Mathematica Bohemica

Mathematica Bohemica, first online, pp. 1-13

Strong endomorphism kernel property for finite Brouwerian semilattices and relative Stone algebras

Jaroslav Guričan, Heghine Ghumashyan

Received March 31, 2022. Published online January 4, 2023.

Abstract: We show that all finite Brouwerian semilattices have strong endomorphism kernel property (SEKP), give a new proof that all finite relative Stone algebras have SEKP and also fully characterize dual generalized boolean algebras which possess SEKP.

Keywords: (strong) endomorphism kernel property; congruence relation; Brouwerian semilattice; Brouwerian algebra; dual generalized Boolean algebra; direct sum; factorable congruences

Classification MSC: 03G25, 06E99, 08A30, 08A35, 08B26

DOI: 10.21136/MB.2023.0050-22

PDF available at: Institute of Mathematics CAS

References:
[1] T. S. Blyth, J. Fang, H. J. Silva: The endomorphism kernel property in finite distributive lattices and de Morgan algebras. Commun. Algebra 32 (2004), 2225-2242. DOI 10.1081/agb-120037216 | MR 2100466 | Zbl 1060.06018
[2] T. S. Blyth, J. Fang, L.-B. Wang: The strong endomorphism kernel property in distributive double $p$-algebras. Sci. Math. Jpn. 76 (2013), 227-234. DOI 10.32219/isms.76.2_227 | MR 3330070 | Zbl 1320.06009
[3] T. S. Blyth, H. J. Silva: The strong endomorphism kernel property in Ockham algebras. Commun. Algebra 36 (2008), 1682-1694. DOI 10.1080/00927870801937240 | MR 2424259 | Zbl 1148.06005
[4] B. A. Davey: Dualities for equational classes of Brouwerian algebras and Heyting algebras. Trans. Am. Math. Soc. 221 (1976), 119-146. DOI 10.1090/S0002-9947-1976-0412063-9 | MR 0412063 | Zbl 0319.06007
[5] G. Fang, J. Fang: The strong endomorphism kernel property in distributive $p$-algebras. Southeast Asian Bull. Math. 37 (2013), 491-497. MR 3134913 | Zbl 1299.06017
[6] J. Fang: The strong endomorphism kernel property in double MS-algebras. Stud. Log. 105 (2017), 995-1013. DOI 10.1007/s11225-017-9722-3 | MR 3704306 | Zbl 1421.06003
[7] J. Fang, Z.-J. Sun: Semilattices with the strong endomorphism kernel property. Algebra Univers. 70 (2013), 393-401. DOI 10.1007/s00012-013-0254-z | MR 3127981 | Zbl 1305.06004
[8] J. Fang, Z. J. Sun: Finite abelian groups with the strong endomorphism kernel property. Acta Math. Sin., Engl. Ser. 36 (2020), 1076-1082. DOI 10.1007/s10114-020-9444-8 | MR 4145699 | Zbl 1484.20093
[9] H. Ghumashyan, J. Guričan: Endomorphism kernel property for finite groups. Math. Bohem. 147 347-358 (2022). DOI 10.21136/MB.2021.0171-20 | MR 4482310 | Zbl 7584129
[10] G. Grätzer: Lattice Theory: Foundation. Birkhäuser, Basel (2011). DOI 10.1007/978-3-0348-0018-1 | MR 2768581 | Zbl 1233.06001
[11] J. Guričan: A note on the endomorphism kernel property. JP J. Algebra Number Theory Appl. 33 (2014), 133-139. Zbl 1302.08004
[12] J. Guričan: Strong endomorphism kernel property for Brouwerian algebras. JP J. Algebra Number Theory Appl. 36 (2015), 241-258. DOI 10.17654/JPANTAJun2015_241_258 | Zbl 1333.06025
[13] J. Guričan, M. Ploščica: The strong endomorphism kernel property for modular $p$-algebras and distributive lattices. Algebra Univers. 75 (2016), 243-255. DOI 10.1007/s00012-016-0370-7 | MR 3515400 | Zbl 1348.06008
[14] E. Halušková: Strong endomorphism kernel property for monounary algebras. Math. Bohem. 143 (2018), 161-171. DOI 10.21136/mb.2017.0056-16 | MR 3831484 | Zbl 1463.08003
[15] E. Halušková: Some monounary algebras with EKP. Math. Bohem. 145 (2020), 401-414. DOI 10.21136/MB.2019.0128-18 | MR 4221842 | Zbl 07286021
[16] J. Hashimoto: Ideal theory for lattices. Math. Jap. 2 (1952), 149-186. MR 0057224 | Zbl 0048.25903
[17] T. Hecht, T. Katriňák: Equational classes of relative Stone algebras. Notre Dame J. Formal Logic 13 (1972), 248-254. DOI 10.1305/ndjfl/1093894723 | MR 0295978 | Zbl 0212.01601
[18] T. Katriňák: Die Kennzeichnung der distributiven pseudokomplementären Halbverbände. J. Reine Angew. Math. 241 (1970), 160-179. (In German.) DOI 10.1515/crll.1970.241.160 | MR 0260629 | Zbl 0192.33503
[19] T. Katriňák: Remarks on the W. C. Nemitz's paper 'Semi-Boolean lattices'. Notre Dame J. Formal Logic 11 (1970), 425-430. DOI 10.1305/ndjfl/1093894072 | MR 0290946 | Zbl 0185.03803
[20] T. Katriňák: Relativ Stonesche Halbverbände sind Verbände. Bull. Soc. R. Sci. Liège 40 (1971), 91-93. (In German.) MR 0288059 | Zbl 0221.06002
[21] T. Katriňák: Die Kennzeichnung der beschränkten Brouwerschen Verbände. Czech. Math. J. 22 (1972), 427-434. (In German.) DOI 10.21136/CMJ.1972.101112 | MR 0309814 | Zbl 0222.06006
[22] P. Köhler: Brouwerian semilattices. Trans. Am. Math. Soc. 268 (1981), 103-126. DOI 10.1090/S0002-9947-1981-0628448-3 | MR 0628448 | Zbl 0473.06003
[23] W. C. Nemitz: Implicative semi-lattices. Trans. Am. Math. Soc. 117 (1965), 128-142. DOI 10.1090/S0002-9947-1965-0176944-9 | MR 0176944 | Zbl 0128.24804
[24] M. Ploščica: Affine completions of distributive lattices. Order 13 (1996), 295-311. DOI 10.1007/BF00338748 | MR 1420402 | Zbl 0907.06013

Affiliations: Jaroslav Guričan (corresponding author), Comenius University Bratislava, Šafárikovo námestie 6, 814 99 Bratislava, Slovakia, e-mail: gurican@fmph.uniba.sk; Heghine Ghumashyan, European University, 10 Davit Anhaght Street, Yerevan 0037, Armenia, e-mail: hgumashyan@mail.ru

[List of online first articles] [Contents of Mathematica Bohemica] [Full text of the older issues of Mathematica Bohemica at DML-CZ]

PDF available at:

Institute of Mathematics CAS
free access