1,720,979 research outputs found
Covers of acts over monoids II
No full textIn 1981 Edgar Enochs conjectured that every module has a at coverand finally proved this, independently with Bican and El Bashir, in 2001.Enochs had in fact considered different types of covers as early as 1963,for example injective and torsion free covers, and since then a great dealof effort has been spent on their study. In 2008, Mahmoudi and Renshawinitiated the study of at covers of acts over monoids but their definitionof cover was slightly different from that of Enochs. Recently, Bailey andRenshaw produced some preliminary results on the `other' type of coverand it is this work that is extended in this paper. We consider free,divisible, torsion free and injective covers and demonstrate that in somecases the results are quite different from the module case
Class-dependent features and multicategory classification
No full textThe problem of pattern classification is considered for the case of multicategory classification where the number of classes, k, is greater than two. Many classification algorithms are in fast 2-class classifiers and are generalised to solve k-class problems. Which classifiers are naturally multicategory and the nature of the generalisation of a 2-class classifier to k classes is not often investigated. A thorough analysis of multicategory classification is given in this thesis which provides a new taxonomy of popular classification algorithms, and goes on to derive these from a probabilistic viewpoint. A clear distinction is made between classifiers that partition the input space and those that partition the set of k classes. Of the classifiers which partition the set of classes, the one-of-n, pairwise, and hierarchical methods of decomposition are shown to be equivalent in the knowledge of the true data distributions. The scaling properties of these algorithms are analysed for increasing k. The effects of learning models on finite data are then investigated to show the practical differences between each decomposition.In classification problems with many classes it is commonly the case that different classes exhibit wildly different properties. In this case it is unreasonable to expect to be able to summarise these properties by using features designed to represent all the classes. In contrast, features should be designed to represent subsets of classes that exhibit common properties without regard to any class outside the subset. The value for classes outside the subset may be meaningless, or simply undefined. The multicategory classification schemes proposed explicitly deal with such class-dependent features, and attractive properties of these classifiers are demonstrated for a real-world handwritten digit recognition application.</p
Covers of acts over monoids and pure epimorphisms
No full textIn 2001 Enochs' celebrated flat cover conjecture was finally proven and the proofs (two different proofs were presented in the same paper [5]) have since generated a great deal of interest among researchers. In particular the results have been recast in a number of other categories and in particular for additive categories (see for example [2], [3], [23] and [24]). In 2008, Mahmoudi and Renshaw considered a similar problem for acts over monoids but used a slightly different definition of cover. They proved that in general their definition was not equivalent to Enochs', except in the projective case, and left open a number of questions regarding the 'other' definition. This 'other' definition is the subject of the present paper and we attempt to emulate some of Enochs' work for the category of acts over monoids and concentrate, in the main, on strongly flat acts. We hope to extend this work to other classes of acts, such as injective, torsion-free, divisible and free, in a future report
A short note on strongly flat covers of acts over monoids
No full textRecently two different concepts of covers of acts over monoids have been studied by a number of authors and many interesting results discovered. One of these concepts is based on coessential epimorphisms and the other is based on Enochs' definition of a flat cover of a module over a ring. Two recent papers have suggested that in the former case, strongly flat covers are not unique. We show that these examples are in fact false and so the question of uniqueness appears to still remain open. In the latter case, we re-present an example due to Kruml that demonstrates that, unlike the case for flat covers of modules, strongly flat covers of S-acts do not always exist
Using Hierarchical Classification to Exploit Context in Pattern Classification for Information Fusion
No full textIn data fusion applications it is important that only the minimum set of relevant features are combined at any one stage in the fusion process. A hierarchical classification methodology is described which handles features at different levels of abstraction to produce a more robust and interpretable classifier. This is achieved by dividing the classes into contextual subgroups, which are further divided to produce a tree structure defining relationships between classes. A novel approach is proposed for the class structure design which is formulated as a constrained search in the structure space. This can be performed via a forward search algorithm driven by a cost function dependent on the performance of the class structure and constraints on the required solution
Extraction of Fuzzy Rules in Command and Control
No full textThis paper described how a neuro-fuzzy construction algorithm was used on data generated in the Anhur research project. The goals of that project included the creation and use of models and rulebases, specifically in these areas: Resource Management, Tactical Situation Assessment, Mission Evaluation, and Infrastructure Reorganization. The models and rulebases were shown and discussed within the context of the methodology developed- SOCIAL: Simulation, Observation, Construction, Incorporation, Adaptation, and Learning
SUBSEMIGROUP, IDEAL AND CONGRUENCE GROWTH OF FREE SEMIGROUPS
No full textThe growth of cofinite subsemigroups of free semigroups is investigated. Lower and upper bounds for the sequence are given and it is shown to have superexponential growth of strict type n(n) for finite free rank greater than 1. Ideal growth is shown to be exponential with strict type 2(n) and congruence growth is shown to be at least exponential. In addition we consider the case when the index is fixed and rank increasing, proving that for subsemigroups and ideals this sequence fits a polynomial of degree the index, whereas for congruences this fits an exponential equation of base the index. We use these results to describe an algorithm for computing values of these sequences and give a table of results for low rank and index
Covers for S-acts and condition (A) for a monoid S
Full text linkA monoid S satisfies Condition (A) if every locally cyclic left S-act is cyclic. This condition first arose in Isbell’s work on left perfect monoids, that is, monoids such that every left S-act has a projective cover. Isbell showed that S is left perfect if and only if every cyclic left S-act has a projective cover and Condition (A) holds. Fountain built on Isbell’s work to show that S is left perfect if and only if it satisfies Condition (A) together with the descending chain condition on principal right ideals, MR. We note that a ring is left perfect (with an analogous definition) if and only if it satisfies MR. The appearance of Condition (A) in this context is therefore monoid specific.Condition (A) has a number of alternative characterisations, in particular, it is equivalent to the ascending chain condition on cyclic subacts of any left S-act. In spite of this, it remains somewhat esoteric. The first aim of this article is to investigate the preservation of Condition (A) under basic semigroup-theoretic constructions.Recently, Khosravi, Ershad and Sedaghatjoo have shown that every left S-act has a strongly flat or Condition (P) cover if and only if every cyclic left S-act has such a cover and Condition (A) holds. Here we find a range of classes of S-acts C such that every left S-act has a cover from C if and only if every cyclic left S-act does and Condition (A) holds. In doing so we find a further characterisation of Condition (A) purely in terms of the existence of covers of a certain kind.Finally, we make some observations concerning left perfect monoids and investigate a class of monoids close to being left perfect, which we name left IPa-perfect
Weak factorization systems for S-acts
No full textThe concept of a weak factorization system has been studied extensively in homotopy theory and has recently found an application in one of the proofs of the celebrated flat cover conjecture, categorical versions of which have been presented by a number of authors including Rosicky. One of the main aims of this paper is to draw attention to this interesting concept and to initiate a study of these systems in relation to flatness of S-acts and related concepts
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