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


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