Applied Categorical Structures – Official Blog

by Lucio, P.; Orejas, F.; Pasarella, E.; Pino, E.

The semantic constructions and results for definite programs do not extend when dealing with negation. The main problem is related to a well-known problem in the area of algebraic specification: if we fix a constraint domain as a given model, its free extension by means of a set of Horn clauses defining a set of new predicates is semicomputable. However, if the language of the extension is richer than Horn clauses its free extension (if it exists) is not necessarily semicomputable. In this paper we present a framework that allows us to deal with these problems in a novel way. This framework is based on two main ideas: a

by Vallette, Bruno

We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad and free properad.

DOI: 10.1007/s10485-008-9130-y
Online Date: 3/7/2008
Print publication date: 2/1/2009
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by Brzeziński, Tomasz; Amin, Ismail; Yousif, Mohamed

DOI: 10.1007/s10485-008-9129-4
Online Date: 3/3/2008
Print publication date: 4/1/2008
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by Corradini, Andrea; Hermann, Frank; Sobociński, Paweł

Subobject transformation systems STS are proposed as a novel formal framework for the analysis of derivations of transformation systems based on the algebraic, double-pushout (DPO) approach. They can be considered as a simplified variant of DPO rewriting, acting in the distributive lattice of subobjects of a given object of an adhesive category. This setting allows a direct analysis of all possible notions of dependency between any two productions without requiring an explicit match. In particular, several equivalent characterizations of independence of productions are proposed, as well as a local Church–Rosser theorem in the setting of STS. Finally, we show how any derivation tree in an ordinary DPO grammar leads to an

by Koubek, V.; Sichler, J.

A concrete category $\mathbb {Q}$ is finite-to-finite (algebraically) almost universal if the category of graphs and graph homomorphisms can be embedded into $\mathbb {Q}$ in such a way that finite $\mathbb {Q}$-objects are assigned to finite graphs and non-constant $\mathbb {Q}$-morphisms between any $\mathbb {Q}$-objects assigned to graphs are exactly those arising from graph homomorphisms. A quasivariety $\mathbb {Q}$ of algebraic systems of a finite similarity type is Q-universal if the lattice of all subquasivarieties of any quasivariety $\mathbb {R}$ of algebraic systems of a finite similarity type is isomorphic to a quotient lattice of a sublattice of the subquasivariety lattice of $\mathbb {Q}$. This paper shows that any finite-to-finite (algebraically) almost universal

by Dube, Themba

Coz-unique frames were defined and characterized by Banaschewski and Gilmour (J Pure Appl Algebra 157:1–22, 2001). In this note we give further characterizations of these frames along the lines of characterizations of absolutely z-embedded spaces obtained by Blair and Hager (Math Z 136:41–52, 1974) on the one hand, and by Hager and Johnson (Canad J Math 20:389–393, 1968) on the other. We also extend to frames certain characterizations of z-embedded spaces; namely, we give a characterization of coz-onto frame homomorphisms in terms of normal covers.

DOI: 10.1007/s10485-008-9125-8
Online Date: 1/29/2008
Print publication date: 2/1/2009
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by Hosseini, S. N.; Mielke, M. V.

In this paper the category, of partial morphisms of a category $\mathcal{C}$ with respect to a certain class $\mathcal{D}$ of subobjects of $\mathcal{C}$ is formed and the universality of monomorphisms of ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\mathcal{C}} }$ is investigated. The main result characterizes ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\mathcal{C}} }$-universality of monos, in terms of $\mathcal{C}$-universality of monos and the existence of local $\mathcal{C}$-implications.

DOI: 10.1007/s10485-007-9123-2
Online Date: 1/29/2008
Print publication date: 10/1/2009
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by Li, Sheng-gang; Li, Hai-yang

Structures of products, coproducts, subobjects, extremal subobjects, quotient objects and extremal quotient objects in (the category of closed-set lattices) and (the category of closed-set lattice pairs) are given.

DOI: 10.1007/s10485-007-9121-4
Online Date: 1/10/2008
Print publication date: 4/1/2009
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by Martínez, Jorge; Zenk, Eric R.

An epireflection ψ is constructed of the category $\mathfrak{KNArS}$ of compact normal joinfit frames, with skeletal maps, in the subcategory $\mathfrak{SPArS}$ consisting of strongly projectable $\mathfrak{KNArS}$-objects. The construction is achieved via a pushout in the category $\mathfrak{FrmS}$ of frames with skeletal maps, and involves rather intimately the regular coreflection of the object to be reflected. Further, if the regular coreflection ρ is applied to the reflection map ψ

by Gumm, H. Peter

We define an out-degree for F-coalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all F-coalgebras, this class has a terminal object, which for many problems can stand in for the terminal F-coalgebra, which need not exist in general. As examples, we derive structure theoretic results about minimal coalgebras, showing that, for instance minimization of coalgebras is functorial, that products of finitely many minimal coalgebras exist and are given by their largest common subcoalgebra, that minimal subcoalgebras have no inner endomorphisms and show how minimal subcoalgebras can be constructed from Moore-automata. Since the elements of minimal subcoalgebras must correspond uniquely to the formulae of any