Geodesic
Logical extensions of the parametric closure operator
Diskretnaya Matematika, Tome 34 (2022) no. 3, pp. 52-62.

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In the present survey, we consider all logical extensions of the parametric closure operator. Such extensions are obtained by augmenting the parametric closure language with arbitrary logical connectives or the universal quantifier. This gives, in addition to the parametric closure operator, the operators of positive and implicative closure, and also the closure operator with complete system of logical connectives, and the conjunctive-quantifier closure operator. We present basic facts on classifications of the sets $P_k$ generated by these operators.
Keywords: operator parametric closure, logical extension of the closure operator. 
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S. S. Marchenkov. Logical extensions of the parametric closure operator. Diskretnaya Matematika, Tome 34 (2022) no. 3, pp. 52-62. http://geodesic.mathdoc.fr/item/DM_2022_34_3_a3/

[1] Golunkov Yu. V., “Polnota sistem funktsii v operatornykh algoritmakh, realizuyuschikh funktsii $k$-znachnoi logiki”, Veroyatnostnye metody i kibernetika, 17 (1980), 23–34 | Zbl

[2] Danilchenko A. F., “O parametricheskoi vyrazimosti funktsii trekhznachnoi logiki”, Algebra i logika, 16:4 (1977), 397–416 | MR

[3] Danilchenko A. F., “Parametricheski zamknutye klassy funktsii trekhznachnoi logiki”, Izvestiya AN MSSR, 2 (1978), 13–20

[4] Danilchenko A. F., Voprosy parametricheskoi vyrazimosti funktsii trekhznachnoi logiki, diss. ... kand. fiz.-matem. nauk, Kishinev, 1979, 141 pp.

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

[6] Marchenkov S. S., “Osnovnye otnosheniya $S$-klassifikatsii funktsii mnogoznachnoi logiki”, Diskretnaya matematika, 8:1 (1996), 99–128 | Zbl

[7] Marchenkov S. S., “$S$-klassifikatsiya funktsii mnogoznachnoi logiki”, Diskretnaya matematika, 9:3 (1997), 125–152 | Zbl

[8] Marchenkov S. S., “O vyrazimosti funktsii mnogoznachnoi logiki v nekotorykh logiko-funktsionalnykh yazykakh”, Diskretnaya matematika, 11:4 (1999), 110–126 | Zbl

[9] Marchenkov S. S., Zamknutye klassy bulevykh funktsii, Fizmatlit, M., 2000, 126 pp.

[10] Marchenkov S. S., $S$-klassifikatsiya funktsii trekhznachnoi logiki, Fizmatlit, M., 2001, 79 pp. | MR

[11] Marchenkov S. S., “Operatory zamykaniya s razvetvleniem po predikatu”, Vestnik Mosk. univ. Seriya 1: Matematika, mekhanika, 6 (2003), 37–39 | Zbl

[12] Marchenkov S. S., “Ekvatsionalnoe zamykanie”, Diskretnaya matematika, 17:2 (2005), 117–126 | Zbl

[13] Marchenkov S. S., “Kriterii pozitivnoi polnoty v trekhznachnoi logike”, Diskret. analiz i issled. operatsii. Seriya 1, 13:3 (2006), 27–39 | Zbl

[14] Marchenkov S. S., “O stroenii ekvatsionalno zamknutykh klassov”, Diskretnaya matematika, 18:4 (2006), 18–30 | MR | Zbl

[15] Marchenkov S. S., “Diskriminatornye pozitivno zamknutye klassy trekhznachnoi logiki”, Diskret. analiz i issled. operatsii. Seriya 1, 14:3 (2007), 53–66 | Zbl

[16] Marchenkov S. S., “O zamknutykh klassakh funktsii $k$-znachnoi logiki, opredelyaemykh odnim endomorfizmom”, Diskret. analiz i issled. operatsii, 16:6 (2009), 52–67 | MR | Zbl

[17] Marchenkov S. S., “Pozitivno zamknutye klassy trekhznachnoi logiki, porozhdaemye odnomestnymi funktsiyami”, Diskretnaya matematika, 21:3 (2009), 37–44 | MR | Zbl

[18] Marchenkov S. S., “Operator zamykaniya v mnogoznachnoi logike, baziruyuschiisya na funktsionalnykh uravneniyakh”, Diskret. analiz i issled. operatsii, 17:4 (2010), 18–31 | Zbl

[19] Marchenkov S. S., “O klassifikatsiyakh funktsii mnogoznachnoi logiki s pomoschyu grupp avtomorfizmov”, Diskret. analiz i issled. operatsii, 18:4 (2011), 66–76 | MR | Zbl

[20] Marchenkov S. S., “FE-klassifikatsiya funktsii mnogoznachnoi logiki”, Vestnik MGU. Seriya 15: Vychisl. matem. i kibern., 2 (2011), 32–39

[21] Marchenkov S. S., “Atomy reshetki pozitivno zamknutykh klassov trekhznachnoi logiki”, Diskretnaya matematika, 24:2 (2012), 79–81

[22] Marchenkov S. S., “Operator pozitivnogo zamykaniya”, Doklady RAN, 442:5 (2012), 598–599 | Zbl

[23] Marchenkov S. S., Funktsionalnye uravneniya diskretnoi matematiki, Fizmatlit, M., 2013, 58 pp.

