Infinite partition monoids

Abstract

Let $\mathcal P_X$ and $\mathcal S_X$ be the partition monoid and symmetric group on an infinite set $X$. We show that $\mathcal P_X$ may be generated by $\mathcal S_X$ together with two (but no fewer) additional partitions, and we classify the pairs $\alpha,\beta\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal S_X\cup\{\alpha,\beta\}$. We also show that $\mathcal P_X$ may be generated by the set $\mathcal E_X$ of all idempotent partitions together with two (but no fewer) additional partitions. In fact, $\mathcal P_X$ is generated by $\mathcal E_X\cup\{\alpha,\beta\}$ if and only if it is generated by $\mathcal E_X\cup\mathcal S_X\cup\{\alpha,\beta\}$. We also classify the pairs $\alpha,\beta\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal E_X\cup\{\alpha,\beta\}$. Among other results, we show that any countable subset of $\mathcal P_X$ is contained in a $4$-generated subsemigroup of $\mathcal P_X$, and that the length function on $\mathcal P_X$ is bounded with respect to any generating set

Ehresmann theory and partition monoids

“…(This latter submonoid played a key role in [15], where it was denoted L X , conflicting with our notation above. Indeed, continuing just for now with the L X notation of [15], and writing R X = L • X = {α ∈ P X : codom(α) = X, coker(α) = ∆}, we have P X = L X R X for infinite X; cf. [15,Remark 11].…”

Idempotents and one-sided units in infinite partial Brauer monoids

“…From Lemma 4.1(iii) we recover the well-known fact that the group of units G X = G(PB X ) is isomorphic to the symmetric group on X; cf. [14,Section 2], [15,Section 2] and [18,Lemma 2.3]. Note also that…”

“…For the proof of the first lemma, we recall again that PB X is a submonoid of the larger partition monoid P X . As before, we will not recall the full definition of P X here; the reader may refer to [14,15], where It is easy to check that Codom(β) = X. Also, since the σ i are involutions, β 2 does not depend on the choices of σ i .…”

Congruences on infinite partition and partial Brauer monoids

“…Congruences on several other families of monoids could potentially be explored using the methods developed here and in [17]: examples include (finite and infinite) twisted diagram monoids [4,9], rook partition monoids [23], monoids of partitioned binary relations [45] and the submonoids of P X and PB X generated by all idempotents and units [13][14][15]. The latter submonoids of P X and PB X are analogous to the submonoid F X of J X discussed above, and the elements of these submonoids may be characterised in terms of a property similar to uniformity of block bijections; see [14,Theorem 6.1] and [15,Theorem 33].…”

“…More background and references on diagram algebras and monoids may be found in the surveys [33,44] or in the introductions to [10,16]. Studies of infinite diagram monoids may be found in [13][14][15].…”