Supremum properties of Galois-type connections

Árpád Száz

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 4, page 569-583
  • ISSN: 0010-2628

Abstract

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In a former paper, motivated by a recent theory of relators (families of relations), we have investigated increasingly regular and normal functions of one preordered set into another instead of Galois connections and residuated mappings of partially ordered sets. A function f of one preordered set X into another Y has been called (1) increasingly   g -normal, for some function g of Y into X , if for any x ∈ X and y ∈ Y we have f ( x ) ≤ y if and only if x ≤ g ( y ) ; (2) increasingly ϕ -regular, for some function ϕ of X into itself, if for any x 1 , x 2 ∈ X we have x 1 ≤ ϕ ( x 2 ) if and only if f ( x 1 ) ≤ f ( x 2 ) . In the present paper, we shall prove that if f is an increasingly regular function of X onto Y , or f is an increasingly normal function of X into Y , then f [ sup ( A ) ] ⊂ sup ( f [ A ] ) for all A ⊂ X . Moreover, we shall also prove some more delicate, but less important supremum properties of such functions.

How to cite

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Száz, Árpád. "Supremum properties of Galois-type connections." Commentationes Mathematicae Universitatis Carolinae 47.4 (2006): 569-583. <http://eudml.org/doc/249849>.

@article{Száz2006,
abstract = {In a former paper, motivated by a recent theory of relators (families of relations), we have investigated increasingly regular and normal functions of one preordered set into another instead of Galois connections and residuated mappings of partially ordered sets. A function $f$ of one preordered set $X$ into another $Y$ has been called (1) increasingly  $g$-normal, for some function $g$ of $Y$ into $X$, if for any $x\in X$ and $y\in Y$ we have $f(x)\le y$ if and only if $x\le g(y)$; (2) increasingly $\varphi $-regular, for some function $\varphi $ of $X$ into itself, if for any $x_\{1\}, x_\{2\}\in X$ we have $x_\{1\}\le \varphi (x_\{2\})$ if and only if $f(x_\{1\})\le f(x_\{2\})$. In the present paper, we shall prove that if $f$ is an increasingly regular function of $X$ onto $Y$, or $f$ is an increasingly normal function of $X$ into $Y$, then $f[\sup (A)]\subset \sup (f[A])$ for all $A\subset X$. Moreover, we shall also prove some more delicate, but less important supremum properties of such functions.},
author = {Száz, Árpád},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {preordered sets; Galois connections (residuated mappings); supremum properties; preordered sets; Galois connections; residuated mappings; supremum properties},
language = {eng},
number = {4},
pages = {569-583},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Supremum properties of Galois-type connections},
url = {http://eudml.org/doc/249849},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Száz, Árpád
TI - Supremum properties of Galois-type connections
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 4
SP - 569
EP - 583
AB - In a former paper, motivated by a recent theory of relators (families of relations), we have investigated increasingly regular and normal functions of one preordered set into another instead of Galois connections and residuated mappings of partially ordered sets. A function $f$ of one preordered set $X$ into another $Y$ has been called (1) increasingly  $g$-normal, for some function $g$ of $Y$ into $X$, if for any $x\in X$ and $y\in Y$ we have $f(x)\le y$ if and only if $x\le g(y)$; (2) increasingly $\varphi $-regular, for some function $\varphi $ of $X$ into itself, if for any $x_{1}, x_{2}\in X$ we have $x_{1}\le \varphi (x_{2})$ if and only if $f(x_{1})\le f(x_{2})$. In the present paper, we shall prove that if $f$ is an increasingly regular function of $X$ onto $Y$, or $f$ is an increasingly normal function of $X$ into $Y$, then $f[\sup (A)]\subset \sup (f[A])$ for all $A\subset X$. Moreover, we shall also prove some more delicate, but less important supremum properties of such functions.
LA - eng
KW - preordered sets; Galois connections (residuated mappings); supremum properties; preordered sets; Galois connections; residuated mappings; supremum properties
UR - http://eudml.org/doc/249849
ER -

References

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  1. Birkhoff G., Lattice Theory, Amer. Math. Soc. Colloq. Publ. 25 Providence, Rhode Island (1967). (1967) Zbl0153.02501MR0598630
  2. Blyth T.S., Janowitz M.F., Residuation Theory, Pergamon Press Oxford (1972). (1972) Zbl0301.06001MR0396359
  3. Boros Z., Száz Á., Infimum and supremum completeness properties of ordered sets without axioms, Tech. Rep., Inst. Math., Univ. Debrecen 2004/4 1-6. 
  4. Boros Z., Száz Á., Finite and conditional completeness properties of generalized ordered sets, Rostock. Math. Kolloq. 59 (2005), 75-86. (2005) Zbl1076.06003MR2169501
  5. Davey B.A., Priestley H.A., Introduction to Lattices and Order, Cambridge University Press Cambridge (2002). (2002) Zbl1002.06001MR1902334
  6. Ganter B., Wille R., Formal Concept Analysis, Springer Berlin (1999). (1999) Zbl0909.06001MR1707295
  7. Pataki G., On the extensions, refinements and modifications of relators, Math. Balkanica (N.S.) 15 (2001), 155-186. (2001) Zbl1042.08001MR1882531
  8. Pickert G., Bemerkungen über Galois-Verbindungen, Arch. Math. 3 (1952), 285-289. (1952) Zbl0047.26402MR0051816
  9. Száz Á., Structures derivable from relators, Singularité 3 (1992), 14-30. (1992) 
  10. Száz Á., Refinements of relators, Tech. Rep., Inst. Math., Univ. Debrecen 1993/76 1-19. 
  11. Száz Á., Upper and lower bounds in relator spaces, Serdica Math. J. 29 (2003), 239-270. (2003) MR2017088
  12. Száz Á., Lower and upper bounds in ordered sets without axioms, Tech. Rep., Inst. Math., Univ. Debrecen 2004/1 1-11. 
  13. Száz Á., The importance of reflexivity, transitivity, antisymmetry and totality in generalized ordered sets, Tech. Rep., Inst. Math., Univ. Debrecen 2004/2 1-15. 
  14. Száz Á., Galois-type connections and closure operations on preordered sets, Tech. Rep., Inst. Math., Univ. Debrecen 2005/1 1-28. 
  15. Száz Á., Galois-type connections on power sets and their applications to relators, Tech. Rep., Inst. Math., Univ. Debrecen 2005/2 1-38. 

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