Hostname: page-component-669899f699-g7b4s Total loading time: 0 Render date: 2025-04-25T23:32:40.146Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong and full dualisability: three-element unary algebras

Part of: Algebraic structures General theory of categories and functors Other classes of algebra

Published online by Cambridge University Press:  09 April 2009

J. G. Pitkethly
J. G. Pitkethly
Affiliation:
La Trobe University, Victoria 3086, Australia e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterise the strongly dualisable three-element unary algebras and show that every fully dualisable three-element unary algebra is strongly dualisable. It follows from the characterisation that, for dualisable three-element unary algebras, strong dualisability is equivalent to a weak form of injectivity.

Keywords

natural dualitystrong dualityfull dualityunary algebras

MSC classification

Secondary: 08C15: Quasivarieties 08A60: Unary algebras 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 2 , October 2002 , pp. 187 - 220
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Clark, D. M. and Davey, B. A., Natural dualities for the working algebraist (Cambridge University Press, Cambridge, 1998).Google Scholar
[2]Clark, D. M., Idziak, P. M., Sabourin, L. R., Szabó, C. and Willard, R., ‘Natural dualities for quasivaneties generated by a finite commutative ring’, Algebra Universalis 46 (2001), 285–320.Google Scholar
[3]Clark, D. M., Davey, B. A. and Pitkethly, J. G., ‘Binary homomorphisms and natural dualities’, J. Pure Appl. Algebra 169 (2002), 1–28.Google Scholar
[4]Clark, D. M., Davey, B. A. and Pitkethly, J. G., ‘The complexity of dualisability: three-element unary algebras’, Internat. J. Algebra Comput., to appear.Google Scholar
[5]Davey, B. A., ‘Dualisability in general and endodualisability in particular’, in: Logic and algebra (Ponrignano, 1994) (eds. Ursini, A. and Aglianò, P.), Lecture Notes in Pure and Appl. Math. 180 (Marcel Dekker, New York, 1996) pp. 437–455.Google Scholar
[6]Hyndman, J., ‘Mono-unary algebras are strongly dualizable’, J. Austral. Math. Soc. 72 (2002), 161–172.Google Scholar
[7]Hyndman, J. and Willard, R., ‘An algebra that is dualizable but not fully dualizable’, J. Pure Appl. Algebra 151 (2000), 31–42.Google Scholar
[8]Lampe, W. A., McNulty, G. F. and Willard, R., ‘Full duality among graph algebras and flat graph algebras’, Algebra Universalis 45 (2001), 311–334.Google Scholar
[9]Priestley, H. A., ‘Ordered sets and duality for distributive lattices’, in: Orders, descriptions and roles (eds. Pouzet, M. and Richard, D.), Ann. Discrete Math. 23 (North-Holland, Amsterdam, 1984) pp. 39–60.Google Scholar
[10]Willard, R., ‘New tools for proving dualizability’, in: Dualities, interpretability and ordered structures (eds. Vaz de Carvalho, J. and Ferreirim, I.), (C.A.U.L., Lisbon, 1999) pp. 69–74.Google Scholar