scholarly journals Some New Intrinsic Topologies on Complete Lattices and the Cartesian Closedness of the Category of Strongly Continuous Lattices

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xiuhua Wu ◽  
Qingguo Li ◽  
Dongsheng Zhao
Keyword(s):  
Complete Lattices ◽  
Continuous Mappings ◽  
Cartesian Closedness ◽  
Continuous Lattices ◽  
Cartesian Closed ◽  
Scott Continuous

We prove some new characterizations of strongly continuous lattices using two new intrinsic topologies and a class of convergences. Lastly we show that the category of strongly continuous lattices and Scott continuous mappings is cartesian closed.

scholarly journals The category of uniform convergence spaces is cartesian closed

1976 ◽  
Vol 15 (3) ◽  
pp. 461-465 ◽  
Author(s):  
R.S. Lee
Keyword(s):  
Uniform Convergence ◽  
Function Spaces ◽  
Convergence Spaces ◽  
Cartesian Closedness ◽  
Cartesian Closed ◽  
And Function

This paper first assigns specific uniform convergence structures to the products and function spaces of pairs of uniform convergence spaces, and then establishes a bijection between corresponding sets of morphisms which yields cartesian closedness.

A completion-invariant extension of the concept of meet continuous lattices

2015 ◽  
Vol 27 (4) ◽  
pp. 530-539
Author(s):  
WENFENG ZHANG ◽  
XIAOQUAN XU
Keyword(s):  
Heyting Algebra ◽  
Continuous Lattice ◽  
Stable Maps ◽  
Complete Lattices ◽  
Invariant Extension ◽  
Closed Sets ◽  
Reflective Subcategory ◽  
Normal Completion ◽  
Continuous Lattices ◽  
Algebra 2

In this paper, the concept of meet F-continuous posets is introduced. The main results are: (1) A poset P is meet F-continuous iff its normal completion is a meet continuous lattice iff a certain system γ(P) which is, in the case of complete lattices, the lattice of all Scott closed sets is a complete Heyting algebra; (2) A poset P is precontinuous iff P is meet F-continuous and quasiprecontinuous; (3) The category of meet continuous lattices with complete homomorphisms is a full reflective subcategory of the category of meet F-continuous posets with cut-stable maps.

scholarly journals Kan injectivity in order-enriched categories

2014 ◽  
Vol 25 (1) ◽  
pp. 6-45 ◽  
Author(s):  
JIŘÍ ADÁMEK ◽  
LURDES SOUSA ◽  
JIŘÍ VELEBIL
Keyword(s):  
Enriched Categories ◽  
Continuous Mappings ◽  
Kan Extensions ◽  
Continuous Lattices

Continuous lattices were characterised by Martín Escardó as precisely those objects that are Kan-injective with respect to a certain class of morphisms. In this paper we study Kan-injectivity in general categories enriched in posets. As an example, ω-CPO's are precisely the posets that are Kan-injective with respect to the embeddings ω ↪ ω + 1 and 0 ↪ 1.For every class $\mathcal{H}$ of morphisms, we study the subcategory of all objects that are Kan-injective with respect to $\mathcal{H}$ and all morphisms preserving Kan extensions. For categories such as Top0 and Pos, we prove that whenever $\mathcal{H}$ is a set of morphisms, the above subcategory is monadic, and the monad it creates is a Kock–Zöberlein monad. However, this does not generalise to proper classes, and we present a class of continuous mappings in Top0 for which Kan-injectivity does not yield a monadic category.

A stable universal domain related to ω

2015 ◽  
Vol 27 (4) ◽  
pp. 540-556 ◽  
Author(s):  
HAORAN ZHAO ◽  
HUI KOU
Keyword(s):  
Cartesian Product ◽  
Continuous Functions ◽  
Domain Theory ◽  
Cartesian Closed Category ◽  
Stable Domain ◽  
Closed Category ◽  
Stable Semantics ◽  
Universal Domain ◽  
Cartesian Closed ◽  
Scott Continuous

In 1978, G. Plotkin noticed that $\mathbb{T}$ω, the cartesian product of ω copies of the three element flat domain of Booleans, is a universal domain, where ‘universal’ means that the retracts of $\mathbb{T}$ω for Scott's continuous semantics are exactly all the ωCC-domains, which with Scott continuous functions form a cartesian closed category. As usual, ‘ω’ is for ‘countably based,’ and here ‘CC’ is for ‘conditionally complete,’ which essentially means that any subset which is pairwise bounded has a least upper bound. Since $\mathbb{T}$ω is also an ωDI-domain (an important structure in stable domain theory), the following problem arises naturally: is there a cartesian closed category C of domains with stable functions such that $\mathbb{T}$ω, or a related structure, is universal in C for Berry’s stable semantics? The aim of this paper is to answer this question. We first investigate the properties of stable retracts. We introduce a new class of domains called conditionally complete DI-domains (CCDI-domain for short) and show that, (1) $\mathbb{T}$ω is an ωCCDI-domain and the category of CCDI-domains (resp. ωCCDI-domains) with stable functions is cartesian closed; (2) [$\mathbb{T}$ω →st$\mathbb{T}$ω] is a stable universal domain in the sense that every ωCCDI-domain is a stable retract of [$\mathbb{T}$ω → st$\mathbb{T}$ω], where [$\mathbb{T}$ω → st$\mathbb{T}$ω] is the stable function space of $\mathbb{T}$ω; (3) in particular, [$\mathbb{T}$ω → st$\mathbb{T}$ω] is not a stable retract of $\mathbb{T}$ω and hence $\mathbb{T}$ω is not universal for Berry’s stable semantics. We remark that this paper is a completion and correction of our earlier report in the Proceedings of the 6th International Symposium on Domain Theory and Its Applications (ISDT2013).

