The p-semisimple property for some generalizations of BCI algebras and its applications

Obojska, Lidia; Walendziak, Andrzej

doi:10.2478/aupcsm-2020-0007

Abstract

This paper presents some generalizations of BCI algebras (the RM, tRM, *RM, RM**, *RM**, aRM**, *aRM**, BCH**, BZ, pre-BZ and pre-BCI algebras). We investigate the p-semisimple property for algebras mentioned above; give some examples and display various conditions equivalent to p-semisimplicity. Finally, we present a model of mereology without antisymmetry (NAM) which could represent a tRM algebra.

References

[1] Aslam, Muhammad, and Allah-Bakhsh Thaheem. “A note on p-semisimple BCI-algebras.” Math. Japon. 36, no. 1 (1991): 39-45. Cited on 79.
Search in Google Scholar Back to article
[2] Clay, Robert Edward. “Relation of Leśniewski’s mereology to Boolean algebra.” J. Symbolic Logic 39 (1974): 638-648. Cited on 91.
Search in Google Scholar Back to article
[3] Hoo, Cheong Seng. “Closed ideals and p-semisimple BCI-algebras.” Math. Japon. 35, no. 6 (1990): 1103-1112. Cited on 79.
Search in Google Scholar Back to article
[4] Hu, Qing Ping, and Xin Li. “On BCH-algebras.” Math. Sem. Notes Kobe Univ. 11, no. 2 (1983): 313-320. Cited on 79.
Search in Google Scholar Back to article
[5] Imai, Yasuyuki, and Kiyoshi Iséki. “On axiom systems of propositional calculi. XIV.” Proc. Japan Acad. 42 (1966): 19-22. Cited on 79.
Search in Google Scholar Back to article
[6] Iorgulescu, Afrodita. “New generalizations of BCI, BCK and Hilbert algebras – Part I.” J. Mult.-Valued Logic Soft Comput. 27, no. 4 (2016): 353-406. Cited on 79 and 80.
Search in Google Scholar Back to article
[7] Iorgulescu, Afrodita. “New generalizations of BCI, BCK and Hilbert algebras – Part II.” J. Mult.-Valued Logic Soft Comput. 27, no. 4 (2016): 407-456. Cited on 82 and 85.
Search in Google Scholar Back to article
[8] Iséki, Kiyoshi. “An algebra related with a propositional calculus.” Proc. Japan Acad. 42 (1966): 26-29. Cited on 79.
Search in Google Scholar Back to article
[9] Kim, Hee Sik, and Hong Goo Park. “On 0-commutative B-algebras.” Sci. Math. Jpn. 62, no. 1 (2005): 7-12. Cited on 79.
Search in Google Scholar Back to article
[10] Lei, Tian De, and Chang Chang Xi. “p-radical in BCI-algebras.” Math. Japon. 30, no. 4 (1985): 511-517. Cited on 79.
Search in Google Scholar Back to article
[11] Leśniewski, Stanisław. Stanislaw Lesniewski: Collected Works - Volumes I and II. Vol. 44 of Nijhoff International Philosophy Series. Dordrecht: Kluwer Academic Publishers, 1992. Cited on 89.
Search in Google Scholar Back to article
[12] Loeb, Iris. “From Mereology to Boolean Algebra: The Role of Regular Open Sets in Alfred Tarski’s Work.” In: The History and Philosophy of Polish Logic: Essays in Honour of Jan Woleński, 259-277. New York: Palgrave Macmillan, 2014. Cited on 91.
Search in Google Scholar Back to article
[13] Meng, Dao Ji. “BCI-algebras and abelian groups.” Math. Japon. 32, no. 5 (1987): 693-696. Cited on 79.
Search in Google Scholar Back to article
[14] Obojska, Lidia. “Some remarks on supplementation principles in the absence of antisymmetry.” Rev. Symb. Log. 6, no. 2 (2013): 343-347. Cited on 91.
Search in Google Scholar Back to article
[15] Obojska, Lidia. U źródeł zbiorów kolektywnych: o mereologii nieantysymetrycznej. Siedlce: Wydawnictwo UPH, 2013. Cited on 90 and 91.
Search in Google Scholar Back to article
[16] Patrignani, Claudia et al. (Particle Data Group). “Review of Particle Physics.” Chin. Phys. C 40, no. 10 (2016): 100001. Cited on 90.
Search in Google Scholar Back to article
[17] Pietruszczak, Andrzej. Metamereologia. Toruń: Wydawnictwo Naukowe Uniwersytetu Mikołaja Kopernika, 2000. Cited on 91.
Search in Google Scholar Back to article
[18] Shohani, Javad, Rajab Ali Borzooei, and Morteza Afshar Jahanashah. “Basic BCI-algebras and abelian groups are equivalent.” Sci. Math. Jpn. 66, no. 2 (2007): 243-245. Cited on 79.
Search in Google Scholar Back to article
[19] Walendziak, Andrzej. “On BF -algebras.” Math. Slovaca 57, no. 2 (2007): 119-128. Cited on 87.
Search in Google Scholar Back to article
[20] Ye, Reifen. “On BZ-algebras.” In: Selected paper on BCI/BCK-algebras and Computer Logics, 21-24. Shaghai: Shaghai Jiaotong University Press, 1991. Cited on 79.
Search in Google Scholar Back to article
[21] Zhang, Qun. “Some other characterizations of p-semisimple BCI-algebras.” Math. Japon. 36, no. 5 (1991): 815-817. Cited on 79.
Search in Google Scholar Back to article

The p-semisimple property for some generalizations of BCI algebras and its applications

Abstract

Paradigm

My account