| Title:
|
Supremum properties of Galois-type connections (English) |
| Author:
|
Száz, Árpád |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
569-583 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 \smallskip (1) increasingly \,$g$-normal, for some function $g$ of $Y$ into $X$, if for any $x\in X$ and $y\in Y$ we have $f(x)\leq y$ if and only if $x\leq g(y)$; \smallskip (2) increasingly $\varphi $-regular, for some function $\varphi$ of $X$ into itself, if for any $x_{1}, x_{2}\in X$ we have $x_{1}\leq \varphi (x_{2})$ if and only if $f(x_{1})\leq f(x_{2})$. \smallskip 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. (English) |
| Keyword:
|
preordered sets |
| Keyword:
|
Galois connections (residuated mappings) |
| Keyword:
|
supremum properties |
| MSC:
|
03E30 |
| MSC:
|
04A05 |
| MSC:
|
06A06 |
| MSC:
|
06A15 |
| MSC:
|
54E15 |
| idZBL:
|
Zbl 1150.06300 |
| idMR:
|
MR2337412 |
| . |
| Date available:
|
2009-05-05T16:59:41Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119618 |
| . |
| Reference:
|
[1] Birkhoff G.: Lattice Theory.Amer. Math. Soc. Colloq. Publ. 25 Providence, Rhode Island (1967). Zbl 0153.02501, MR 0598630 |
| Reference:
|
[2] Blyth T.S., Janowitz M.F.: Residuation Theory.Pergamon Press Oxford (1972). Zbl 0301.06001, MR 0396359 |
| Reference:
|
[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. |
| Reference:
|
[4] Boros Z., Száz Á.: Finite and conditional completeness properties of generalized ordered sets.Rostock. Math. Kolloq. 59 (2005), 75-86. Zbl 1076.06003, MR 2169501 |
| Reference:
|
[5] Davey B.A., Priestley H.A.: Introduction to Lattices and Order.Cambridge University Press Cambridge (2002). Zbl 1002.06001, MR 1902334 |
| Reference:
|
[6] Ganter B., Wille R.: Formal Concept Analysis.Springer Berlin (1999). Zbl 0909.06001, MR 1707295 |
| Reference:
|
[7] Pataki G.: On the extensions, refinements and modifications of relators.Math. Balkanica (N.S.) 15 (2001), 155-186. Zbl 1042.08001, MR 1882531 |
| Reference:
|
[8] Pickert G.: Bemerkungen über Galois-Verbindungen.Arch. Math. 3 (1952), 285-289. Zbl 0047.26402, MR 0051816 |
| Reference:
|
[9] Száz Á.: Structures derivable from relators.Singularité 3 (1992), 14-30. |
| Reference:
|
[10] Száz Á.: Refinements of relators.Tech. Rep., Inst. Math., Univ. Debrecen 1993/76 1-19. |
| Reference:
|
[11] Száz Á.: Upper and lower bounds in relator spaces.Serdica Math. J. 29 (2003), 239-270. MR 2017088 |
| Reference:
|
[12] Száz Á.: Lower and upper bounds in ordered sets without axioms.Tech. Rep., Inst. Math., Univ. Debrecen 2004/1 1-11. |
| Reference:
|
[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. |
| Reference:
|
[14] Száz Á.: Galois-type connections and closure operations on preordered sets.Tech. Rep., Inst. Math., Univ. Debrecen 2005/1 1-28. |
| Reference:
|
[15] Száz Á.: Galois-type connections on power sets and their applications to relators.Tech. Rep., Inst. Math., Univ. Debrecen 2005/2 1-38. |
| . |
