Independence algebras, basis algebras and the distributivity condition

Bentz, Wolfram; Gould, Victoria

http://hdl.handle.net/10400.2/13253

Use this identifier to reference this record.

Name:	Description:	Size:	Format:
BG Distr Prop1804.03177.pdf		292.45 KB	Adobe PDF	Download

Send Feedback

Authors

Bentz, Wolfram

Gould, Victoria

Abstract(s)

Stable basis algebras were introduced by Fountain and Gould and developed in a series of articles. They form a class of universal algebras, extending that of independence algebras, and reflecting the way in which free modules over well-behaved domains generalise vector spaces. If a stable basis algebra B satisfies the distributivity condition (a condition satisfied by all the previously known examples), it is a reduct of an independence algebra A. Our first aim is to give an example of an independence algebra not satisfying the distributivity condition. Gould showed that if a stable basis algebra B with the distributivity condition has finite rank, then so does the independence algebra A of which it is a reduct, and that in this case the endomorphism monoid End(B) of B is a left order in the endomorphism monoid End(A) of A. We complete the picture by determining when End(B) is a right, and hence a two-sided, order in End(A). In fact (for rank at least 2), this happens precisely when every element of End(A) can be written as α]β where α, β ∈ End(B), α] is the inverse of α in a subgroup of End(A) and α and β have the same kernel. This is equivalent to End(B) being a special kind of left order in End(A) known as straight.

Description

Preprint de "W. Bentz and V. Gould, “Independence Algebras, Basis Algebras and the Distributivity Condition”, Acta Mathematica Hungarica 162 (2020), 419–444."