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...ebraic) if every element x 2 P is the directed supremum of those elements (resp., those compact elements) which approximate x. The wellbehaved properties of Scott continuous functions can be found in =-=[9]-=- or many related literatures, and the full subcategory of DCPO which consists of all continuous domains (resp. algebraic domains) with bottoms is denoted by CONT? (resp., ALG?). One of the most intere...

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...mains), the category of FS-domains (resp., algebraic FSdomains). Notice that the last two are exactly all maximal Cartesian closed subcategories of CONT? (cf. [9]). Also the category of Scott domains =-=[5]-=-, the category of bifinite domain are cccs in ALG?. Moreover, the category of bifinite domains and the category of algebraic L-domains are two maximal cccs in ALG?. Although the category CONT? has man...

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...rmation Sciences 171 (2005) 173–187 that date type and denotation of programs are important notions in computer programming languages. In domain theory, dcpo is exactly a generalization of data types =-=[13]-=-. A generalization of denotation of programs in domain theory is Scott continuous function, namely, a mapping between two domains which preserves orders and directed suprema. Categories consisting of ...

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...article. Some stable categories have already beensproved to be Cartesian closed, among which are the category SLP (resp., SLC, SLA) ofL-domains (resp., continuous L-domains, algebraic L-domains) (cf. =-=[1,4,17]-=-), the category of distributive L-domains (cf. [14]), and the well-known category DI of dI-domains (cf. [2]), the category of meet cpos (resp., distributive meet cpos, connected meet cpo) (cf. [15]), ...

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...1]) is a cpo in which every principal ideal is a complete lattice (equivalently, every nonempty bounded subset has an infimum). A bounded-complete domain (bc-domain for short, or complete semilattice =-=[10]-=-) is a cpo in which every bounded pair of elements has supremum (equivalently, every nonempty subset has an infimum). An xalgebraic bc-domain D is called a dI-domain if D satisfies Axiom (d) ð8a; b; c...

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... domain with a basis BðEÞ, and f ; g 2D !s EŠ. If f 6 g and for every x 2 D, b 2 BðEÞ with b 6 f ðxÞ, mðf ; x; bÞ mðg; x; bÞ, then f 6 sg. 3. Exponentials in the full subcategories of SLP Lemma 3.1 =-=[11]-=-. A category C having all finite products (including terminal object) is Cartesian closed provided that each pair of objects A, B in C has an exponential B A , that is, an object B A and a morphism ev...

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...ifinite L-domains (cf. [16]), and the category of stable L-domains (cf. [3]). DI is also the largest Cartesian closed category within the category of x-algebraic bc-domains with stable functions (cf. =-=[12]-=-). For two categories A and B, if we use A@B to denote that A is a full subcategory of B, then we have DI @ SLA @ SLC @ SLP. The purpose of this paper is to examine which stable category C satisfying ...

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...d to be Cartesian closed, among which are the category SLP (resp., SLC, SLA) ofL-domains (resp., continuous L-domains, algebraic L-domains) (cf. [1,4,17]), the category of distributive L-domains (cf. =-=[14]-=-), and the well-known category DI of dI-domains (cf. [2]), the category of meet cpos (resp., distributive meet cpos, connected meet cpo) (cf. [15]), the category of stable bifinite L-domains (cf. [16]...

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...ns (cf. [2]), the category of meet cpos (resp., distributive meet cpos, connected meet cpo) (cf. [15]), the category of stable bifinite L-domains (cf. [16]), and the category of stable L-domains (cf. =-=[3]-=-). DI is also the largest Cartesian closed category within the category of x-algebraic bc-domains with stable functions (cf. [12]). For two categories A and B, if we use A@B to denote that A is a full...

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...s called distributive if for any a, b, c 2 P, the equation a ^ðb _ cÞ ða ^ bÞ_ða ^ cÞ holds whenever both sides are well-defined. By the definition of distributive dcpo, we have Proposition 4.2 (see =-=[6]-=-). For a bc-domain P, the following are equivalent: (1) P is distributive; (2) 8a; b; c 2 P, b " c ) a ^ðb _ cÞ ða ^ bÞ_ða ^ cÞ; (3) 8a; b; c 2 P, fa; b; cg " ) a ^ðb _ cÞ ða ^ bÞ_ða ^ cÞ. Therefore...

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...article. Some stable categories have already beensproved to be Cartesian closed, among which are the category SLP (resp., SLC, SLA) ofL-domains (resp., continuous L-domains, algebraic L-domains) (cf. =-=[1,4,17]-=-), the category of distributive L-domains (cf. [14]), and the well-known category DI of dI-domains (cf. [2]), the category of meet cpos (resp., distributive meet cpos, connected meet cpo) (cf. [15]), ...

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... is the largest lower bound of f and g. Suppose that h 2D !s EŠ, h 6 sf and h 6 sg. Then hðxÞ h ðx 0 Þ^f ðxÞ, hðxÞ hðx 0 Þ^gðxÞð8x 6 x 0 2 D. Therefore hðxÞ hðx 0 Þ^ðf ^ gÞðxÞ, by Proposition 2.2 =-=[2]-=- which implies h 6 sf ^ g. Corollary4.5. Let D, E be distributive and meet-continuous bc-domains, then D !s EŠ is a distributive and meet-continuous bc-domain. Lemma 4.6. The category MBC of meet-con...

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...n spaces.s178 N. Liu, S.-G. Li / Information Sciences 171 (2005) 173–187 Remark 3.3. In any full subcategory of DCPO or SLP, a morphism is isomorphism if and only if it is an order isomorphism. Smith =-=[7]-=- had shown that in any full subcategory of CONT, Cartesian products are the categorical products and exponentials are the spaces of Scott continuous functions. Similarly, we have the following result:...

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...tegory CONT? has many interesting properties, the fullness for subcategory is not always necessary and appropriate, because too many morphisms may bring about errors as pointed out in Plotkin’s paper =-=[8]-=-. For this consideration, Berry [2] defined the stable function (a notion strictly stronger than Scott continuous function) to replace the notion of Scott continuous function. Domain category with sta...

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...[14]), and the well-known category DI of dI-domains (cf. [2]), the category of meet cpos (resp., distributive meet cpos, connected meet cpo) (cf. [15]), the category of stable bifinite L-domains (cf. =-=[16]-=-), and the category of stable L-domains (cf. [3]). DI is also the largest Cartesian closed category within the category of x-algebraic bc-domains with stable functions (cf. [12]). For two categories A...

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...article. Some stable categories have already beensproved to be Cartesian closed, among which are the category SLP (resp., SLC, SLA) ofL-domains (resp., continuous L-domains, algebraic L-domains) (cf. =-=[1,4,17]-=-), the category of distributive L-domains (cf. [14]), and the well-known category DI of dI-domains (cf. [2]), the category of meet cpos (resp., distributive meet cpos, connected meet cpo) (cf. [15]), ...