an:05896424
Zbl 1237.08003
Chajda, Ivan; L??nger, Helmut
Genomorphisms of monounary algebras
EN
Chajda, I. (ed.) et al., Proceedings of the 79th workshop on general algebra ``79. Arbeitstagung Allgemeine Algebra'', 25th conference of young algebraists, Palack?? University Olomouc, Olomouc, Czech Republic, February 12--14, 2010. Klagenfurt: Verlag Johannes Heyn (ISBN 978-3-7084-0407-3/pbk). Contributions to General Algebra 19, 25-32 (2010).
2010
a
08A60 08A35
genomorphism; isogenomorphism; monounary algebra; induced quasiorder; component; cycle
Genomorphisms were first applied in theoretical computer science. The notion of genomorphism was introduced by \textit{E. K. Blum} and \textit{D. R. Estes} [Algebra Univers. 7, 143--161 (1977; Zbl 0386.08003)] as follows.
Let \({\mathcal A}=(A,F)\), \({\mathcal B}=(B,G)\) be algebras (not necessarily of the same type), \(h: A\to B\). The mapping \(h\) is called a genomorphism of \({\mathcal A}\) into \({\mathcal B}\), if \(\mathrm{ker} h\in \mathrm{Con} {\mathcal A}\) (\(h\) is congruential) and for each \(n\)-ary \(f\in F\), \(a_1,\dots,a_n\in A\), the element \(h(f(a_1,\dots,a_n))\) belongs to a subalgebra of \({\mathcal B}\) generated by \(\{h(a_1),\dots,h(a_n)\}\) (\(h\) is generative). An isogenomorphism is a bijective genomorphism; it is said to be invertible if \(h^{-1}\) is an isogenomorphism of \({\mathcal B}\) onto \({\mathcal A}\). In the mentioned paper it was shown that each genomorphism is the composition of an isogenomorphism and a homomorphism, thus the present paper is devoted mostly to isogenomorphisms. The authors first characterize generative mappings and then invertible isogenomorphisms between monounary algebras. Further they provide some constructions which, applied to a monounary algebra, yield an isogenomorphic copy where the identity mapping is the corresponding isogenomorphism.
For the entire collection see [Zbl 1201.08001].
Danica Jakubikov??-Studenovsk?? (Ko??ice)
Zbl 0386.08003