Kybernetika 53 no. 5, 892-910, 2017

Congruences and homomorphisms on Omega-algebras

Elijah Eghosa Edeghagba, Branimir Šešelja and Andreja TepavčevićDOI: 10.14736/kyb-2017-5-0892


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.


congruence, lattice-valued algebra, homomorphism


06D72, 08A72


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