| Title: | Strong endomorphism kernel property for finite Brouwerian semilattices and relative Stone algebras (English) | 
| Author: | Guričan, Jaroslav | 
| Author: | Ghumashyan, Heghine | 
| Language: | English | 
| Journal: | Mathematica Bohemica | 
| ISSN: | 0862-7959 (print) | 
| ISSN: | 2464-7136 (online) | 
| Volume: | 149 | 
| Issue: | 1 | 
| Year: | 2024 | 
| Pages: | 13-25 | 
| Summary lang: | English | 
| . | 
| Category: | math | 
| . | 
| Summary: | 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. (English) | 
| Keyword: | (strong) endomorphism kernel property | 
| Keyword: | congruence relation | 
| Keyword: | Brouwerian semilattice | 
| Keyword: | Brouwerian algebra | 
| Keyword: | dual generalized Boolean algebra | 
| Keyword: | direct sum | 
| Keyword: | factorable congruences | 
| MSC: | 03G25 | 
| MSC: | 06E99 | 
| MSC: | 08A30 | 
| MSC: | 08A35 | 
| MSC: | 08B26 | 
| idZBL: | Zbl 07830540 | 
| idMR: | MR4715553 | 
| DOI: | 10.21136/MB.2023.0050-22 | 
| . | 
| Date available: | 2024-03-13T10:16:05Z | 
| Last updated: | 2024-12-13 | 
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152289 | 
| . | 
| Reference: | [1] Blyth, T. S., Fang, J., Silva, H. J.: The endomorphism kernel property in finite distributive lattices and de Morgan algebras.Commun. Algebra 32 (2004), 2225-2242. Zbl 1060.06018, MR 2100466, 10.1081/agb-120037216 | 
| Reference: | [2] Blyth, T. S., Fang, J., Wang, L.-B.: The strong endomorphism kernel property in distributive double $p$-algebras.Sci. Math. Jpn. 76 (2013), 227-234. Zbl 1320.06009, MR 3330070, 10.32219/isms.76.2_227 | 
| Reference: | [3] Blyth, T. S., Silva, H. J.: The strong endomorphism kernel property in Ockham algebras.Commun. Algebra 36 (2008), 1682-1694. Zbl 1148.06005, MR 2424259, 10.1080/00927870801937240 | 
| Reference: | [4] Davey, B. A.: Dualities for equational classes of Brouwerian algebras and Heyting algebras.Trans. Am. Math. Soc. 221 (1976), 119-146. Zbl 0319.06007, MR 0412063, 10.1090/S0002-9947-1976-0412063-9 | 
| Reference: | [5] Fang, G., Fang, J.: The strong endomorphism kernel property in distributive $p$-algebras.Southeast Asian Bull. Math. 37 (2013), 491-497. Zbl 1299.06017, MR 3134913 | 
| Reference: | [6] Fang, J.: The strong endomorphism kernel property in double MS-algebras.Stud. Log. 105 (2017), 995-1013. Zbl 1421.06003, MR 3704306, 10.1007/s11225-017-9722-3 | 
| Reference: | [7] Fang, J., Sun, Z.-J.: Semilattices with the strong endomorphism kernel property.Algebra Univers. 70 (2013), 393-401. Zbl 1305.06004, MR 3127981, 10.1007/s00012-013-0254-z | 
| Reference: | [8] Fang, J., Sun, Z. J.: Finite abelian groups with the strong endomorphism kernel property.Acta Math. Sin., Engl. Ser. 36 (2020), 1076-1082. Zbl 1484.20093, MR 4145699, 10.1007/s10114-020-9444-8 | 
| Reference: | [9] Ghumashyan, H., Guričan, J.: Endomorphism kernel property for finite groups.Math. Bohem. 147 347-358 (2022). Zbl 7584129, MR 4482310, 10.21136/MB.2021.0171-20 | 
| Reference: | [10] Grätzer, G.: Lattice Theory: Foundation.Birkhäuser, Basel (2011). Zbl 1233.06001, MR 2768581, 10.1007/978-3-0348-0018-1 | 
| Reference: | [11] Guričan, J.: A note on the endomorphism kernel property.JP J. Algebra Number Theory Appl. 33 (2014), 133-139. Zbl 1302.08004 | 
| Reference: | [12] Guričan, J.: Strong endomorphism kernel property for Brouwerian algebras.JP J. Algebra Number Theory Appl. 36 (2015), 241-258. Zbl 1333.06025, 10.17654/JPANTAJun2015_241_258 | 
| Reference: | [13] Guričan, J., Ploščica, M.: The strong endomorphism kernel property for modular $p$-algebras and distributive lattices.Algebra Univers. 75 (2016), 243-255. Zbl 1348.06008, MR 3515400, 10.1007/s00012-016-0370-7 | 
| Reference: | [14] Halušková, E.: Strong endomorphism kernel property for monounary algebras.Math. Bohem. 143 (2018), 161-171. Zbl 1463.08003, MR 3831484, 10.21136/mb.2017.0056-16 | 
| Reference: | [15] Halušková, E.: Some monounary algebras with EKP.Math. Bohem. 145 (2020), 401-414. Zbl 07286021, MR 4221842, 10.21136/MB.2019.0128-18 | 
| Reference: | [16] Hashimoto, J.: Ideal theory for lattices.Math. Jap. 2 (1952), 149-186. Zbl 0048.25903, MR 0057224 | 
| Reference: | [17] Hecht, T., Katriňák, T.: Equational classes of relative Stone algebras.Notre Dame J. Formal Logic 13 (1972), 248-254. Zbl 0212.01601, MR 0295978, 10.1305/ndjfl/1093894723 | 
| Reference: | [18] Katriňák, T.: Die Kennzeichnung der distributiven pseudokomplementären Halbverbände.J. Reine Angew. Math. 241 (1970), 160-179 German. Zbl 0192.33503, MR 0260629, 10.1515/crll.1970.241.160 | 
| Reference: | [19] Katriňák, T.: Remarks on the W. C. Nemitz's paper 'Semi-Boolean lattices'.Notre Dame J. Formal Logic 11 (1970), 425-430. Zbl 0185.03803, MR 0290946, 10.1305/ndjfl/1093894072 | 
| Reference: | [20] Katriňák, T.: Relativ Stonesche Halbverbände sind Verbände.Bull. Soc. R. Sci. Liège 40 (1971), 91-93 German. Zbl 0221.06002, MR 0288059 | 
| Reference: | [21] Katriňák, T.: Die Kennzeichnung der beschränkten Brouwerschen Verbände.Czech. Math. J. 22 (1972), 427-434 German. Zbl 0222.06006, MR 0309814, 10.21136/CMJ.1972.101112 | 
| Reference: | [22] Köhler, P.: Brouwerian semilattices.Trans. Am. Math. Soc. 268 (1981), 103-126. Zbl 0473.06003, MR 0628448, 10.1090/S0002-9947-1981-0628448-3 | 
| Reference: | [23] Nemitz, W. C.: Implicative semi-lattices.Trans. Am. Math. Soc. 117 (1965), 128-142. Zbl 0128.24804, MR 0176944, 10.1090/S0002-9947-1965-0176944-9 | 
| Reference: | [24] Ploščica, M.: Affine completions of distributive lattices.Order 13 (1996), 295-311. Zbl 0907.06013, MR 1420402, 10.1007/BF00338748 | 
| . |