Independence algebras, basis algebras and the distributivity condition
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INDEPENDENCE ALGEBRAS, BASIS ALGEBRAS AND THE DISTRIBUTIVITY CONDITION W. BENTZ1,∗ and V. GOULD2 1
Department of Physics and Mathematics, University of Hull, Kingston-upon-Hull HU6 7RX, UK e-mail: [email protected] 2
Department of Mathematics, University of York, York YO10 5DD, UK e-mail: [email protected]
(Received November 15, 2019; revised May 22, 2020; accepted May 24, 2020)
Abstract. 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.
1. Introduction and preliminaries The second author introduced the study of the endomorphism monoids of universal algebras called v ∗ -algebras, which she named independence algebras. These algebras appear first in an article of Narkiewicz [30] and were inspired by Marczewski’s study of notions of independence, initiated in [28] (see [20] and the survey article [33]). Such algebras may be defined via properties of the closure operator �−� which takes a subset of an algebra to the ∗ Corresponding
author. Key words and phrases: independence algebra, basis algebra, v∗ -algebra, reduct, order. Mathematics Subject Classification: 08A05, 20M20, 20M25.
c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, ´ Budapest 0236-5294/$20.00 Akade ´miai Kiado , Budapest, Hungary
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W. W. BENTZ BENTZ and and V. V. GOULD GOULD
subalgebra it generates. In an independence algebra, �−� must satisfy the exchange property, which guarantees that we have a well behaved notion of rank for subalgebras and hence for endomorphisms, generalising that of the dimension of a vector space. Further, independence algebras are relatively free. Precise definitions and further details may be found in [17]. We remark that sets, vector spaces and free acts over any group are examples of independence algebras. A full classification, which we will draw upon for this article, is given by Urbanik in [33]. As explicated in [2], inde
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