| Title:
|
Congruences and homomorphisms on $\Omega $-algebras (English) |
| Author:
|
Eghosa Edeghagba, Elijah |
| Author:
|
Šešelja, Branimir |
| Author:
|
Tepavčević, Andreja |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
53 |
| Issue:
|
5 |
| Year:
|
2017 |
| Pages:
|
892-910 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The topic of the paper are $\Omega$-algebras, where $\Omega$ is a complete lattice. In this research we deal with congruences and homomorphisms. An $\Omega$-algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an $\Omega$-valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce $\Omega$-valued congruences, corresponding quotient $\Omega$-algebras and $\Omega$-homomorphisms and we investigate connections among these notions. We prove that there is an $\Omega$-homomorphism from an $\Omega$-algebra to the corresponding quotient $\Omega$-algebra. The kernel of an $\Omega$-homomorphism is an $\Omega$-valued congruence. When dealing with cut structures, we prove that an $\Omega$-homomorphism determines classical homomorphisms among the corresponding quotient structures over cut subalgebras. In addition, an $\Omega$-congruence determines a closure system of classical congruences on cut subalgebras. Finally, identities are preserved under $\Omega$-homomorphisms. (English) |
| Keyword:
|
lattice-valued algebra |
| Keyword:
|
congruence |
| Keyword:
|
homomorphism |
| MSC:
|
06D72 |
| MSC:
|
08A72 |
| idZBL:
|
Zbl 06861631 |
| idMR:
|
MR3750110 |
| DOI:
|
10.14736/kyb-2017-5-0892 |
| . |
| Date available:
|
2018-02-26T11:51:02Z |
| Last updated:
|
2018-05-25 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147100 |
| . |
| Reference:
|
[1] Ajmal, N., Thomas, K. V.: Fuzzy lattices..Inform. Sci. 79 (1994), 271-291. MR 1282402, 10.1016/0020-0255(94)90124-4 |
| Reference:
|
[2] Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles..Kluwer Academic/Plenum Publishers, New York 2002. 10.1007/978-1-4615-0633-1 |
| Reference:
|
[3] Bělohlávek, R., Vychodil, V.: Algebras with fuzzy equalities..Fuzzy Sets and Systems 157 (2006), 161-201. MR 2186221, 10.1016/j.fss.2005.05.044 |
| Reference:
|
[4] Bělohlávek, R., Vychodil, V.: Fuzzy Equational Logic..Studies in Fuzziness and Soft Computing, Springer 186 (2005), pp. 139-170. 10.1007/11376422_3 |
| Reference:
|
[5] Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: Fuzzy identities with application to fuzzy semigroups..Inform. Sci. 266 (2014), 148-159. MR 3165413, 10.1016/j.ins.2013.11.007 |
| Reference:
|
[6] Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: Fuzzy equational classes are fuzzy varieties..Iranian J. Fuzzy Systems 10 (2013), 1-18. MR 3135796 |
| Reference:
|
[7] Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: Fuzzy equational classes..In: Fuzzy Systems (FUZZ-IEEE), 2012 IEEE International Conference, pp. 1-6. MR 3135796, 10.1109/fuzz-ieee.2012.6251259 |
| Reference:
|
[8] Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: $E$-fuzzy groups..Fuzzy Sets and Systems 289 (2016), 94-112. MR 3454464, 10.1016/j.fss.2015.03.011 |
| Reference:
|
[9] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra..Grauate Texts in Mathematics, 1981. Zbl 0478.08001, MR 0648287, 10.1007/978-1-4613-8130-3 |
| Reference:
|
[10] Czédli, G., Erné, M., Šešelja, B., Tepavčević, A.: Characteristic triangles of closure operators with applications in general algebra..Algebra Univers. 62 (2009), 399-418. MR 2670173, 10.1007/s00012-010-0059-2 |
| Reference:
|
[11] Demirci, M.: Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations. Part I: Fuzzy functions and their applications. Part II: Vague algebraic notions. Part III: Constructions of vague algebraic notions and vague arithmetic operations..Int. J. General Systems 32 (2003), 3, 123-155, 157-175, 177-201. MR 1967128, 10.1080/0308107031000090765 |
| Reference:
|
[12] Demirci, M.: A theory of vague lattices based on many-valued equivalence relations I: general representation results..Fuzzy Sets and Systems 151 (2005), 437-472. MR 2126168, 10.1016/j.fss.2004.06.017 |
| Reference:
|
[13] Demirci, M.: A theory of vague lattices based on many-valued equivalence relations II: Complete lattices..Fuzzy Sets and Systems 151 (2005), 473-489. MR 2126169, 10.1016/j.fss.2004.06.004 |
| Reference:
|
[14] Nola, A. Di, Gerla, G.: Lattice valued algebras..Stochastica 11 (1987), 137-150. MR 0990882 |
| Reference:
|
[15] Edeghagba, E. E., Šešelja, B., Tepavčević, A.: Omega-Lattices..Fuzzy Sets and Systems 311 (2017), 53-69. MR 3597106, 10.1016/j.fss.2016.10.011 |
| Reference:
|
[16] Fourman, M. P., Scott, D. S.: Sheaves and logic..In: Applications of Sheaves (M. P. Fourman, C. J. Mulvey and D. S. Scott, eds.), Lecture Notes in Mathematics, 753, Springer, Berlin, Heidelberg, New York 1979, pp. 302-401. MR 0555551, 10.1007/bfb0061824 |
| Reference:
|
[17] Goguen, J. A.: $L$-fuzzy sets..J. Math. Anal. Appl. 18 (1967), 145-174. Zbl 0145.24404, MR 0224391, 10.1016/0022-247x(67)90189-8 |
| Reference:
|
[18] Gottwald, S.: Universes of fuzzy sets and axiomatizations of fuzzy set theory. Part II: Category theoretic approaches..Studia Logica 84 (2006) 1, 23-50, 1143-1174. MR 2271287, 10.1007/s11225-006-9001-1 |
| Reference:
|
[19] Höhle, U.: Quotients with respect to similarity relations..Fuzzy Sets and Systems 27 (1988), 31-44. MR 0950448, 10.1016/0165-0114(88)90080-2 |
| Reference:
|
[20] Höhle, U.: Fuzzy sets and sheaves. Part I: Basic concepts..Fuzzy Sets and Systems 158 (2007), 11, 1143-1174. MR 2314674, 10.1016/j.fss.2006.12.009 |
| Reference:
|
[21] Höhle, U., Šostak, A. P.: Axiomatic foundations of fixed-basis fuzzy topology..Springer US, 1999, pp. 123-272. MR 1788903, 10.1007/978-1-4615-5079-2_5 |
| Reference:
|
[22] Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic..Prentice Hall, New Jersey 1995. MR 1329731 |
| Reference:
|
[23] Šešelja, B., Tepavčević, A.: On Generalizations of fuzzy algebras and congruences..Fuzzy Sets and Systems 65 (1994), 85-94. MR 1294042, 10.1016/0165-0114(94)90249-6 |
| Reference:
|
[24] Šešelja, B., Tepavčević, A.: Fuzzy identities..In: Proc. 2009 IEEE International Conference on Fuzzy Systems, pp. 1660-1664. 10.1109/fuzzy.2009.5277317 |
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