Details

Autor(en) / Beteiligte

• Bentz, Wolfram
• Gould, Victoria

Titel

Independence algebras, basis algebras and the distributivity condition

Ist Teil von

• Acta mathematica Hungarica, 2020-12, Vol.162 (2), p.419-440

Ort / Verlag

Cham: Springer

Erscheinungsjahr

2020

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

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