Decomposability of the Finitely Generated Free Hoop Residuation Algebra

Abstract

In this paper we prove that, for n > 1, the n-generated free algebra $F_{{\mathcal{V}}}(n)$ in any locally finite subvariety ${\mathcal{V}}$ of HoRA can be written in a unique nontrivial way as Ł₂ ×  A′, where A′ is a directly indecomposable algebra in ${\mathcal{V}}$ . More precisely, we prove that the unique nontrivial pair of factor congruences of $F_{{\mathcal{V}}}(n)$ is given by the filters $[{\mathcal{J}})$ and $F_\mathcal {V}(n) - (\mathcal {J}]$ , where the element ${\mathcal {J}}$ is recursively defined from the term $j(x, y) =(((x \rightarrow y) \rightarrow y) \rightarrow x) \rightarrow x$ introduced by W. H. Cornish. As an additional result we obtain a characterization of minimal irreducible filters of $F_{{\mathcal{V}}}(n)$ in terms of its coatoms.