Extending Algebraic Operations to D-Completions

In this article we show how separately continuous algebraic operations on T"0-spaces and the laws that they satisfy, both identities and inequalities, can be extended to the D-completion, that is, the universal monotone convergence space completion. Indeed we show that the operations can be extended to the lattice of closed sets, but in this case it is only the linear identities that admit extension. Via the Scott topology, the theory is shown to be applicable to dcpo-completions of posets. We also explore connections with the construction of free algebras in the context of monotone convergence spaces.

[1]  Klaus Keimel,et al.  D-completions and the d-topology , 2009, Ann. Pure Appl. Log..

[2]  H. Herrlich,et al.  Category theory at work , 1991 .

[3]  Michael W. Mislove,et al.  Monoids over domains , 2006, Mathematical Structures in Computer Science.

[4]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

[5]  Yuri Leonidovich Ershov,et al.  Theory of Domains and Nearby (Invited Paper) , 1993, Formal Methods in Programming and Their Applications.

[6]  George Grätzer,et al.  Identities for globals (complex algebras) of algebras , 1988 .

[7]  Yuri Leonidovich Ershov,et al.  The Bounded-Complete Hull of an Alpha-Space , 1997, Theor. Comput. Sci..

[8]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[9]  Glynn Winskel,et al.  Distributing probability over non-determinism , 2006, Mathematical Structures in Computer Science.

[10]  Zoltán Ésik,et al.  Scott induction and equational proofs , 1995, MFPS.

[11]  Alan Bundy,et al.  Constructing Induction Rules for Deductive Synthesis Proofs , 2006, CLASE.

[12]  Jimmie D. Lawson.  T0-spaces and pointwise convergence , 1985 .

[13]  Michael B. Smyth,et al.  Power Domains and Predicate Transformers: A Topological View , 1983, ICALP.

[14]  N. D. Gautam.  The validity of equations of complex algebras , 1957, Arch. Math. Log..

[15]  Klaus Keimel,et al.  Topological cones: functional analysis in a T0-setting , 2008 .

[16]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[17]  Matthias Schröder,et al.  Compactly generated domain theory , 2006, Mathematical Structures in Computer Science.

[18]  Steven J. Vickers,et al.  Presenting Dcpos and Dcpo Algebras , 2008, MFPS.