On the action of the implicative closure operator on the set of partial functions of the multivalued logic

S. S. Marchenkov

Diskretnaya Matematika, Tome 32 (2020) no. 1, pp. 60-73.

Résumé

On the set $P_k^*$ of partial functions of the $k$-valued logic, we consider the implicative closure operator, which is the extension of the parametric closure operator via the logical implication. It is proved that, for any $k\geqslant 2$, the number of implicative closed classes in $P_k^*$ is finite. For any $k\geqslant 2$, in $P_k^*$ two series of implicative closed classes are defined. We show that these two series exhaust all implicative precomplete classes. We also identify all 8 atoms of the lattice of implicative closed classes in $P_3^*$.

Détail

Keywords: implicative closure operator, partial functions of multivalued logic.

Bibliographie

[1] Alekseev V.B., Voronenko A. A., “On some closed classes in partial two-valued logic”, Discrete Math. Appl., 4:5 (1994) | DOI | MR | Zbl

[2] Kuznetsov A.V., “O sredstvakh dlya obnaruzheniya nevyvodimosti i nevyrazimosti”, Logicheskii vyvod, Nauka, M., 1979, 5–33

[3] Lo Chzhukai, “Maksimalnye zamknutye klassy v mnozhestve chastichnykh funktsii mnogoznachnoi logiki”, Kiberneticheskii sbornik, no. 25, Mir, M., 1988, 131–141

[4] Lo Chzhukai, “Teoriya polnoty dlya chastichnykh funktsii mnogoznachnoi logiki'”, Kiberneticheskii sbornik, no. 25, Mir, M., 1988, 142–161

[5] Marchenkov S.S., “On expressibility of functions of many-valued logic in some logical-functional languages”, Discrete Math. Appl., 9:6 (1999), 563–581 | DOI | DOI | MR | Zbl

[6] Marchenkov S.S., “Ekvatsionalno zamknutye klassy chastichnykh bulevykh funktsii”, Diskretn. analiz i issled. oper., 15:1 (2008), 82–97 | MR | Zbl

[7] Marchenkov S.S., “The closure operator with the equality predicate branching on the set of partial Boolean functions”, Discrete Math. Appl., 18:4 (2008), 381–389 | DOI | DOI | MR | Zbl

[8] Marchenkov S.S., “Operator E-zamykaniya na mnozhestve chastichnykh funktsii mnogoznachnoi logiki”, Matematicheskie voprosy kibernetiki, no. 18, Fizmatlit, M., 2013, 227–238

[9] Marchenkov S.S., “Positively closed classes of three-valued logic”, J. Appl. Industr. Math., 8:2 (2014), 256–266 | DOI | MR | Zbl

[10] Marchenkov S.S., “Extensions of the positive closure operator by using logical connectives”, J. Appl. Industr. Math., 12:4 (2018), 678–683 | DOI | DOI | MR | Zbl

[11] Marchenkov S.S., Popova A.A., “Positively closed classes of partial Boolean functions”, Moscow Univ. Comput. Math. Cyber., 32 (2008), 147–151 | DOI | MR | Zbl

[12] Romov B.A., “O maksimalnykh podalgebrakh algebry chastichnykh funktsii mnogoznachnoi logiki”, Kibernetika, 1 (1980), 28–36 | MR | Zbl

[13] Romov B.A., “Algebry chastichnykh funktsii i ikh invarianty”, Kibernetika, 2 (1981), 1–11 | Zbl

[14] Romov B.A., “O probleme polnoty v algebre chastichnykh funktsii konechnoznachnoi logiki”, Kibernetika, 1 (1990), 102–106 | MR | Zbl

[15] Freivald R.V., “Kriterii polnoty dlya chastichnykh funktsii algebry logiki i mnogoznachnykh logik”, DAN SSSR, 167:6 (1966), 1249–1250 | Zbl

[16] Freivald R.V., “Funktsionalnaya polnota dlya ne vsyudu opredelennykh funktsii algery logiki”, Diskretnyi analiz, 8 (1966), 55–68 | Zbl

[17] Fleischer I., Rosenberg I.G., “The Galois connection between partial functions and relations”, Pacific J. Math., 79:1 (1978), 93–98 | DOI | MR

[18] Haddad L., Rosenberg I., “Critére général de complétude pour les algébres partielles finies”, C. r. Acad. sci., 304:17 (1987), 507–509 | MR | Zbl

[19] Rosenberg I.G., “Galois theory for partial algebras”, Lecture Notes in Math., 1004 (1983), 257–272 | DOI | MR | Zbl