Faithfully ⋆-representing semigroups and groupoids with involution by infinite matrices

Abstract

The set M ∞(Z) of countably infinite square matrices over Z forms a (partial, non-associative) groupoid. The authors, with Davies, showed in 1996 that every finite or countable groupoid embeds in M ∞(Z). This is a non-associative analogue of the elementary result that every finite or countable semigroup embeds in row-finite matrices over Z, although the proof is very different. This article considers corresponding questions about ⋆-embeddings of groupoids (G, ⋆) with involution ⋆. The two main results here are that (i) there is an improper involution ⊛ on M ∞(Z) such that every countable (G, ⋆) embeds in (M ∞(Z), ⊛), and (ii) every countable partial ⋆-groupoid obtained by omitting all products g ⋆ g from (G, ⋆) embeds in (M ∞(Z), T), where T is the transpose. It is also noted that, even in the finite associative case, not every finite semigroup with proper involution ⋆-embeds in an M n (Z) with a proper involution. However, with respect to a version of ⊛ acting on square matrices of finite even order, as an analogue of (i) apparently new even for the finite associative case, it is shown that (iii) every finite ⋆-semigroup embeds into some (M 2n (Z), ⊛). All these results have corresponding versions for k- and (k, ⋆)-algebras. As an application, these embedding results are used to provide a much simpler and more general way of finding examples, like those constructed by Sivakumar in 2006, of infinite matrices having uncountably many distinct groups and Moore–Penrose generalized inverses which are not classical inverses.