Independence algebras

Free idempotent generated semigroups and endomorphism monoids of independence algebras

“…In [18] the second author answered the question 'What then do vector spaces and sets have in common which forces End V and T n to support a similar pleasing structure?'. To do so, she investigated a class of universal algebras, v * -algebras, which she called independence algebras: the class includes sets, vector spaces and free group acts.…”

“…Alternative proofs of [14] were given by Araújo and Mitchell [1], Dawlings [8] and Djoković [9], and the result was generalized to finite dimensional vector spaces over division rings by Laffey [26]. Given the common properties shared by full transformation monoids and matrix monoids, Gould [18] and Fountain and Lewin [17] studied the endomorphism monoid End A of an independence algebra A. In Sect.…”

A Description of Normal Semigroups of Endomorphisms of Proper Independence Algebras

“…For independence algebras we recommend [3] and [4] as references, and for general semigroup theory we recommend [5].…”

“…It is clear from Lemma 1 that any algebra with [EP] has a basis. Furthermore, for such an algebra, bases may be characterised as minimal generating sets or maximal independent sets, and all bases for A have the same cardinality [4,Proposition 3.3]. This cardinal is called the rank of A and is written rank(A).…”

Relatively free algebras with weak exchange properties

“…Thus if A satisfies (T), so that PC is a closure operator, X is directly independent if and only if it is independent with respect to PC. Following [7] we say that X is independent if X is independent with respect to the subalgebra operator, that is, if for all x e X, x £ (X \ {x}).…”