Embedding in factorisable restriction monoids

Abstract

Each restriction semigroup is proved to be embeddable in a factorisable restriction monoid, or, equivalently, in an almost factorisable restriction semigroup. It is also established that each restriction semigroup has a proper cover which is embeddable in a semidirect product of a semilattice by a group.

Matrix semigroups over semirings

“…For each i ∈ [n], A i,i = A i,i α i,i , and E i,n = A i,n (A n,n ) −1 . From (13), it follows that α i,i = α i,n = 1 1 1. Similarly, it is easy to see that β i,i = 1 1 1 and β 1,i = 1 1 1 for all 1 ≤ i ≤ n.…”

“…Lemma 6.9. Let A, B ∈ U T n (L * ) and express A i,j and B i,j as in (13) and (14). Then the same elements α i,j and β i,j appear in the corresponding descriptions of A ⋄ and B • .…”

“…For such semigroups the H := R ∩ L -classes of idempotents are subsemigroups, indeed, unipotent monoids (monoids possessing a single idempotent). A substantial theory exists for Fountain semigroups in which unipotent monoids, arising from the H -classes of idempotents, or as quotients, play the role held by maximal subgroups in the theory of regular semigroups (see, for example, [13]). In Section 6 we see that not only are R and L not left and right compatible, respectively, but are as far from being so as possible, in a sense we will make precise (Proposition 6.10).…”

“…Triangular matrix semigroups with full diagonal are Fountain. Proposition 3 13. gives an indication that the failure of U T n (S) to be Fountain lies in the degenerate behaviour of matrices with diagonal entries equal to 0.…”