Multipliers, closure endomorphisms and quasi-decompositions of a Hilbert algebra. (English) Zbl 1082.03056

Cīrulis, Jānis

Chajda, I. (ed.) et al., Proceedings of the 68th workshop on general algebra “68. Arbeitstagung Allgemeine Algebra”, Dresden, Germany, June 10–13, 2004 and of the summer school 2004 on general algebra and ordered sets, Malá Morávka, Czech Republic, September 5–11, 2004. Klagenfurt: Verlag Johannes Heyn (ISBN 3-7084-0163-8/pbk). Contributions to General Algebra 16, 25-34 (2005).

The author considers properties of a (right) multiplier on a Hilbert algebra and gives some fundamental results. A right multiplier \(\varphi\) on a Hilbert algebra \(A\) is a map from \(A\) into itself satisfying \(\varphi (x\to y)=x\to\varphi(y)\). He shows that the class of all multipliers on a Hilbert algebra forms a Boolean lattice. As one of the main theorems of this paper, he proves that for any Hilbert algebra \(A\), there is a one-to-one correspondence between the class \(M_0\) of all isotonic multipliers on \(A\) and the class of all decompositions of the algebra \(A\). The result is so interesting that it is to be developed further.
For the entire collection see [Zbl 1067.08002].

Reviewer: Michiro Kondo (Inzai)

MSC:

03G25	Other algebras related to logic
06A15	Galois correspondences, closure operators (in relation to ordered sets)

Keywords:

multiplier; Hilbert algebra; decomposition