Geodesic
Congruences and homomorphisms on $\Omega $-algebras
Kybernetika, Tome 53 (2017) no. 5, pp. 892-910
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.14736/kyb-2017-5-0892
Classification : 06D72, 08A72
Keywords: lattice-valued algebra; congruence; homomorphism 
@article{10_14736_kyb_2017_5_0892,
     author = {Eghosa Edeghagba, Elijah and \v{S}e\v{s}elja, Branimir and Tepav\v{c}evi\'c, Andreja},
     title = {Congruences and homomorphisms on $\Omega $-algebras},
     journal = {Kybernetika},
     pages = {892--910},
     year = {2017},
     volume = {53},
     number = {5},
     doi = {10.14736/kyb-2017-5-0892},
     mrnumber = {3750110},
     zbl = {06861631},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2017-5-0892/}
}
TY  - JOUR
AU  - Eghosa Edeghagba, Elijah
AU  - Šešelja, Branimir
AU  - Tepavčević, Andreja
TI  - Congruences and homomorphisms on $\Omega $-algebras
JO  - Kybernetika
PY  - 2017
SP  - 892
EP  - 910
VL  - 53
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.14736/kyb-2017-5-0892/
DO  - 10.14736/kyb-2017-5-0892
LA  - en
ID  - 10_14736_kyb_2017_5_0892
ER  - 
%0 Journal Article
%A Eghosa Edeghagba, Elijah
%A Šešelja, Branimir
%A Tepavčević, Andreja
%T Congruences and homomorphisms on $\Omega $-algebras
%J Kybernetika
%D 2017
%P 892-910
%V 53
%N 5
%U http://geodesic.mathdoc.fr/articles/10.14736/kyb-2017-5-0892/
%R 10.14736/kyb-2017-5-0892
%G en
%F 10_14736_kyb_2017_5_0892
Eghosa Edeghagba, Elijah; Šešelja, Branimir; Tepavčević, Andreja. Congruences and homomorphisms on $\Omega $-algebras. Kybernetika, Tome 53 (2017) no. 5, pp. 892-910. doi: 10.14736/kyb-2017-5-0892

[1] Ajmal, N., Thomas, K. V.: Fuzzy lattices. Inform. Sci. 79 (1994), 271-291. | DOI | MR

[2] Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic/Plenum Publishers, New York 2002. | DOI

[3] Bělohlávek, R., Vychodil, V.: Algebras with fuzzy equalities. Fuzzy Sets and Systems 157 (2006), 161-201. | DOI | MR

[4] Bělohlávek, R., Vychodil, V.: Fuzzy Equational Logic. Studies in Fuzziness and Soft Computing, Springer 186 (2005), pp. 139-170. | DOI

[5] Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: Fuzzy identities with application to fuzzy semigroups. Inform. Sci. 266 (2014), 148-159. | DOI | MR

[6] Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: Fuzzy equational classes are fuzzy varieties. Iranian J. Fuzzy Systems 10 (2013), 1-18. | MR

[7] Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: Fuzzy equational classes. In: Fuzzy Systems (FUZZ-IEEE), 2012 IEEE International Conference, pp. 1-6. | DOI | MR

[8] Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: $E$-fuzzy groups. Fuzzy Sets and Systems 289 (2016), 94-112. | DOI | MR

[9] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra. Grauate Texts in Mathematics, 1981. | DOI | MR | Zbl

[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. | DOI | MR

[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. | DOI | MR

[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. | DOI | MR

[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. | DOI | MR

[14] Nola, A. Di, Gerla, G.: Lattice valued algebras. Stochastica 11 (1987), 137-150. | MR

[15] Edeghagba, E. E., Šešelja, B., Tepavčević, A.: Omega-Lattices. Fuzzy Sets and Systems 311 (2017), 53-69. | DOI | MR

[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. | DOI | MR

[17] Goguen, J. A.: $L$-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145-174. | DOI | MR | Zbl

[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. | DOI | MR

[19] Höhle, U.: Quotients with respect to similarity relations. Fuzzy Sets and Systems 27 (1988), 31-44. | DOI | MR

[20] Höhle, U.: Fuzzy sets and sheaves. Part I: Basic concepts. Fuzzy Sets and Systems 158 (2007), 11, 1143-1174. | DOI | MR

[21] Höhle, U., Šostak, A. P.: Axiomatic foundations of fixed-basis fuzzy topology. Springer US, 1999, pp. 123-272. | DOI | MR

[22] Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, New Jersey 1995. | MR

[23] Šešelja, B., Tepavčević, A.: On Generalizations of fuzzy algebras and congruences. Fuzzy Sets and Systems 65 (1994), 85-94. | DOI | MR

[24] Šešelja, B., Tepavčević, A.: Fuzzy identities. In: Proc. 2009 IEEE International Conference on Fuzzy Systems, pp. 1660-1664. | DOI

Cité par Sources :