Geodesic
A minimal non-extendable partial semigroup
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 1, pp. 31-39.

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

This article discusses partial semigroups with a finite number of elements. Any partial semigroup can be extended to a full semigroup by adding elements to it, for example, a zero semigroup, in an external semigroup way. The author of the article is interested in the question of continuation of a partial semigroup without adding any elements to it in an internal semigroup way. The aim of this work is to find an internally non-extendable partial semigroup with a minimal number of elements. With increasing the number of elements in the set the number of partial groupoids on this set increases exponentially, and the number of partial semigroups among these partial groupoids is not known in advance. In order to find such partial semigroups it is necessary to use a computer or the Internet. In the Internet (GAP package) there are stored all the semigroups up to isomorphism and antiisomorphism on the set consisting of no more than 8 elements; that is why it will be enough to get partial semigroups out of semigroups with zero by deleting zero. The possibility of continuation of a partial semigroup in an internal semigroup way was checked out by a computer. As a result, it was revealed that all the partial semigroups on the set consisting of no more than 4 elements can be extended in an internal semigroup way to full ones. On the 5-element set, there is only one partial semigroup up to isomorphism and antiisomorphism, which can not be extended to a full semigroup. 
@article{ISU_2017_17_1_a2,
     author = {A. O. Petrikov},
     title = {A minimal non-extendable partial semigroup},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {31--39},
     publisher = {mathdoc},
     volume = {17},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2017_17_1_a2/}
}
TY  - JOUR
AU  - A. O. Petrikov
TI  - A minimal non-extendable partial semigroup
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2017
SP  - 31
EP  - 39
VL  - 17
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ISU_2017_17_1_a2/
LA  - ru
ID  - ISU_2017_17_1_a2
ER  - 
%0 Journal Article
%A A. O. Petrikov
%T A minimal non-extendable partial semigroup
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2017
%P 31-39
%V 17
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ISU_2017_17_1_a2/
%G ru
%F ISU_2017_17_1_a2
A. O. Petrikov. A minimal non-extendable partial semigroup. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 1, pp. 31-39. http://geodesic.mathdoc.fr/item/ISU_2017_17_1_a2/

[1] Lyapin E. S., “The possibility of semigroup continuation of a partial groupoid”, Soviet Math. (Iz. VUZ), 33:12 (1989), 82–85 | MR | Zbl

[2] Lyapin E. S., “Inner semigroup continuation of some semigroup amalgams”, Soviet Math. (Iz. VUZ), 37:11 (1993), 18–24 | MR | Zbl

[3] Goralcik P., Koubek V., “On completing partial groupoids to semigroups”, Intern. J. Algebra Comput., 16:3 (2006), 551–562 | DOI | MR | Zbl

[4] Petrikov A. O., “Partial Semigroups and Green's Relations”, Electronic Information Systems, 2014, no. 3(3), 65–72 (in Russian)

[5] Rozen V. V., Partial operations in ordered sets, Saratov Univ. Press, Saratov, 1973, 123 pp. (in Russian)

[6] Kurosh A. G., Lectures on general algebra, Nauka, M., 1973, 400 pp. (in Russian)

[7] Clifford A., Preston G., The algebraic theory of semigroups, v. 1, American Mathematical Society, Providence, R.I., 1964, 224 pp.

[8] Lyapin E. S., Evseev A. E., Partial algebaic operations, Obrazovanie, S. Petersburg, 1991, 163 pp. (in Russian)

[9] Cayley A., “On the Theory of Groups”, American Journal of Mathematics, 11:2 (1889), 139–157 | DOI | MR

[10] The GAP Group, (data obrascheniya: accessed 14 August 2016) http://www.gap-system.org