A Study on Cubic H-Relations in a Topological Universe Viewpoint

Jeong-Gon Lee; Kul Hur; Xueyou Chen

doi:10.3390/math8040482

A Study on Cubic H-Relations in a Topological Universe Viewpoint

Mathematics ◽

10.3390/math8040482 ◽

2020 ◽

Vol 8 (4) ◽

pp. 482

Author(s):

Jeong-Gon Lee ◽

Kul Hur ◽

Xueyou Chen

Keyword(s):

Concrete Category ◽

Cartesian Closed

We introduce the concrete category CRel P ( H ) [resp. CRel R ( H ) ] of cubic H-relational spaces and P-preserving [resp. R-preserving] mappings between them and study it in a topological universe viewpoint. In addition, we prove that it is Cartesian closed over Set . Next, we introduce the subcategory CRel P , R ( H ) [resp. CRel R , R ( H ) ] of CRel P ( H ) [resp. CRel R ( H ) ] and investigate it in the sense of a topological universe. In particular, we obtain exponential objects in CRel P , R ( H ) [resp. CRel R , R ( H ) ] quite different from those in CRel P ( H ) [resp. CRel R ( H ) ].

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Function spaces based on L-sets

Filomat ◽

10.2298/fil2011815f ◽

2020 ◽

Vol 34 (11) ◽

pp. 3815-3834

Author(s):

Jinming Fang ◽

Yueli Yue

Keyword(s):

Function Spaces ◽

Concrete Category ◽

Convergence Spaces ◽

Cartesian Closed

For a commutative, integral, and divisible quantale L, a concept of top L-convergence spaces based on L-sets other than crisp sets is proposed by using a kind of L-filters, namely limited L-filters defined in the paper. Our main result is the existence of function spaces in the the concrete category of top L-convergence spaces over the slice category Set#L rather than the category Set of sets, such that the concrete category of top L-convergence spaces over the slice category Set#L is Cartesian closed. In order to support the existence of top L-convergence spaces, some nontrivial examples of limited L-filters and top L-convergence spaces are presented also.

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Dynamic game semantics

Mathematical Structures in Computer Science ◽

10.1017/s0960129520000250 ◽

2020 ◽

pp. 1-60

Author(s):

Norihiro Yamada ◽

Samson Abramsky

Keyword(s):

Programming Languages ◽

Functional Programming ◽

Operational Semantics ◽

Dynamic Game ◽

Standard Interpretation ◽

Functional Programming Languages ◽

Game Semantics ◽

Wide Range ◽

Cartesian Closed Categories ◽

Cartesian Closed

Abstract The present work achieves a mathematical, in particular syntax-independent, formulation of dynamics and intensionality of computation in terms of games and strategies. Specifically, we give game semantics of a higher-order programming language that distinguishes programmes with the same value yet different algorithms (or intensionality) and the hiding operation on strategies that precisely corresponds to the (small-step) operational semantics (or dynamics) of the language. Categorically, our games and strategies give rise to a cartesian closed bicategory, and our game semantics forms an instance of a bicategorical generalisation of the standard interpretation of functional programming languages in cartesian closed categories. This work is intended to be a step towards a mathematical foundation of intensional and dynamic aspects of logic and computation; it should be applicable to a wide range of logics and computations.

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