Related papers: Natural Monoids and Non-commutative Arithmetics
Arithmetical invariants---such as sets of lengths, catenary and tame degrees---describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the…
An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of…
Let $\mathfrak{M}_n$ be the multiplicative monoid of $n \times n$ matrices over a finite field. The monoid algebra $\mathbf{C}[\mathfrak{M}_n]$ has been studied for several decades. One of the important early results is Kov\'acs' theorem…
A magmoid is a non-empty set with a partial binary operation; group-like magmoids generalize group-like magmas such as semigroups, monoids and groups. In this article, we first consider the many ways in which the notions of associative…
Let A be an alphabet and W be a set of words in the free monoid A*. Let S(W) denote the Rees quotient over the ideal of A* consisting of all words that are not subwords of words in W. We call a set of words W finitely based if the monoid…
Let $M$ be a monoid that is embeddable in a group. We consider the topos $\mathbf{PSh}(M)$ of sets equipped with a right $M$-action, and we study the subtoposes that are of monoid type, i.e. the subtoposes that are again of the form…
We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, {\em if-then-else} and…
We introduce a new family of monoids, which we call gap absorbing monoids. Every gap absorbing monoid is an ideal extension of a free commutative monoid. For a gap absorbing monoid $S$ we study its set of atoms and Betti elements, which…
We define a random Liouville function (\lambda_Q) which depends on a random set (Q) of primes and prove that (A_Q = \{n \in \mathbb{N} | \lambda_Q(n) = -1 \}) is normal almost everywhere. This fact enables us to generate a family of normal…
We generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field. The normal crossings condition considered by Goward is weakened to the condition that…
A submonoid A of N^d has a natural order defined by a <= a + b for elements a and b of A. The Frobenius complex is the order complex of an open interval of A with respect to this order. In this paper, the homotopy type of the Frobenius…
We study functions from a unique factorization monoid to a field. The set of all such functions is a commutative ring isomorphic to a ring of formal power series over the field, with indeterminates indexed by the prime elements of the…
Let $S \subset \mathbb{N}_0$ be a numerical monoid and let $\mathcal P_{\mathrm{fin}} (S)$, resp $\mathcal P_{\mathrm{fin},0}(S)$, denote the power monoid, resp. the restricted power monoid, of $S$, that is the set of all finite nonempty…
This work is dedicated to the results were got in the model theory of the regular polygons. We give the characterization of the monoids with axiomatizable and model complete class of regular polygons. We describe the monoids with complete…
Given a monoid $H$ (written multiplicatively), the family $\mathcal{P}_{\mathrm{fin},1}(H)$ of all non-empty finite subsets of $H$ containing the identity element $1_H$ is itself a monoid, called the reduced finitary power monoid of $H$,…
We consider first-order logic with monoidal quantifiers over words. We show that all languages with a neutral letter, definable using the addition numerical predicate are also definable with the order predicate as the only numerical…
In this note, we define the Burnside ring of a monoid, generalizing the construction for groups. After giving foundational definitions, we characterize transitive M-sets and their automorphisms, then prove a structure theorem for a broad…
Let $M$ be a cancellative and commutative monoid. A submonoid $N$ of $M$ is called an undermonoid if the Grothendieck groups of $M$ and $N$ coincide. For a given property $\mathfrak{p}$, we are interested in providing an answer to the…
Recent results have generalized Gowers' Theorem (to Lupini's Theorem) and the Furstenberg-Katznelson theorem, both infinite dimensional Ramsey Theorems. The framework of arXiv:1611.06600 provides a machine which accepts (almost…
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as…