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46
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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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.
Notes on restriction semigroups and related structures Formerly . . .
, 2010
"... The aim of these notes is to provide a single reference source for basic definitions and results concerning classes of semigroups (and, indeed, of semigroupoids) related to those we refer to as left restriction or weakly left Eample. We give the ‘York’ perspective on these classes of semigroups. ..."
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The aim of these notes is to provide a single reference source for basic definitions and results concerning classes of semigroups (and, indeed, of semigroupoids) related to those we refer to as left restriction or weakly left Eample. We give the ‘York’ perspective on these classes of semigroups. We present a comprehensive account of the relations R ∗ and R̃E and show how many classes of interest to us, including that of left restriction semigroups, arise as natural generalisations of inverse semigroups. Little of this material is new, but some parts of it lie in the realms of folklore, hence one reason for this article. We give a list of original sources, but with no claim to comprehensiveness more references will be added.
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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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.
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 7 (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.
Partial maps with domain and range: extending Schein’s representation
 Commun. Algebra
"... Abstract. The semigroup of all partial maps on a set under the operation of composition admits a number of operations relating to the domain and range of a partial map. Of particular interest are the operations R and L returning the identity on the domain of a map and on the range of a map respectiv ..."
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Abstract. The semigroup of all partial maps on a set under the operation of composition admits a number of operations relating to the domain and range of a partial map. Of particular interest are the operations R and L returning the identity on the domain of a map and on the range of a map respectively. Schein (1970a) gave an axiomatic characterisation of the semigroups with R and L representable as systems of partial maps; the class is a finitely axiomatisable quasivariety closely related to ample semigroups (which were introduced—as type A semigroups—by Fountain, 1979). We provide an account of Schein’s result (which until now appears only in Russian) and extend Schein’s method to include the binary operations of intersection, of greatest common range restriction, and some unary operations relating to the set of fixed points of a partial map. Unlike the case of semigroups with R and L, a number of the possibilities can be equationally axiomatised. 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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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.
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.
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 6 (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.
RESTRICTION SEMIGROUPS AND INDUCTIVE CONSTELLATIONS
, 2009
"... 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 T ..."
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Cited by 6 (3 self)
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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
Restriction categories III: colimits, partial limits, and extensivity
 Mathematical Structures in Computer Science
, 2007
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