Previous |  Up |  Next


lattice-valued algebra; congruence; homomorphism
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.
[1] Ajmal, N., Thomas, K. V.: Fuzzy lattices. Inform. Sci. 79 (1994), 271-291. DOI 10.1016/0020-0255(94)90124-4 | MR 1282402
[2] Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic/Plenum Publishers, New York 2002. DOI 10.1007/978-1-4615-0633-1
[3] Bělohlávek, R., Vychodil, V.: Algebras with fuzzy equalities. Fuzzy Sets and Systems 157 (2006), 161-201. DOI 10.1016/j.fss.2005.05.044 | MR 2186221
[4] Bělohlávek, R., Vychodil, V.: Fuzzy Equational Logic. Studies in Fuzziness and Soft Computing, Springer 186 (2005), pp. 139-170. DOI 10.1007/11376422_3
[5] Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: Fuzzy identities with application to fuzzy semigroups. Inform. Sci. 266 (2014), 148-159. DOI 10.1016/j.ins.2013.11.007 | MR 3165413
[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
[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 10.1109/fuzz-ieee.2012.6251259 | MR 3135796
[8] Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: $E$-fuzzy groups. Fuzzy Sets and Systems 289 (2016), 94-112. DOI 10.1016/j.fss.2015.03.011 | MR 3454464
[9] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra. Grauate Texts in Mathematics, 1981. DOI 10.1007/978-1-4613-8130-3 | MR 0648287 | Zbl 0478.08001
[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 10.1007/s00012-010-0059-2 | MR 2670173
[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 10.1080/0308107031000090765 | MR 1967128
[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 10.1016/j.fss.2004.06.017 | MR 2126168
[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 10.1016/j.fss.2004.06.004 | MR 2126169
[14] Nola, A. Di, Gerla, G.: Lattice valued algebras. Stochastica 11 (1987), 137-150. MR 0990882
[15] Edeghagba, E. E., Šešelja, B., Tepavčević, A.: Omega-Lattices. Fuzzy Sets and Systems 311 (2017), 53-69. DOI 10.1016/j.fss.2016.10.011 | MR 3597106
[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 10.1007/bfb0061824 | MR 0555551
[17] Goguen, J. A.: $L$-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145-174. DOI 10.1016/0022-247x(67)90189-8 | MR 0224391 | Zbl 0145.24404
[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 10.1007/s11225-006-9001-1 | MR 2271287
[19] Höhle, U.: Quotients with respect to similarity relations. Fuzzy Sets and Systems 27 (1988), 31-44. DOI 10.1016/0165-0114(88)90080-2 | MR 0950448
[20] Höhle, U.: Fuzzy sets and sheaves. Part I: Basic concepts. Fuzzy Sets and Systems 158 (2007), 11, 1143-1174. DOI 10.1016/j.fss.2006.12.009 | MR 2314674
[21] Höhle, U., Šostak, A. P.: Axiomatic foundations of fixed-basis fuzzy topology. Springer US, 1999, pp. 123-272. DOI 10.1007/978-1-4615-5079-2_5 | MR 1788903
[22] Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, New Jersey 1995. MR 1329731
[23] Šešelja, B., Tepavčević, A.: On Generalizations of fuzzy algebras and congruences. Fuzzy Sets and Systems 65 (1994), 85-94. DOI 10.1016/0165-0114(94)90249-6 | MR 1294042
[24] Šešelja, B., Tepavčević, A.: Fuzzy identities. In: Proc. 2009 IEEE International Conference on Fuzzy Systems, pp. 1660-1664. DOI 10.1109/fuzzy.2009.5277317
Partner of
EuDML logo