Studia Logica

, Volume 77, Issue 2, pp 153–180

Free Łukasiewicz and Hoop Residuation Algebras

  • Joel Berman
  • W. J. Blok
Article

DOI: 10.1023/B:STUD.0000037125.49866.50

Cite this article as:
Berman, J. & Blok, W.J. Studia Logica (2004) 77: 153. doi:10.1023/B:STUD.0000037125.49866.50

Abstract

Hoop residuation algebras are the {→, 1}-subreducts of hoops; they include Hilbert algebras and the {→, 1}-reducts of MV-algebras (also known as Wajsberg algebras). The paper investigates the structure and cardinality of finitely generated free algebras in varieties of k-potent hoop residuation algebras. The assumption of k-potency guarantees local finiteness of the varieties considered. It is shown that the free algebra on n generators in any of these varieties can be represented as a union of n subalgebras, each of which is a copy of the {→, 1}-reduct of the same finite MV-algebra, i.e., of the same finite product of linearly ordered (simple) algebras. The cardinality of the product can be determined in principle, and an inclusion-exclusion type argument yields the cardinality of the free algebra. The methods are illustrated by applying them to various cases, both known (varieties generated by a finite linearly ordered Hilbert algebra) and new (residuation reducts of MV-algebras and of hoops).

free algebrafree spectrumBCK-algebrahoophoop residuation algebraŁukasiewicz algebraMV-algebraŁukasiewicz residuation algebraHilbert algebra

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Joel Berman
    • 1
  • W. J. Blok
    • 1
  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoU.S.A.