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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.

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Faithful Functors From Cancellative Categories to Cancellative Monoids With an Application to Abundant Semigroups

Gould

Kambites

2005

Int. J. Algebra Comput.

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We prove that any small cancellative category admits a faithful functor to a cancellative monoid. We use our result to show that any primitive ample semigroup is a full subsemigroup of a Rees matrix semigroup [Formula: see text] where M is a cancellative monoid and P is the identity matrix. On the other hand a consequence of a recent result of Steinberg is that it is undecidable whether a finite ample semigroup embeds as a full subsemigroup of an inverse semigroup.

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Cited by 3 publications

References 4 publications

Endomorphisms of relatively free algebras with weak exchange properties

Endomorphisms of relatively free algebras with weak exchange properties

Independence algebras, basis algebras and the distributivity condition

Faithful Functors From Cancellative Categories to Cancellative Monoids With an Application to Abundant Semigroups

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