Narrow Results

In this paper, we explore a class of numerical semigroups initiated by Kunz and Waldi containing two coprime numbers $p<q$, which we call KW semigroups. We characterize KW numerical semigroups by their principal matrices. We present a necessary and sufficient criterion for a matrix to be the principal matrix of a KW semigroup. An explicit description of the minimal resolutions of numerical semigroups in the same class with small embedding dimensions $3$ and $4$ is given. We give a generalization of this notion to three dimensions using lattice paths under a plane and present some preliminary results and questions.

The category ${STROP}_m$ of supertropical monoids, whose morphisms are transmissions, has the full-reflective subcategory $STROP$ of commutative semirings. In this setup, quotients are determined directly by equivalence relations, as ideals are not applicable for monoids, leading to a new approach to factorization theory. To this end, tangible factorization into irreducibles is obtained through fiber contractions and their hierarchy. Fiber contractions also provide different quotient structures, associated with covers and types of splitting covers.

We consider fragments of first-order logic over finite words. In particular, we deal with first-order logic with a restricted number of variables and with the lower levels of the alternation hierarchy. We use the algebraic approach to show decidability of expressibility within these fragments. As a byproduct, we survey several characterizations of the respective fragments. We give complete proofs for all characterizations and we provide all necessary background. Some of the proofs seem to be new and simpler than those which can be found elsewhere. We also give a proof of Simon's theorem on factorization forests restricted to aperiodic monoids because this is simpler and sufficient for our purpose.

We are interested in regular expressions and transducers that represent word relations in an alphabet-invariant way — for example, the set of all word pairs $u,v$ where $v$ is a prefix of $u$ independently of what the alphabet is. Current software systems of formal language objects do not have a mechanism to define such objects. We define transducers in which transition labels involve what we call set specifications, some of which are alphabet invariant. In fact, we give a more broad definition of automata-type objects, called labelled graphs, where each transition label can be any string, as long as that string represents a subset of a certain monoid. Then, the behavior of the labelled graph is a subset of that monoid. We do the same for regular expressions. We obtain extensions of a few classic algorithmic constructions on ordinary regular expressions and transducers at the broad level of labelled graphs and in such a way that the computational efficiency of the extended constructions is not sacrificed. For transducers with set specs we obtain further algorithms that can be applied to questions about independent regular languages as well as a decision question about synchronous transducers.

A new category of algebro-geometric objects is defined which contains the category of monoid schemes, or schemes over 𝔽₁, and the category of Grothendieck schemes as subcategories.

A theory of the semidirect product of categories and the derived category of a category morphism is presented. In order to include division (≺) in this theory, the traditional setting of these constructions is expanded to include relational arrows. In this expanded setting, a relational morphism φ : M → N of categories determines an optimal decomposition

The theory of the semidirect product of varieties of categories, V * W, is developed. Associated with each variety V of categories is the collection of relational morphisms whose derived category belongs to V. The semidirect product of varieties and the composition of classes of the form are shown to stand in the relationship

Finally, it is demonstrated that all the results in the article concerning varieties of categories have pseudovariety and monoidal versions. This allows us to furnish a straightforward proof that

We show that if H is a Fitting pseudovariety of groups and V is a local pseudovariety of monoids, then LH ⓜ V is local if either V contains the six element Brandt monoid, or H is a non-trivial pseudovariety of groups closed under extension.

The monoid of extensive transformations of a chain of order four is shown to be finitely based and a finite basis for this monoid is given. This completes the description of the equational property of the monoids of all full extensive transformations, partial extensive transformations, and partial order-preserving extensive transformations over any finite chain.

It is shown that if 𝖵 is a local monoidal pseudovariety of semigroups, then 𝖪 ⓜ 𝖵, 𝖣 ⓜ 𝖵 and are local. Other operators of the form 𝖹 ⓜ(_) are considered. In the process, results about the interplay between operators 𝖹 ⓜ(_) and (_) _* 𝖣_k are obtained.

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.

A non-finitely based variety of algebras is said to be a limit variety if all its proper subvarieties are finitely based. Recently, Marcel Jackson published two examples of finitely generated limit varieties of aperiodic monoids with central idempotents and questioned whether or not they are unique. The present article answers this question affirmatively.

The authors of Berg et al. [J. Algebra348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for $ℂ{ℳ}$, where ${ℳ}$ is any finite ${ℛ}$-trivial monoid. Their method relies on a technical result stating that ${ℛ}$-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an ${ℛ}$-trivial monoid may be realized by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where ${ℒ}$-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for $\mathbb{Z}{ℳ}$, after which we prove that it also works for $R{ℳ}$ where $R$ is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if $R$ is any field. In this respect our result constitutes a considerable generalization of the results in Berg et al. [J. Algebra348 (2011) 446–461]. Moreover, the system of idempotents for $R{ℳ}$ is obtained from the one our algorithm yields for $\mathbb{Z}{ℳ}$ in a straightforward manner. In other words, for any finite ${ℛ}$-trivial monoid ${ℳ}$ our algorithm only has to be performed for $\mathbb{Z}{ℳ}$, after which a system of idempotents follows for any ring with a given system of idempotents.

In this paper, we introduce (Rees) artinian $S$-acts as those acts that satisfy the descending chain condition on their (Rees) congruences. Then, we show that (Rees) artinian $S$-acts are those acts whose factor acts are all finitely (Rees) cogenerated. In addition, we continue the study of (Rees) noetherian $S$-acts, namely, those acts whose (Rees) congruences are finitely generated. Moreover, as a useful tool for the investigation of chain conditions, we prove that the properties of being (Rees) noetherian and artinian are inherited in Rees short exact sequences. Next, we specifically consider the chain conditions on monoids. Finally, we prove that every right Rees artinian, commutative monoid is right Rees noetherian.

We characterize a class of option pricing models by their algebraic structure. Option prices are monoids, that is, operators endowed with the commutativity and associativity property and an identity element. If the price of the underlying asset is bounded, the operator corresponds to the concept of t-conorm, while if it is defined on the positive real line, the operator is a pseudo-addition. These operators have the same no-arbitrage properties as the classical option pricing models but are also associative. Each model in this class is characterized by a univariate increasing function that is defined as the generator of the model. The generator encodes a synthetic representation of the probability structure of the underlying asset. We provide no-arbitrage conditions for the generators and practical guidelines to construct them.