Related papers: Equations over free inverse monoids with idempoten…
Every abelian (and even every nilpotent) group contains a solution of any finite unimodular system of equations over itself. However, this is not true for infinite systems. We deduced a criterion for a periodic abelian group to contain a…
We consider the following practical question: given a finite algebra A in a finite language, can we efficiently decide whether the variety generated by A has a difference term? We answer this question (positively) in the idempotent case and…
We compute the rate of exponential growth of the free inverse monoid of rank $r$ (and hence an upper bound on the corresponding rate for all $r$-generated inverse monoids and semigroups). This turns out to be an algebraic number strictly…
It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper we prove that the set of…
The Diophantine problem for a monoid $M$ is the decision problem to decide whether any given system of equations has a solution in $M$. In this note, we give a simple example of a context-free, word-hyperbolic, finitely presented, special…
Monoids generated by elements of order two appear in numerous places in the literature. For example, Coxeter reflection groups in geometry, Kuratowski monoids in topology, various monoids generated by regular operations in language theory…
Larrauri and \v{Z}ivn\'y [ICALP'25/ACM ToCL'24] recently established a complete complexity classification of the problem of solving a system of equations over a monoid $N$ assuming that a solution exists over a monoid $M$, where both…
Let $F$ be a free group of finite rank. We say that the monomorphism problem in $F$ is decidable if for any two elements $u$ and $v$ in $F$, there is an algorithm that determines whether there exists a monomorphism of $F$ that sends $u$ to…
Let $M(A,I)$ be a free partially commutative monoid with involution and $G(A,I)$ be its quotient group, e.g. a right-angled Artin or Coxeter group. Given a system of word equations over $M(A,I)$ with recognizable constraints with input size…
We investigate systems of equations and the first-order theory of one-relator monoids. We describe a family $\mathcal{F}$ of one-relator monoids of the form $\langle A\mid w=1\rangle$ where for each monoid $M$ in $\mathcal{F}$, the…
Recently, in weighted automata theory the weight structure of strong bimonoids has found much interest; they form a generalization of semirings and are closely related to near-semirings studied in algebra. Here, we define polynomials over a…
Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and…
A set-theoretical solution of the pentagon equation on a non-empty set $X$ is a function $s:X\times X\to X\times X$ satisfying the relation $s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}$, with $s_{12}=s\times \,id_X$, $s_{23}=id_X \times \, s$…
We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which seem to be necessary to make the…
Monadic decomposability is a notion of variable independence, which asks whether a given formula in a first-order theory is expressible as a Boolean combination of monadic predicates in the theory. Recently, Veanes et al. showed the…
It is known that singular values of idempotent matrices are either zero or larger or equal to one \cite{HouC63}. We state exactly how many singular values greater than one, equal to one, and equal to zero there are. Moreover, we derive a…
We study one-variable equations over the lamplighter group $\MZ_2 \wr \MZ$. While the decidability of arbitrary equations over $L_2$ remains open, we prove that the Diophantine problem for single equations in one variable is decidable. Our…
$ $We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference…
We study a class of projective transformations of spectraplexes associated with self-dual cones and, on this basis, propose a polynomial-time algorithm for convex feasibility problems with positive definite constraints. At each iteration of…
This paper tackles the problem of the existence of solutions for recursive systems of Horn clauses with second-order variables interpreted as integer relations, and harnessed by quantifier-free difference bounds arithmetic. We start by…