Geodesic
$E$-closed sets of hyperfunctions on two-element set
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 2, pp. 231-241.

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

Hyperfunctions are functions that are defined on a finite set and return all non-empty subsets of the considered set as their values. This paper deals with the classification of hyperfunctions on a two-element set. We consider the composition and the closure operator with the equality predicate branching ($E$-operator). $E$-closed sets of hyperfunctions are sets that are obtained using the operations of adding dummy variables, identifying variables, composition, and $E$-operator. It is shown that the considered classification leads to a finite set of closed classes. The paper presents all 78 $E$-closed classes of hyperfunctions, among which there are 28 pairs of dual classes and 22 self-dual classes. The inclusion diagram of the $E$-closed classes is constructed, and for each class its generating system is obtained.
Keywords: closure, equality predicate, hyperfunction, closed set
Mots-clés : composition. 
@article{JSFU_2020_13_2_a9,
     author = {Vladimir I. Panteleyev and Leonid V. Riabets},
     title = {$E$-closed sets of hyperfunctions on two-element set},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {231--241},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2020_13_2_a9/}
}
TY  - JOUR
AU  - Vladimir I. Panteleyev
AU  - Leonid V. Riabets
TI  - $E$-closed sets of hyperfunctions on two-element set
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2020
SP  - 231
EP  - 241
VL  - 13
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2020_13_2_a9/
LA  - en
ID  - JSFU_2020_13_2_a9
ER  - 
%0 Journal Article
%A Vladimir I. Panteleyev
%A Leonid V. Riabets
%T $E$-closed sets of hyperfunctions on two-element set
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2020
%P 231-241
%V 13
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2020_13_2_a9/
%G en
%F JSFU_2020_13_2_a9
Vladimir I. Panteleyev; Leonid V. Riabets. $E$-closed sets of hyperfunctions on two-element set. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 2, pp. 231-241. http://geodesic.mathdoc.fr/item/JSFU_2020_13_2_a9/

[1] D. Lau, Function algebras on finite sets. A basic course on many-valued logic and clone theory, Springer, 2006 | MR | Zbl

[2] H.Machida, Multiple-Valued Logic, 8 (2002), 495–501 | DOI | MR | Zbl

[3] H.Machida, J.Pantovic, “On Maximal Hyperclones on $\{0, 1\}$ — a new approach”, Proceedings of 38th IEEE International Symposium on Multiple-Valued Logic, 2008, 32–37

[4] B.A.Romov, “Hyperclones on a Finite Set”, Multiple-Valued Logic, 3 (1998), 285–300 | Zbl

[5] V.I.Panteleyev, “Completeness Criterion for Additional Defined Boolean Functions”, Vestnik Sam. Gos. Univ., 68 (2009), 60–79 (in Russian)

[6] S.V.Zamaratskaya, V.I.Panteleyev, “Maximal Clones of Ultrafunctions of Rank 2”, The Bulletin of Irkutsk State University. Series Mathematics, 15 (2016), 26–37 (in Russian) | Zbl

[7] S.V.Zamaratskaya, V.I.Panteleyev, “Classification and Types of Bases of All Ultrafunctions on Two-Element Set”, The Bulletin of Irkutsk State University. Series Mathematics, 16 (2016), 58–70 (in Russian) | Zbl

[8] S.S.Marchenkov, Discrete Math. Appl., 9:6 (1999), 563–581 | DOI | MR | Zbl

[9] S.S.Marchenkov, “Closure Operators with Predicate Branching”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2003, no. 6, 37–39 (in Russian) | MR | Zbl

[10] S.S.Marchenkov, “The Closure Operator With the Equality Predicate Branching on the Set of Partial Boolean Functions”, Discrete Math. Appl., 18:4 (2008), 381–389 | DOI | MR | Zbl

[11] S.S.Marchenkov, “E-closed Operator in the Set of Partial Many-Valued Logic Functions”, Mathematical problems in cybernetics, 2013, no. 19, 227–238 (in Russian)

[12] S.A.Matveev, “Construction of All E-closed Classes of Partial Boolean Functions”, Mathematical problems in cybernetics, 2013, no. 18, 239–244 (in Russian)

[13] L.V.Ryabets, “Parametric Closed Classes of Hyperfunctions on Two-Element Set”, The Bulletin of Irkutsk State University. Series Mathematics, 17 (2016), 46–61 (in Russian) | Zbl

[14] L.V.Ryabets, “Parametric and Positive Closed Operators on the Set of Hyperfunctions on Two-Element Set”, Intelligent systems. Theory and applications, 20:3 (2016), 79–84 (in Russian)

[15] V.I.Panteleyev, L.V.Ryabets, “The Closure Operator with the Equality Predicate Branching on the Set of Hyperfunctions on Two-Element Set”, The Bulletin of Irkutsk State University. Series Mathematics, 10 (2014), 93–105 (in Russian) | MR | Zbl