Results 1  10
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21
Free ample monoids
 Internat. J. Algebra Comput
, 2007
"... Abstract. We show that the free weakly Eample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y ∗ T of a m ..."
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Cited by 12 (9 self)
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Abstract. We show that the free weakly Eample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y ∗ T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X ∗ acts doubly on the semilattice Y of idempotents of FIM(X) and that FAM(X) is embedded in Y ∗ X ∗. Finally we show that every weakly Eample monoid has a proper ample cover. 1.
The Extensive Completion Of A Distributive Category
 Theory Appl. Categ
, 2001
"... A category with finite products and finite coproducts is said to be distributive if the canonical map AB+AC # A (B +C) is invertible for all objects A, B, and C. Given a distributive category D , we describe a universal functor D # D ex preserving finite products and finite coproducts, for wh ..."
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Cited by 7 (1 self)
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A category with finite products and finite coproducts is said to be distributive if the canonical map AB+AC # A (B +C) is invertible for all objects A, B, and C. Given a distributive category D , we describe a universal functor D # D ex preserving finite products and finite coproducts, for which D ex is extensive; that is, for all objects A and B the functor D ex /A D ex /B # D ex /(A + B) is an equivalence of categories. As an application, we show that a distributive category D has a full distributive embedding into the product of an extensive category with products and a distributive preorder. 1.
The HasCasl prologue: categorical syntax and semantics of the partial λcalculus
 COMPUT. SCI
, 2006
"... We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λcalculus. Generalizing Lambek’s classical equivalence between the simply typed λcalculus and cartesian closed categories, we ..."
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Cited by 6 (4 self)
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We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λcalculus. Generalizing Lambek’s classical equivalence between the simply typed λcalculus and cartesian closed categories, we establish an equivalence between partial cartesian closed categories (pccc’s) and partial λtheories. Building on these results, we define (settheoretic) notions of intensional Henkin model and syntactic λalgebra for Moggi’s partial λcalculus. These models are shown to be equivalent to the originally described categorical models in pccc’s via the global element construction. The semantics of HasCasl is defined in terms of syntactic λalgebras. Correlations between logics and classes of categories facilitate reasoning both on the logical and on the categorical side; as an application, we pinpoint unique choice as the distinctive feature of topos logic (in comparison to intuitionistic higherorder logic of partial functions, which by our results is the logic of pccc’s with equality). Finally, we give some applications of the modeltheoretic equivalence result to the semantics of HasCasl and its relation to firstorder Casl.
FREE ADEQUATE SEMIGROUPS
, 902
"... Abstract. We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup ar ..."
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Cited by 6 (1 self)
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Abstract. We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial “folding ” operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are Jtrivial and never finitely generated as semigroups, and that those which are finitely generated as (2, 1,1)algebras have decidable word problem. 1.
STABLE MEET SEMILATTICE FIBRATIONS AND FREE RESTRICTION CATEGORIES
"... Abstract. The construction of a free restriction category can be broken into two steps: the construction of a free stable semilattice fibration followed by the construction of a free restriction category for this fibration. Restriction categories produced from such fibrations are “unitary”, in a sen ..."
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Cited by 5 (2 self)
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Abstract. The construction of a free restriction category can be broken into two steps: the construction of a free stable semilattice fibration followed by the construction of a free restriction category for this fibration. Restriction categories produced from such fibrations are “unitary”, in a sense which generalizes that from the theory of inverse semigroups. Characterization theorems for unitary restriction categories are derived. The paper ends with an explicit description of the free restriction category on a directed graph. 1.
Restriction categories III: colimits, partial limits, and extensivity
 Mathematical Structures in Computer Science
, 2007
"... ..."
Extensions and Covers for Semigroups whose Idempotents form a Left Regular Band’, Semigroup Forum
"... Abstract. Proper extensions that are “injective on Lrelated idempotents” of Runipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgeb ..."
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Cited by 4 (4 self)
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Abstract. Proper extensions that are “injective on Lrelated idempotents” of Runipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a λsemidirect product of a left regular band by an Runipotent or by a glrac semigroup, respectively. An example of such is the generalized Szendrei expansion. As a consequence of our embedding, we are able to give a structure theorem for proper left restriction semigroups. Further, we show that any glrac semigroup S has a proper cover that is a semidirect product of a left regular band by a monoid, and if S is left restriction, the left regular band may be taken to be a semilattice. Dedicated to the memory of our friend, Prof. Douglas Munn 1.
An Equational Notion of Lifting Monad
 TITLE WILL BE SET BY THE PUBLISHER
, 2003
"... We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category ..."
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Cited by 3 (1 self)
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We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the equational properties of partial maps as induced by partial map classifiers. The representation theorem also provides a tool for transferring nonequational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right. 1
Restriction semigroups and inductive constellations’, preprint at www.usersyork.ac.uk/∼varg1
"... 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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Cited by 3 (3 self)
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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
DIFFERENTIAL RESTRICTION CATEGORIES
"... Abstract. We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of Rn in a way that is completely algebraic. We also give other models for the resulting structure ..."
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Cited by 2 (2 self)
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Abstract. We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of Rn in a way that is completely algebraic. We also give other models for the resulting structure, discuss what it means for a partial map to be additive or linear, and show that differential restriction structure can be lifted through various completion operations.