A symmetrical presentation for the singular part of the symmetric inverse monoid

Abstract

We give a presentation for In \ Sn, the semigroup of all singular injective partial transformations on an n-element set, in terms of a particularly symmetrical generating set. Our generators are all restrictions of transpositions, and the relations we obtain display all the symmetry possessed by In \ Sn.

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

“…In this case, we denote β by i j , meaning β maps i to j and fixes every other point. Defining relations were given for Sing n with respect to the generating set F in [38]; see also [37,39,42,100] for presentations of other singular semigroups.…”

Presentations for rook partition monoids and algebras and their singular ideals

“…However, rather than duplicating the method for RP n , we instead use Theorem 3.14 as a stepping stone towards proving Theorem 4.7 in Section 4, leading to a shorter proof. It seems that this is the first time this method (of first finding a presentation for the singular subsemigroup) has been used, though we note some similarities to the author's recent work on (singular) symmetric inverse semigroups [22]. Now define a new alphabet O = {o 1 , .…”

“…(Although this appears to be the first instance of such a "singular first" approach, we note that a somewhat similar method was used in the author's recent work on (singular) symmetric inverse semigroups in [22].) Our approach makes crucial use of the author's presentations for the (ordinary) partition monoid P n [20] and its singular ideal P n \ S n [21]; these are stated in Section 2, along with various definitions, notations, background information, and illustrative examples.…”

Presentations for (singular) partition monoids: a new approach

“…We conclude this section by stating the presentation for I n \ S n from [9], as this will play a crucial role in our proof of Theorems 2•1 and 2•2. (Recall that I n was defined above.)…”