Geodesic
Transformation semigroups preserving a binary relation
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 7, pp. 129-135 
I. B. Kozhukhov; V. A. Yaroshevich. Transformation semigroups preserving a binary relation. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 7, pp. 129-135. http://geodesic.mathdoc.fr/item/FPM_2008_14_7_a9/
@article{FPM_2008_14_7_a9,
     author = {I. B. Kozhukhov and V. A. Yaroshevich},
     title = {Transformation semigroups preserving a~binary relation},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {129--135},
     year = {2008},
     volume = {14},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2008_14_7_a9/}
}
TY  - JOUR
AU  - I. B. Kozhukhov
AU  - V. A. Yaroshevich
TI  - Transformation semigroups preserving a binary relation
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2008
SP  - 129
EP  - 135
VL  - 14
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/FPM_2008_14_7_a9/
LA  - ru
ID  - FPM_2008_14_7_a9
ER  - 
%0 Journal Article
%A I. B. Kozhukhov
%A V. A. Yaroshevich
%T Transformation semigroups preserving a binary relation
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2008
%P 129-135
%V 14
%N 7
%U http://geodesic.mathdoc.fr/item/FPM_2008_14_7_a9/
%G ru
%F FPM_2008_14_7_a9

Voir la notice de l'article provenant de la source Math-Net.Ru

We investigate the semigroups of full and partial transformations of a set $X$ which preserve a binary relation $\sigma$ defined on $X$. We consider in detail the case where $\sigma$ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of $\sigma$ for the semigroup of binary relations. It is proved that subsets of $\mathrm B(X)$ preserving $\sigma$ are semigroups in each case. We give the condition of regularity of $\mathrm B_\sigma(X)$ in the case where $\sigma(X)$ is a quasi-order.

[1] Aizenshtat A. Ya., “Regulyarnye polugruppy endomorfizmov uporyadochennykh mnozhestv”, Uchenye zapiski Leningradskogo gos. ped. in-ta im. A. I. Gertsena, 387 (1968), 3–11 | MR

[2] Gluskin L. M., “Polugruppy izotonnykh preobrazovanii”, Uspekhi mat. nauk, 16:5 (1961), 157–162 | MR | Zbl

[3] Yaroshevich V. A., “Polugruppy chastichnykh izotonnykh preobrazovanii kvaziuporyadochennykh mnozhestv”, Sib. mat. zhurn. (to appear)

[4] Adams M. E., Gould M., “Posets whose monoids of order-preserving maps are regular”, Order, 6:2 (1989), 195–201 | DOI | MR | Zbl

[5] Bötcher M., Knauer U., “Endomorphism spectra of graphs”, Discrete Math., 109 (1992), 45–57 | DOI | MR

[6] Bötcher M., Knauer U., “Postscript: ‘Endomorphism spectra of graphs’ ”, Discrete Math., 270 (2003), 329–331 | DOI | MR

[7] Kim V. I., Kozhukhov I. B., “Weakly regular semigroups of isotone transformations”, Filomat (to appear)

[8] Molchanov V. A., “Semigroups of mappings on graphs”, Semigroup Forum, 27 (1983), 155–199 | DOI | MR | Zbl