## Weak homomorphisms in some classes of algebras.(English)Zbl 0608.08002

Given two algebras $${\mathcal A}=(A,F)$$ and $${\mathcal B}=(B,G)$$, a mapping $$\phi$$ : $$A\to B$$ is a semi-weak homomorphism if for each operation $$f\in F$$ there is term g of $${\mathcal B}$$ such that $(*)\quad \phi (f(a_ 1,...,a_ n))=g(\phi (a_ 1),...,\phi (a_ n))$ for all $$a_ 1,...,a_ n\in A$$. If, in addition, for each $$g\in G$$ there is a term f of $${\mathcal A}$$ such that (*) holds, $$\phi$$ is said to be a weak homomorphism. These concepts are investigated for three classes of algebras: semigroups, lattices and median algebras. It is presented a complete description of surjective semi-weak homomorphisms for semigroups satisfying the identity $$x^ 2.y=y.x^ 2$$. For modular median algebras, the only surjective semi-weak homomorphisms are usual homomorphisms. For lattices, a bijection $$\phi$$ is a semi-weak homomorphism iff $$\phi$$ is either an isomorphism or a dual isomorphism.
Reviewer: I.Chajda

### MSC:

 08A35 Automorphisms and endomorphisms of algebraic structures 20M15 Mappings of semigroups 06B05 Structure theory of lattices 06C05 Modular lattices, Desarguesian lattices