# Supremum properties of Galois-type connections

Commentationes Mathematicae Universitatis Carolinae (2006)

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

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topSzáz, 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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