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Locally connected recursion categories
, 2006
"... Abstract. A recursion category is locally connected if connected domains are jointly epimorphic. New proofs of the existence of noncomplemented and recursively inseparable domains are given in a locally connected category. The use of local connectedness to produce categorical analogs of undecidable ..."
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Abstract. A recursion category is locally connected if connected domains are jointly epimorphic. New proofs of the existence of noncomplemented and recursively inseparable domains are given in a locally connected category. The use of local connectedness to produce categorical analogs of undecidable problems is new; the approach allows us to relax the hypotheses under which the results were originally proved. The results are generalized to nonlocally
Restriction Categories and MCategories
, 2010
"... This paper gives an exposition of the relationship between restriction categories and Mcategories, both of which are formulations of partial maps. The categories of restriction categories and Mcategories are both developed as 2categories. The 2equivalence between the 2category of Mcategories a ..."
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This paper gives an exposition of the relationship between restriction categories and Mcategories, both of which are formulations of partial maps. The categories of restriction categories and Mcategories are both developed as 2categories. The 2equivalence between the 2category of Mcategories and the full sub2category of split restriction categories is described in detail. 1
Centro de Álgebra da Universidade de Lisboa, Portugal. RESTRICTION SEMIGROUPS AND INDUCTIVE CONSTELLATIONS
"... Abstract. The EhresmannScheinNambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended ..."
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Abstract. The EhresmannScheinNambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended the ESN Theorem to the classes of ample, weakly ample and weakly Eample semigroups. A semigroup in any of these classes must contain a semilattice of idempotents, but need not be regular. It is significant here that these classes are each defined by a set of conditions and their leftright duals. Recently, a class of semigroups has come to the fore that is a onesided version of the class of weakly Eample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the
ON THE FUNCTOR ℓ 2
"... and both categories are enriched over algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous lin ..."
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and both categories are enriched over algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous linear maps between Hilbert spaces. 1.
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
, 2012
"... Restriction categories provide a convenient abstract formulation of partial functions. However, restriction categories can have a variety of structures such as finite partial products (cartesianess), joins, meets, and ranges which are of interest in computability theory, semigroup theory, topology, ..."
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Restriction categories provide a convenient abstract formulation of partial functions. However, restriction categories can have a variety of structures such as finite partial products (cartesianess), joins, meets, and ranges which are of interest in computability theory, semigroup theory, topology, and algebraic geometry. This thesis studies these structures. For finite partial products (cartesianess), a construction to add finite partial products to an arbitrary restriction category freely is provided. For joins, we introduce the notion of join restriction categories, describe a construction for the join completion of a restriction category, and show the completeness of join restriction categories in partial map categories using Madhesive categories and Mgaps. As the join completion for inverse semigroups is wellknown in semigroup theory, we show the relationships between the join completion for restriction categories and the join completion for inverse semigroups by providing adjunctions among restriction categories, join restriction categories, inverse categories, and join inverse categories.
DOI:10.1017/S0013091509001230 RETRACTS OF TREES AND FREE LEFT ADEQUATE SEMIGROUPS
, 2009
"... Abstract Recent research of the author has studied edgelabelled directed trees under a natural multiplication operation. The class of all such trees (with a fixed labelling alphabet) has an algebraic interpretation, as a free object in the class of adequate semigroups. We consider here a natural su ..."
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Abstract Recent research of the author has studied edgelabelled directed trees under a natural multiplication operation. The class of all such trees (with a fixed labelling alphabet) has an algebraic interpretation, as a free object in the class of adequate semigroups. We consider here a natural subclass of these trees, defined by placing a restriction on edge orientations, and show that the resulting algebraic structure is a free object in the class of left adequate semigroups. Through this correspondence we establish some structural and algorithmic properties of free left adequate semigroups and monoids, and
What is a differential partial combinatory algebra?
, 2011
"... In this thesis we combine Turing categories with Cartesian left additive restriction categories and again with differential restriction categories. The result of the first combination is a new structure which models nondeterministic computation. The result of the second combination is a structure wh ..."
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In this thesis we combine Turing categories with Cartesian left additive restriction categories and again with differential restriction categories. The result of the first combination is a new structure which models nondeterministic computation. The result of the second combination is a structure which models the notion of linear resource consumption. We also study the structural background required to understand what new features Turing structure should have in light of addition and differentiation – most crucial to this development is the way in which idempotents split. For the combination of Turing categories with Cartesian left additive restriction categories we will also provide a model.
ACTIONS AND PARTIAL ACTIONS OF INDUCTIVE CONSTELLATIONS
"... Abstract. Inductive constellations are onesided analogues of inductive categories which correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids. In this paper, we define the notions of the action and partial action of ..."
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Abstract. Inductive constellations are onesided analogues of inductive categories which correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids. In this paper, we define the notions of the action and partial action of an inductive constellation on a set, before introducing the Szendrei expansion of an inductive constellation, which is modelled closely on that defined by Gilbert (2005) for inductive groupoids. The main result of the paper is a theorem which uses this Szendrei expansion to link the actions and partial actions of inductive constellations, and is analogous to results previously proved by various authors for groups, monoids, and other objects.