scholarly journals OnFS+-Domains

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Yayan Yuan ◽  
Jibo Li
Keyword(s):  
Continuous Functions ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
New Construction ◽  
Cartesian Closed ◽  
Scott Continuous

We introduce a new construction—FS+-domain—and prove that the category withFS+-domains as objects and Scott continuous functions as morphisms is a Cartesian closed category. We obtain that the Plotkin powerdomainPP(L)over anFS-domainLis anFS+-domain.

WHEN IS A CATEGORY OF MANY-SORTED PARTIAL ALGEBRAS CARTESIAN-CLOSED?

1995 ◽  
Vol 06 (01) ◽  
pp. 51-66 ◽  
Author(s):  
M. MONSERRAT ◽  
F. ROSSELLÓ ◽  
J. TORRENS
Keyword(s):  
Operation Symbol ◽  
Functor Category ◽  
Natural Categories ◽  
First Case ◽  
Cartesian Closedness ◽  
Partial Algebras ◽  
The One ◽  
Cartesian Closed

In this paper we study the cartesian closedness of the five most natural categories with objects all partial many-sorted algebras of a given signature. In particular, we prove that, from these categories, only the usual one and the one having as morphisms the closed homomorphisms can be cartesian closed. In the first case, it is cartesian closed exactly when the signature contains no operation symbol, in which case such a category is a slice category of sets. In the second case, it is cartesian closed if and only if all operations are unary. In this case, we identify it as a functor category and we show some relevant constructions in it, such as its subobjects classifier or the exponentials.

Free Finitary Algebras in a Cocomplete Cartesian Closed Category

1972 ◽  
Vol 15 (3) ◽  
pp. 373-374 ◽  
Author(s):  
C. Howlett ◽  
D. Schumacher
Keyword(s):  
Full Subcategory ◽  
Left Adjoint ◽  
Finite Sets ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Cartesian Closedness ◽  
Finite Products ◽  
Cartesian Closed

In [2] Volger proved that the underlying functor of a category of set-valued models of an r-ary theory has a left adjoint. We want to show that his proof remains valid if instead of set valued models of an r-ary theory models of a finitary theory with values in an arbitrary cocomplete cartesian closed category are considered. As Volger for sets we show for any cocomplete cartesian closed category T C that for every finitary theory (S being a skeleton of the full subcategory of finite sets) the restriction of the left adjoint of on C(s) is a functor in ; here brackets around the exponent indicate as usual a restriction to functors which preserve finite products. We are very much indebted to the referee for pointing out that our proof of the last statement is only based on the properties of C mentioned above and the fact that S has and T preserves finite products. With this in mind and retaining only that part of the cartesian closedness which is relevant for the following considerations we can state the following.

Failure of Cartesian closedness in NF

1992 ◽  
Vol 57 (2) ◽  
pp. 555-556 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Category Theory ◽  
Cartesian Closedness ◽  
New Foundations ◽  
Definition Of ◽  
Cartesian Closed ◽  
Ordered Pairs

On any reasonable definition of functions, neither the category of sets nor the category of small categories is cartesian closed in New Foundations (NF). The latter category is sometimes proposed as a foundation for category theory since it is among its own objects. Our result shows it is a poor one.In NF, as in other set theories, a "function" f from a set A to a set B is defined to be a set f of ordered pairs 〈x, y〉 with x in A and y in B, such that (a) if 〈x, y〉 ∈ f and 〈x, y′〉 ∈ f then y = y′, and (b) for every x in A there is some y in B with 〈x, y〉 ∈ f. But in NF different definitions of ordered pairs give significantly different functions. I say a reasonable definition must give:1. The formula z = 〈x, y〉 is stratifiable.2. For every set S there is a set {〈x, x〉 ∣ x ∈ S}.3. If f is a function from A to B, and g one from B to C, there is a set {〈x, z〉∣(∃y)〈x, y〉∈ f & 〈y, z〉∈ g}.Principles 2 and 3 are needed for identity functions and composites. By principle 1, any sets A and B have a set A × B of all ordered pairs 〈x, y〉 with x in A and y in B, but it does not follow that functions exist making A × B a categorical product of A and B.

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