuple in $G$ satisfies $w$ if and only if it generates a solvable subgroup. In particular, if $G$ itself is not solvable, then it cannot…
A generalized lexicographical order on infinite words is defined by choosing for each position a total order on the alphabet. This allows to define generalized Lyndon words. Every word in the free monoid can be factorized in a unique way as…
We study the maximum multiplicity $\mathcal{M}(k,n)$ of a simple transposition $s_k=(k \: k+1)$ in a reduced word for the longest permutation $w_0=n \: n-1 \: \cdots \: 2 \: 1$, a problem closely related to much previous work on sorting…
We show that whenever $m \geq 1$ and $M_1, \dots, M_m$ are nonamenable factors in a large class of von Neumann algebras that we call $\mathcal C_{(\text{AO})}$ and which contains all free Araki-Woods factors, the tensor product factor $M_1…
We study FO+, a fragment of first-order logic on finite words, where monadic predicates can only appear positively. We show that there is a FO-definable language that is monotone in monadic predicates but not definable in FO+. This provides…