Abstract
Let A be an algebra (of an arbitrary finitary type), and let γ be a binary term. A pair (a, b) of elements of A will be called a γ-eligible pair if for each x in the subalgebra generated by {a, b} such that x is distinct from a there exists an element y in A such that b = xyγ. We say that A is a γ-closed algebra if for each γ-eligible pair (a, b) there is an element c with b = acγ. We call A a closed algebra if it is γ-closed for all binary terms γ that do not induce a projection.
Let T be a unital subring of the field of real numbers. Equipped with all the binary operations \({(x, y) \mapsto (1- p)x+py}\) for \({p \in T}\) and 0 < p < 1, T becomes a mode, that is, an idempotent algebra in which any two term functions commute. In fact, the mode T is a (generalized) barycentric algebra. Let \({\mathcal{Q}(T)}\) denote the quasivariety generated by this mode.
Our main theorem asserts that each mode of \({\mathcal{Q}(T)}\) extends to a minimal closed cancellative mode, which is unique in a reasonable sense. In fact, we prove a slightly stronger statement. As corollaries, we obtain a purely algebraic description of the usual topological closure of convex sets, and we exemplify how to use the main theorem to show that certain open convex sets are not isomorphic.
Similar content being viewed by others
References
Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Graduate Texts in Mathematics, vol. 78. Springer, New York (1981). The Millennium Edition, http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html
Herstein I.N.: Topics in Algebra, 2nd edn. Xerox College Publishing, Lexington (1975)
Ignatov V.V.: Quasivarieties of convexors. Izv. Vyssh. Uchebn. Zaved. Mar. 29, 12–14 (1985) (Russian)
Ježek J., Kepka T.: Semigroup representations of commutative idempotent abelian groupoids. Comment. Math. Univ. Carolinae 16, 487–500 (1975)
Ježek J., Kepka T.: Medial Groupoids. Academia, Praha (1983)
Mac Lane S.: Categories for the Working Mathematician. Springer, New York (1971)
Mal’cev A.I.: Algebraic Systems. Springer, Berlin (1973)
Matczak K., Romanowska A.: Quasivarieties of cancellative commutative binary modes. Studia Logica 78, 321–335 (2004)
Matczak K., Romanowska A.B., Smith J.D.H.: Dyadic polygons. Internat. J. Algebra Comput. 21, 387–408 (2011)
Neumann W.D.: On the quasivariety of convex subsets of affine spaces. Arch. Math. (Basel) 21, 11–16 (1970)
Pszczoła K., Romanowska A., Smith J.D.H.: Duality for some free modes. Discuss. Math. Gen. Algebra Appl. 23, 45–62 (2003)
Romanowska A.B., Smith J.D.H.: Modal Theory. Heldermann, Berlin (1985)
Romanowska A.B., Smith J.D.H.: On the structure of barycentric algebras. Houston J. Math. 16, 431–448 (1990)
Romanowska A.B., Smith J.D.H.: On the structure of semilattice sums. Czechoslovak Math. J. 41, 24–43 (1991)
Romanowska, A.B., Smith, J.D.H.: Embedding sums of cancellative modes into functorial sums of affine spaces. In: Abe, J.M., Tanaka, S. (eds.) Unsolved Problems on Mathematics for the 21st Century. A Tribute to Kiyoshi Iseki’s 80th Birthday, pp. 127–139. OS Press, Amsterdam (2001)
Romanowska A.B., Smith J.D.H.: Modes. World Scientific, Singapore (2002)
Smith J.D.H.: Mal’cev Varieties. Springer, Berlin (1976)
van Lint H.H.: Introduction to Coding Theory. Springer, New York (1982)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by J. Berman.
This research was supported by the NFSR of Hungary (OTKA), grant numbers K77432 and K83219, by TÁMOP-4.2.1/B-09/1/KONV-2010-0005, and by the Warsaw University of Technology under grant number 504G/1120/0054/000. Part of the work on this paper was conducted during the visit of the second author to Iowa State University, Ames Iowa, in Summer 2010.
Rights and permissions
About this article
Cite this article
Czédli, G., Romanowska, A.B. An algebraic closure for barycentric algebras and convex sets. Algebra Univers. 68, 111–143 (2012). https://doi.org/10.1007/s00012-012-0195-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-012-0195-y