scholarly journals Functoriality of the Schmidt Construction

2021 ◽  
Author(s):  
Juan Climent Vidal ◽  
Enric Cosme Llópez
Keyword(s):  
Left Adjoint ◽  
Sigma Algebras ◽  
Recursion Theorem ◽  
Comma Category ◽  
Object Mapping ◽  
Initial Object

After proving, in a purely categorial way, that the inclusion functor InAlg(Σ) from Alg(Σ), the category of many-sorted Σ-algebras, to PAlg(Σ), the category of many-sorted partial Σ-algebras, has a left adjoint FΣ, the (absolutely) free completion functor, we recall, in connection with the functor FΣ, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next we define a category Cmpl(Σ), of Σ-completions, and prove that FΣ, labeled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this we associate to an ordered pair (α,f), where α=(K,γ,α) is a morphism of Σ-completions from F=(C,F,η) to G=(D,G,ρ) and f a homomorphism in D from the partial Σ-algebra A to the partial Σ-algebra B, a homomorphism ΥαG,0(f):Schα(f)B. We then prove that there exists an endofunctor, ΥαG,0, of Mortw(D), the twisted morphism category of D, thus showing the naturalness of the previous construction. Afterwards we prove that, for every Σ-completion G=(D,G,ρ), there exists a functor ΥG from the comma category (Cmpl(Σ)↓G) to End(Mortw(D)), the category of endofunctors of Mortw(D), such that ΥG,0, the object mapping of ΥG, sends a morphism of Σ-completion in Cmpl(Σ) with codomain G, to the endofunctor ΥαG,0.

scholarly journals Finitary -adhesive categories

2014 ◽  
Vol 24 (4) ◽  
Author(s):  
KARSTEN GABRIEL ◽  
BENJAMIN BRAATZ ◽  
HARTMUT EHRIG ◽  
ULRIKE GOLAS
Keyword(s):  
Finite Number ◽  
Petri Net ◽  
Local Church ◽  
Comma Category ◽  
High Level ◽  
Transformation Systems ◽  
Initial Object ◽  
Critical Pairs

Finitary$\mathcal{M}$-adhesive categories are$\mathcal{M}$-adhesive categories with finite objects only, where$\mathcal{M}$-adhesive categories are a slight generalisation of weak adhesive high-level replacement (HLR) categories. We say an object is finite if it has a finite number of$\mathcal{M}$-subobjects. In this paper, we show that in finitary$\mathcal{M}$-adhesive categories we not only have all the well-known HLR properties of weak adhesive HLR categories, which are already valid for$\mathcal{M}$-adhesive categories, but also all the additional HLR requirements needed to prove classical results including the Local Church-Rosser, Parallelism, Concurrency, Embedding, Extension and Local Confluence Theorems, where the last of these is based on critical pairs. More precisely, we are able to show that finitary$\mathcal{M}$-adhesive categories have a unique$\mathcal{E}$-$\mathcal{M}$factorisation and initial pushouts, and the existence of an$\mathcal{M}$-initial object implies we also have finite coproducts and a unique$\mathcal{E}$′-$\mathcal{M}$pair factorisation. Moreover, we can show that the finitary restriction of each$\mathcal{M}$-adhesive category is a finitary$\mathcal{M}$-adhesive category, and finitarity is preserved under functor and comma category constructions based on$\mathcal{M}$-adhesive categories. This means that all the classical results are also valid for corresponding finitary$\mathcal{M}$-adhesive transformation systems including several kinds of finitary graph and Petri net transformation systems. Finally, we discuss how some of the results can be extended to non-$\mathcal{M}$-adhesive categories.

scholarly journals Absolutely Free Multialgebras

2021 ◽  
Author(s):  
Marcelo Esteban Coniglio ◽  
Guilherme Vicentin de Toledo
Keyword(s):  
Algebraic Logic ◽  
Forgetful Functor ◽  
Left Adjoint ◽  
Paraconsistent Logics ◽  
Mapping Property ◽  
Strong Basis ◽  
Generating Set ◽  
Sigma Algebras ◽  
Unique Homomorphism ◽  
Natural Way

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in exploring the foundations of multialgebras applied to the study of logic systems. It is well known from universal algebra that, for every signature \(\Sigma\), there exist algebras over \(\Sigma\) which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of \(\Sigma\)-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of \(\Sigma\)-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor \(\mathcal{U}\), from the category of \(\Sigma\)-multialgebras to Set, does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that \(\mathcal{U}\) does not have a left adjoint.

scholarly journals Recollements, comma categories and morphic enhancements

2021 ◽  
pp. 1-25
Author(s):  
Xiao-Wu Chen ◽  
Jue Le
Keyword(s):  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Comma Category ◽  
The Right ◽  
Projective Lines ◽  
Triangle Functor

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$ , there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$ . Examples related to inflation categories and weighted projective lines are discussed.

