NDLI: An algebraic closure for barycentric algebras and convex sets

Content Provider	Springer Nature Link
Author	Romawska, A. B. Czédli, Gábor
Copyright Year	2012
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.
Ending Page	143
Page Count	33
Starting Page	111
File Format	PDF
ISSN	00025240
e-ISSN	14208911
Journal	algebra universalis
Issue Number	1
Volume Number	68
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2012-08-07
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	mode algebraic closure convex set Axiomatic and generalized convexity entropic groupoid Algebra barycentric algebra entropic algebra closed algebra medial groupoid
Content Type	Text
Resource Type	Article
Subject	Logic Algebra and Number Theory

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