In this paper, we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results of Barnabei et al. on admissible pairs of Dyck paths. We compute a presentation and determine the minimal dimension of an effective representation for the double Catalan monoid. We also determine the minimal dimension of an effective representation for the 0-Hecke monoid.

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Double Catalan monoids

We define the category of partitioned binary relations and show that it contains many classical diagram categories, including categories of binary relations, maps, injective maps, partitions, (oriented) Brauer diagrams and (oriented) Temperley-Lieb diagrams. We construct a one-parameter deformation of the category of partitioned binary relations and show that it gives rise to classical one-parameter deformations of partition, Brauer and Temperley-Lieb categories. Finally, we describe a factorization of partitioned binary relations into a product of certain idempotents and pairs of usual binary relations.

Partitioned Binary Relations

For the factor-powerFP(Sn) of the symmetric groupSn, we describe regular elements, maximal subgroups, isolated and fully isolated subsemigroups, and also maximal nilpotent subsemigroups whose zero elements coincide with the zero element of the semigroupFP(Sn).

Structure of subsemigroups of factor-powers of finite symmetric groups

http://www2.math.uu.se/%7Emazor/PREPRINTS/SEMI/gan6.pdf

On classification of maximal nilpotent subsemigroups

We develop a general approach to the study of maximal nilpotent subsemigroups of finite semigroups. This approach can be used to recover many known classifications of maximal nilpotent subsemigroups, in particular, for the symmetric inverse semigroup, the symmetric semigroup, and the factor power of the symmetric group. We also apply this approach to obtain a classification of maximal nilpotent subsemigroups in the semigroup of binary relations and in certain 0-simple finite semigroups.

To a transformation semigroup (S, M) we assign a new semigroupFP(S) called the factor-power of the semigroup (S, M). Then we apply this construction to the symmetric groupSn. Some combinatorial properties of the semigroupFP(Sn) are studied; in particular, we investigate its relationship with the semigroup of 2-stochastic matrices of ordern and the structure of its idempotents. The idempotents are used in characterizingFP(Sn) as an extremal subsemigroup of the semigroupBn of all binary relations of ann-element set and also in the proof of the fact thatFP(Sn) contains almost all elements ofBn.

V. S. Mazorchuk

Papers

Factor-powers of finite symmetric groups