scholarly journals Separation Axioms in Intuitionistic Fuzzy Topological Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Amit Kumar Singh ◽  
Rekha Srivastava
Keyword(s):  
Topological Space ◽  
Left Adjoint ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Separation Axioms ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

In this paper we have studied separation axiomsTi,i=0,1,2in an intuitionistic fuzzy topological space introduced by Coker. We also show the existence of functorsℬ:IF-Top→BF-Topand𝒟:BF-Top→IF-Topand observe that𝒟is left adjoint toℬ.

scholarly journals Multiellipsoidal Mapping Algorithm

2018 ◽  
Vol 8 (8) ◽  
pp. 1239 ◽  
Author(s):  
Carlos Villaseñor ◽  
Nancy Arana-Daniel ◽  
Alma Alanis ◽  
Carlos Lopez-Franco ◽  
Javier Gomez-Avila
Keyword(s):  
Point Clouds ◽  
Mapping Algorithm ◽  
Robotic Mapping ◽  
Wide Range ◽  
Important Challenge ◽  
Memory Cost ◽  
Environmental Representation ◽  
Object Shapes ◽  
Quadratic Surfaces ◽  
Object Mapping

The robotic mapping problem, which consists in providing a spatial model of the environment to a robot, is a research topic with a wide range of applications. One important challenge of this problem is to obtain a map that is information-rich (i.e., a map that preserves main structures of the environment and object shapes) yet still has a low memory cost. Point clouds offer a highly descriptive and information-rich environmental representation; accordingly, many algorithms have been developed to approximate point clouds and lower the memory cost. In recent years, approaches using basic and “simple” (i.e., using only planes or spheres) geometric entities for approximating point clouds have been shown to provide accurate representations at low memory cost. However, a better approximation can be implemented if more complex geometric entities are used. In the present paper, a new object-mapping algorithm is introduced for approximating point clouds with multiple ellipsoids and other quadratic surfaces. We show that this algorithm creates maps that are rich in information yet low in memory cost and have features suitable for other robotics problems such as navigation and pose estimation.

Algorithm of Multi-camera Object Handoff Based on Object Mapping

2017 ◽  
pp. 575-582
Author(s):  
Jianrong Cao ◽  
Xuemei Sun ◽  
Zhenyu Li ◽  
Yameng Wang
Keyword(s):  
Object Mapping

scholarly journals Concatenation of Finite Sequences

2019 ◽  
Vol 27 (1) ◽  
pp. 1-13
Author(s):  
Rafał Ziobro
Keyword(s):  
Finite Sequence ◽  
The Other ◽  
Other Hand ◽  
Finite Sequences ◽  
Initial Object

Summary The coexistence of “classical” finite sequences [1] and their zero-based equivalents finite 0-sequences [6] in Mizar has been regarded as a disadvantage. However the suggested replacement of the former type with the latter [5] has not yet been implemented, despite of several advantages of this form, such as the identity of length and domain operators [4]. On the other hand the number of theorems formalized using finite sequence notation is much larger then of those based on finite 0-sequences, so such translation would require quite an effort. The paper addresses this problem with another solution, using the Mizar system [3], [2]. Instead of removing one notation it is possible to introduce operators which would concatenate sequences of various types, and in this way allow utilization of the whole range of formalized theorems. While the operation could replace existing FS2XFS, XFS2FS commands (by using empty sequences as initial elements) its universal notation (independent on sequences that are concatenated to the initial object) allows to “forget” about the type of sequences that are concatenated on further positions, and thus simplify the proofs.

scholarly journals High resolution remote sensing image segmentation based on graph theory and fractal net evolution approach

2015 ◽  
Vol XL-7/W4 ◽  
pp. 197-201 ◽  
Author(s):  
Y. Yang ◽  
H. T. Li ◽  
Y. S. Han ◽  
H. Y. Gu
Keyword(s):  
Remote Sensing ◽  
Image Segmentation ◽  
Graph Theory ◽  
High Resolution ◽  
Spatial Information ◽  
Remote Sensing Image ◽  
Sensing Applications ◽  
Initial Object ◽  
Object Layer ◽  
Evolution Approach

Image segmentation is the foundation of further object-oriented image analysis, understanding and recognition. It is one of the key technologies in high resolution remote sensing applications. In this paper, a new fast image segmentation algorithm for high resolution remote sensing imagery is proposed, which is based on graph theory and fractal net evolution approach (FNEA). Firstly, an image is modelled as a weighted undirected graph, where nodes correspond to pixels, and edges connect adjacent pixels. An initial object layer can be obtained efficiently from graph-based segmentation, which runs in time nearly linear in the number of image pixels. Then FNEA starts with the initial object layer and a pairwise merge of its neighbour object with the aim to minimize the resulting summed heterogeneity. Furthermore, according to the character of different features in high resolution remote sensing image, three different merging criterions for image objects based on spectral and spatial information are adopted. Finally, compared with the commercial remote sensing software eCognition, the experimental results demonstrate that the efficiency of the algorithm has significantly improved, and the result can maintain good feature boundaries.

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