Reiterman’s Theorem on Finite Algebras for a Monad

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e., classes of finite algebras closed under finite products, subalgebras and quotients. In this article, Reiterman’s theorem is generalized to finite Eilenberg-Moore algebras for a monad T on a category D: we prove that a class of finite T -algebras is a pseudovariety iff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinite monad T ^ associated to the monad T , which gives a categorical view of the construction of the space of profinite terms.

Report on the finiteness of silting objects

We discuss the finiteness of (two-term) silting objects. First, we investigate new triangulated categories without silting object. Second, we study two classes of $\tau$ -tilting-finite algebras and give the numbers of their two-term silting objects. Finally, we explore when $\tau$ -tilting-finiteness implies representation-finiteness and obtain several classes of algebras in which a $\tau$ -tilting-finite algebra is representation-finite.

Automated Generation of EQ-Algebras through Genetic Algorithms

This article introduces an approach to the automated generation of special algebras through genetic algorithms. These algorithms can be also used for a broader variety of applications in mathematics. We describe the results of research aiming at automated production of such algebras with the help of evolutionary techniques. Standard approach is not relevant due to the time complexity of the task, which is superexponential. Our research concerning the usage of genetic algorithms enabled the problem to be solvable in reasonable time and we were able to produce finite algebras with special properties called EQ-algebras. EQ-algebras form an alternate truth–value structure for new fuzzy logics. We present the algorithms and special versions of genetic operators suitable for this task. Then we performed experiments with application EQ-Creator are discussed with proper statistical analysis through ANOVA. The genetic approach enables to automatically generate algebras of sufficient extent without superexponential complexity. Our main results include: that elitism is necessary at least for several parent members, a high mutation ratio must be set, optional axioms fulfilment increases computing time significantly, optional properties negatively affect convergence, and colorfulness was defined to prevent trivial solutions (evolution tends to the simplest way of achieving results).

The structure of the observable algebra determined by a Hopf *-subalgebra in Hopf spin models

Let H be a finite dimensional Hopf C*-algebra, H1 a Hopf*-subalgebra of H. This paper focuses on the observable algebra AH1 determined by H1 in nonequilibrium Hopf spin models, in which there is a copy of H1 on each lattice site, and a copy of ? on each link, where ? denotes the dual of H. Furthermore, using the iterated twisted tensor product of finite +*-algebras, one can prove that the observable algebraAH1 is *-isomorphic to the C*-inductive limit ... o H1 o ? o H1 o ? o H1 o ... .

Weak Separation Problem for Tree Languages

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors. They consist of two monoids, the horizontal and the vertical, with an action of the vertical monoid on the horizontal monoid, and a complementary axiom of faithfulness. A pseudovariety is a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. By looking at the syntactic congruence for monoids and as the natural extension in the case of forest algebras, we could define a version of syntactic congruence of a subset of the free forest algebra, not just a forest language. Let [Formula: see text] be a finite alphabet and [Formula: see text] be a pseudovariety of finite forest algebras. A language [Formula: see text] is [Formula: see text]-recognizable if its syntactic forest algebra belongs to [Formula: see text]. Separation is a classical problem in mathematics and computer science. It asks whether, given two sets belonging to some class, it is possible to separate them by another set of a smaller class. Suppose that a forest language [Formula: see text] and a forest [Formula: see text] are given. We want to find if there exists any proof for that [Formula: see text] does not belong to [Formula: see text] just by using [Formula: see text]-recognizable languages, i.e. given such [Formula: see text] and [Formula: see text], if there exists a [Formula: see text]-recognizable language [Formula: see text] which contains [Formula: see text] and does not contain [Formula: see text]. In this paper, we present how one can use profinite forest algebra to separate a forest language and a forest term and also to separate two forest languages.

THE COMBINATORICS OF TENSOR PRODUCTS OF HIGHER AUSLANDER ALGEBRAS OF TYPE A

We consider maximal non-l-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Plücker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type A. We show that a higher preprojective algebra of the tensor product of two d-representation-finite algebras has a d-precluster-tilting subcategory. Finally, we relate mutations of these collections to a form of tilting for these algebras.