Abstract
Interval algebras are a class of Boolean algebras with a linearly ordered set of generators. This class of algebras is not hereditary, i.e., not closed under taking subalgebras. We investigate the problem of finding a natural subclass of this class that is hereditary. For example, we prove that σ-centered subalgebras of interval algebras of size less than \({\mathfrak{b}}\) are interval algebras themselves. We state a dual form of our result saying that continuous zero-dimensional images of ordered compacta of weight less than \({\mathfrak{b}}\) are themselves ordered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bekkali M.: Pseudo tree algebras. Notre Dame J. Formal Logic 42, 101–108 (2001)
Bekkali, M.: Open problems in Boolean algebras over partially ordered sets. BLAST 2010, June 2–6, 2010, University of Colorado, Boulder. http://spot.colorado.edu/~szendrei/BLAST2010/bekkali_problems.pdf Accessed 10 Aug 2014
Heindorf L.: On subalgebras of Boolean interval algebras. Proc. Amer. Math. Soc. 125, 2265–2274 (1997)
Koppelberg, S., Monk, J.D.: Pseudo-trees and Boolean algebras. Order 8, 359–374 (1991/92)
Kunen, K., Vaughan, J.E. (eds.): Handbook of Set-Theoretic Topology. North-Holland, Amsterdam (1984)
Monk, J.D., Bonnet, R. (eds.): Handbook of Boolean algebras, vol.2. North-Holland, Amsterdam (1989)
Nikiel J.: Orderability properties of a zero-dimensional space which is a continuous image of an ordered compactum. Topology Appl. 31, 269–276 (1989)
Nikiel J., Purisch S., Treybig L.B.: Separable zero-dimensional spaces which are continuous images of ordered compacta. Houston J. Math. 24, 45–56 (1998)
Nikiel J.: On continuous images of arcs and compact orderable spaces. Topology Proc. 14, 163–193 (1989)
Odintsov A.A.: Separable images of ordered compacta. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3, 35–38 (1989)
Todorčević, S.: Trees and linearly ordered sets. In: Kunen, K., Vaughan, J.E. (eds.) Handbook of set-theoretic topology, pp. 235–293. North-Holland, Amsterdam (1984)
Mostowski, A., Tarski, A.: Boolesche Ringe mit geordneter Basis. Fund. Math. 32, 69–86 (1939) (German)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by A. Dow.
To the memory of Richard Laver.
This research was done during the Spring of 2014 when the first author was visiting the University of Toronto. The first author wishes to thank the Department of Mathematics of University of Toronto for its hospitality.
Rights and permissions
About this article
Cite this article
Bekkali, M., Todorcevic, S. Algebras that are hereditarily interval. Algebra Univers. 73, 87–95 (2015). https://doi.org/10.1007/s00012-014-0315-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-014-0315-y