Congruences and homomorphisms on Ω -algebras

Elijah Eghosa Edeghagba; Branimir Šešelja; Andreja Tepavčević

Kybernetika (2017)

  • Volume: 53, Issue: 5, page 892-910
  • ISSN: 0023-5954

Abstract

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The topic of the paper are Ω -algebras, where Ω is a complete lattice. In this research we deal with congruences and homomorphisms. An Ω -algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an Ω -valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce Ω -valued congruences, corresponding quotient Ω -algebras and Ω -homomorphisms and we investigate connections among these notions. We prove that there is an Ω -homomorphism from an Ω -algebra to the corresponding quotient Ω -algebra. The kernel of an Ω -homomorphism is an Ω -valued congruence. When dealing with cut structures, we prove that an Ω -homomorphism determines classical homomorphisms among the corresponding quotient structures over cut subalgebras. In addition, an Ω -congruence determines a closure system of classical congruences on cut subalgebras. Finally, identities are preserved under Ω -homomorphisms.

How to cite

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Eghosa Edeghagba, Elijah, Šešelja, Branimir, and Tepavčević, Andreja. "Congruences and homomorphisms on $\Omega $-algebras." Kybernetika 53.5 (2017): 892-910. <http://eudml.org/doc/294602>.

@article{EghosaEdeghagba2017,
abstract = {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.},
author = {Eghosa Edeghagba, Elijah, Šešelja, Branimir, Tepavčević, Andreja},
journal = {Kybernetika},
keywords = {lattice-valued algebra; congruence; homomorphism},
language = {eng},
number = {5},
pages = {892-910},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Congruences and homomorphisms on $\Omega $-algebras},
url = {http://eudml.org/doc/294602},
volume = {53},
year = {2017},
}

TY - JOUR
AU - Eghosa Edeghagba, Elijah
AU - Šešelja, Branimir
AU - Tepavčević, Andreja
TI - Congruences and homomorphisms on $\Omega $-algebras
JO - Kybernetika
PY - 2017
PB - Institute of Information Theory and Automation AS CR
VL - 53
IS - 5
SP - 892
EP - 910
AB - 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.
LA - eng
KW - lattice-valued algebra; congruence; homomorphism
UR - http://eudml.org/doc/294602
ER -

References

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  1. Ajmal, N., Thomas, K. V., 10.1016/0020-0255(94)90124-4, Inform. Sci. 79 (1994), 271-291. MR1282402DOI10.1016/0020-0255(94)90124-4
  2. Bělohlávek, R., 10.1007/978-1-4615-0633-1, Kluwer Academic/Plenum Publishers, New York 2002. DOI10.1007/978-1-4615-0633-1
  3. Bělohlávek, R., Vychodil, V., 10.1016/j.fss.2005.05.044, Fuzzy Sets and Systems 157 (2006), 161-201. MR2186221DOI10.1016/j.fss.2005.05.044
  4. Bělohlávek, R., Vychodil, V., 10.1007/11376422_3, Studies in Fuzziness and Soft Computing, Springer 186 (2005), pp. 139-170. DOI10.1007/11376422_3
  5. Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A., 10.1016/j.ins.2013.11.007, Inform. Sci. 266 (2014), 148-159. MR3165413DOI10.1016/j.ins.2013.11.007
  6. Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A., Fuzzy equational classes are fuzzy varieties., Iranian J. Fuzzy Systems 10 (2013), 1-18. MR3135796
  7. Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A., 10.1109/fuzz-ieee.2012.6251259, In: Fuzzy Systems (FUZZ-IEEE), 2012 IEEE International Conference, pp. 1-6. MR3135796DOI10.1109/fuzz-ieee.2012.6251259
  8. Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A., 10.1016/j.fss.2015.03.011, Fuzzy Sets and Systems 289 (2016), 94-112. MR3454464DOI10.1016/j.fss.2015.03.011
  9. Burris, S., Sankappanavar, H. P., 10.1007/978-1-4613-8130-3, Grauate Texts in Mathematics, 1981. Zbl0478.08001MR0648287DOI10.1007/978-1-4613-8130-3
  10. Czédli, G., Erné, M., Šešelja, B., Tepavčević, A., 10.1007/s00012-010-0059-2, Algebra Univers. 62 (2009), 399-418. MR2670173DOI10.1007/s00012-010-0059-2
  11. Demirci, M., 10.1080/0308107031000090765, Int. J. General Systems 32 (2003), 3, 123-155, 157-175, 177-201. MR1967128DOI10.1080/0308107031000090765
  12. Demirci, M., 10.1016/j.fss.2004.06.017, Fuzzy Sets and Systems 151 (2005), 437-472. MR2126168DOI10.1016/j.fss.2004.06.017
  13. Demirci, M., 10.1016/j.fss.2004.06.004, Fuzzy Sets and Systems 151 (2005), 473-489. MR2126169DOI10.1016/j.fss.2004.06.004
  14. Nola, A. Di, Gerla, G., Lattice valued algebras., Stochastica 11 (1987), 137-150. MR0990882
  15. Edeghagba, E. E., Šešelja, B., Tepavčević, A., 10.1016/j.fss.2016.10.011, Fuzzy Sets and Systems 311 (2017), 53-69. MR3597106DOI10.1016/j.fss.2016.10.011
  16. Fourman, M. P., Scott, D. S., 10.1007/bfb0061824, 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. MR0555551DOI10.1007/bfb0061824
  17. Goguen, J. A., 10.1016/0022-247x(67)90189-8, J. Math. Anal. Appl. 18 (1967), 145-174. Zbl0145.24404MR0224391DOI10.1016/0022-247x(67)90189-8
  18. Gottwald, S., 10.1007/s11225-006-9001-1, Studia Logica 84 (2006) 1, 23-50, 1143-1174. MR2271287DOI10.1007/s11225-006-9001-1
  19. Höhle, U., 10.1016/0165-0114(88)90080-2, Fuzzy Sets and Systems 27 (1988), 31-44. MR0950448DOI10.1016/0165-0114(88)90080-2
  20. Höhle, U., 10.1016/j.fss.2006.12.009, Fuzzy Sets and Systems 158 (2007), 11, 1143-1174. MR2314674DOI10.1016/j.fss.2006.12.009
  21. Höhle, U., Šostak, A. P., 10.1007/978-1-4615-5079-2_5, Springer US, 1999, pp. 123-272. MR1788903DOI10.1007/978-1-4615-5079-2_5
  22. Klir, G., Yuan, B., Fuzzy Sets and Fuzzy Logic., Prentice Hall, New Jersey 1995. MR1329731
  23. Šešelja, B., Tepavčević, A., 10.1016/0165-0114(94)90249-6, Fuzzy Sets and Systems 65 (1994), 85-94. MR1294042DOI10.1016/0165-0114(94)90249-6
  24. Šešelja, B., Tepavčević, A., 10.1109/fuzzy.2009.5277317, In: Proc. 2009 IEEE International Conference on Fuzzy Systems, pp. 1660-1664. DOI10.1109/fuzzy.2009.5277317

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