[24] Marchenkov S. S., Osnovy teorii bulevykh funktsii, Fizmatlit, M., 2014, 135 pp.

[25] Marchenkov S. S., “Pozitivno zamknutye klassy trekhznachnoi logiki”, Diskret. analiz i issled. operatsii, 21:1 (2014), 67–83 | MR | Zbl

[26] Marchenkov S. S., “Ob operatore zamykaniya po perechisleniyu v mnogoznachnoi logike”, Vestnik MGU. Seriya 15: Vychisl. matem. i kibern., 2 (2015), 32–39

[27] Marchenkov S. S., “Operatory zamykaniya s pozitivnymi svyazkami i kvantorami”, Vestnik MGU. Seriya 15: Vychisl. matem. i kibern., 1 (2017), 33–38

[28] Marchenkov S. S., “O rasshireniyakh operatora parametricheskogo zamykaniya s pomoschyu logicheskikh svyazok”, Izvestiya vysshikh uchebnykh zavedenii. Povolzhskii region. Fiz.-mat. nauki, 1 (2017), 22–31

[29] Marchenkov S. S., “Kriterii polnoty dlya operatora zamykaniya po perechisleniyu v trekhznachnoi logike”, Diskretnaya matematika, 30:4 (2018), 47–54

[30] Marchenkov S. S., “Rasshireniya operatora pozitivnogo zamykaniya s pomoschyu logicheskikh svyazok”, Diskret. analiz i issled. operatsii, 25:4 (2018), 46–58 | Zbl

[31] Marchenkov S. S., Silnye operatory zamykaniya, Fizmatlit, M., 2017, 94 pp.

[32] Marchenkov S. S., Chernyshev A. V., “O podschete chisla funktsii s zadannym endomorfizmom”, Vestnik MGU. Seriya 15: Vychisl. matem. i kibern., 4 (2018), 26–32

[33] Nguen Van Khoa, “O strukture samodvoistvennykh zamknutykh klassov trekhznachnoi logiki $P_3$”, Diskretnaya matematika, 4:4 (1992), 82–95 | MR

[34] Nguen Van Khoa, “O semeistvakh zamknutykh klassov $k$-znachnoi logiki, sokhranyaemykh vsemi avtomorfizmami”, Diskretnaya matematika, 5:4 (1993), 87–108 | MR

[35] Podolko D. K., “Ob odnom semeistve klassov funktsii, zamknutykh otnositelno usilennoi operatsii superpozitsii”, Vestnik MGU. Seriya 1: Matematika, mekhanika, 2 (2015), 27–31 | MR

[36] Podolko D. K., “O klassakh funktsii mnogoznachnoi logiki, zamknutykh otnositelno usilennoi operatsii superpozitsii”, Matematicheskie voprosy kibernetiki, 19, Fizmatlit, M., 2019, 199–278

[37] Solovev V. D., “Zamknutye klassy v $k$-znachnoi logike s operatsiei razvetvleniya po predikatam”, Diskretnaya matematika, 2:4 (1990), 82–95

[38] Taimanov V. A., “O funktsionalnykh sistemakh $k$-znachnoi logiki s operatsiyami programmnogo tipa”, DAN SSSR, 268:6 (1983), 1307–1310 | MR | Zbl

[39] Tarasova O. S., “Klassy $k$-znachnoi logiki, zamknutye otnositelno rasshirennoi operatsii superpozitsii”, Vestnik MGU. Seriya 1: Matematika, mekhanika, 6 (2001), 54–57 | Zbl

[40] Yanov Yu. I., Muchnik A. A., “O suschestvovanii $k$-znachnykh zamknutykh klassov, ne imeyuschikh bazisa”, DAN SSSR, 127:1 (1959), 44–46 | Zbl

[41] Barris S., Willard R., “Finitely many primitive positive clones”, Proc. Amer. Math. Soc., 101:3 (1987), 427–430 | DOI | MR

[42] Danil'čenko A. F., “On parametrical expressibility of the functions of $k$-valued logic”, Colloq. Math. Soc. J. Bolyai, 28 (1981), 147–159 | MR | Zbl

[43] Hermann M., “On Boolean primitive positive clones”, Discrete Mathematics, 308 (2008), 3151–3162 | DOI | MR | Zbl

[44] Snow J. W., “Generating primitive positive clones”, Algebra universalis, 44 (2000), 169–185 | DOI | MR | Zbl

[45] Szabó L., “Concrete representation of structures of universal algebras I”, Acta Sci. Math. (Szeged), 40 (1978), 175–184 | MR | Zbl

[46] Szabó L., “On the lattice of clones acting bicentrally”, Acta Cybernet., 6 (1984), 381–388 | MR