Hecke-Kiselman monoids of small cardinality

Abstract

In this paper, we give a characterization of digraphs Q,|Q| ≤ 4 such that the associated Hecke-Kiselman monoid H Q is finite. In general, a necessary condition for H Q to be a finite monoid is that Q is acyclic and its Coxeter components are Dynkin diagram. We show, by constructing examples, that such conditions are not sufficient.Date: June 25, 2018.

Growth alternative for Hecke-Kiselman monoids

“…In what follows, Θ will denote an oriented graph. According to Theorem 3 in [13] and Lemma 2.6 in [2], we know that GKdim(A Θ ) = 0 if and only if Θ is acyclic. Consequently, we start with the case when Θ is a cycle of length 3.…”

“…Applying arguments similar to those used in the previous case, we see that xa 1 is of one of the forms: Statement (1) is verified. We proceed to prove (2). Take a reduced word w of the form (3.2).…”

“…The monoid HK Θ has been studied in particular in [2], [3], [7], [9]. So far, the emphasis has been concentrated on three problems, which still remain open in full generality: on the description of all graphs Θ, for which the monoid HK Θ is finite; on the existence of faithful representations of the monoid HK Θ in the multiplicative semigroup M n (Z) of matrices over the ring of integers; and whether HK Θ is always a J -trivial monoid.…”

The Finite Basis Problem for Kiselman Monoids

“…Such an interpretation may shed new light on complexity-theoretical aspects of the theory of semigroup identities as a whole and of the finite basis problem in particular; see [12] for an impressive instance of this approach. It is to expect, however, that Questions 1 and 2 may be rather hard-for comparison, recall that in general it is still unknown for which graphs (V n , Θ) the Hecke-Kiselman monoid HK Θ is finite, and even for n = 4, the classification of graphs with finite Hecke-Kiselman monoids has turned out to be non-trivial, see [1].…”

Normal form in Hecke-Kiselman monoids associated with simple oriented graphs

“…Understanding which mixed graphs Γ yield finite Hecke-Kiselman monoids is a difficult problem and the only nontrivial results so far seem to be [1] and [10,11]. In the same vein, a characterization of reduced expressions of elements as words in the idempotent generators are only known in the Kiselman case Γ = Γ n [9] or when Γ is an unoriented graph and one may reduce to standard Coxeter combinatorics.…”