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Decomposability of the Finitely Generated Free Hoop Residuation Algebra

Marta A. Zander¹

(1)	Departamento de Matemàtica, Universidad Nacional del Sur, Av. Alem 1253, Bahìa Blanca, Argentina

Received: 28 August 2006 Published online: 28 February 2008

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.

Keywords Hoop residuation algebras - free algebras - decomposability

Presented by Daniele Mundici

Marta A. Zander
Email: mzander@criba.edu.ar

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