On presentations of Brauer-type monoids

Abstract

We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.

Motzkin monoids and partial Brauer monoids

“…Analogous results have been obtained for many other important families of semigroups, such as full linear monoids [26,29,38], endomorphism monoids of various algebraic structures [15,18,30,31,37], sandwich semigroups [13,14] and, more recently, certain families of diagram monoids [22,24,25,58].The current article mainly concerns the partial Brauer monoid PB n and the Motzkin monoid M n (see Section 2 for definitions). There is a growing body of literature on partial Brauer monoids and algebras; 1 see for example [7,16,43,46,55,60,[64][65][66]. Motzkin monoids and algebras are a more recent phenomenon [7,9,17,46,66,70], and are planar versions of the partial Brauer monoids and algebras (see Section 2); the relationship between PB n and M n is therefore analogous to that between the Brauer monoid B n and the Jones monoid J n .…”

“…The current article mainly concerns the partial Brauer monoid PB n and the Motzkin monoid M n (see Section 2 for definitions). There is a growing body of literature on partial Brauer monoids and algebras; 1 see for example [7,16,43,46,55,60,[64][65][66]. Motzkin monoids and algebras are a more recent phenomenon [7,9,17,46,66,70], and are planar versions of the partial Brauer monoids and algebras (see Section 2); the relationship between PB n and M n is therefore analogous to that between the Brauer monoid B n and the Jones monoid J n .…”

Generators and relations for partition monoids and algebras

“…The partition algebra arises naturally when studying Schur-Weyl duality in the representation theory of the symmetric group, and a thorough exposition may be found in the survey-style article of Halverson and Ram [19] as well as an extensive list of references; see also the more recent studies [10,12,15]. A number of prominent semigroup examples may also be thought of as diagram algebras, including the (full) transformation semigroups (see for example [20] or [21]), and the symmetric and dual symmetric inverse semigroups (see [24] and [16] respectively); see also [14,23] for some other examples. In fact, Wilcox [29] realized the partition algebra as a twisted semigroup algebra of the partition monoid P n also considered in [19]; this point of view allows much information, such as cellularity [18] of the algebra, to be deduced from various aspects of the structure of the monoid.…”

Presentations for rook partition monoids and algebras and their singular ideals

“…15 and 4.14), and show that RP n \ S n is generated by its idempotents (Proposition 3.1 and Remark 3.2). Our approach is quite different to previous studies of (singular) diagram semigroups and algebras [7,20,21,49,56,71], in the sense that we first obtain a presentation for the singular rook partition monoid RP n \ S n (the subject of Section 3), and then use this to bootstrap up to a number of presentations for the (full) rook partition monoid RP n (in Section 4).…”