Multipliers in implicative algebras. (English) Zbl 0634.03067

By a multiplier in an implicative (i.e. Hilbert) algebra A we mean a mapping \(\phi\) : \(A\to A\) such that \(\phi (a\to b)=a\to \phi (b)\) holds for all a, b in A. We describe some elementary properties of multipliers themselves as well as of their kernels and fixed point sets. In an implicative semilattice, every isotonic multiplier turns out to be a closure endomorphism (and vice versa); this case was considered by the author in some details in Latv. Mat. Ezheg. 30, 136-149 (1986; Zbl 0621.06002). See also W. H. Cornish, Math. Semin. Notes, Kobe Univ. 8, 157-169 (1980; Zbl 0465.03029).
Reviewer: J.Cirulis


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