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ON KISELMAN QUOTIENTS OF 0-HECKE MONOIDS
"... Abstract. Combining the definition of 0-Hecke monoids with that of Kiselman semigroups, we define what we call Kiselman quotients of 0-Hecke monoids associated with simply laced Dynkin diagrams. We classify these monoids up to isomorphism, determine their idempotents and show that they are J-trivial ..."
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Abstract. Combining the definition of 0-Hecke monoids with that of Kiselman semigroups, we define what we call Kiselman quotients of 0-Hecke monoids associated with simply laced Dynkin diagrams. We classify these monoids up to isomorphism, determine their idempotents and show that they are J-trivial. For type A we show that Catalan numbers appear as the maximal cardinality of our monoids, in which case the corresponding monoid is isomorphic to the monoid of all order-preserving and order-decreasing total transformations on a finite chain. This, in particular, implies a presentation for the latter monoid. Finally, we construct various representations of these monoids by matrices, total transformations and binary relations. 1. Definitions and description of the results Let Γ be a simply laced Dynkin diagram (or a disjoint union of simply laced Dynkin diagrams). Then the 0-Hecke monoid HΓ associated with
EFFECTIVE DIMENSION OF FINITE SEMIGROUPS
"... Abstract. In this paper we discuss various aspects of the problem of determining the minimal dimension of an injective linear representation of a finite semigroup over a field. We outline some general techniques and results, and apply them to numerous examples. 1. ..."
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Abstract. In this paper we discuss various aspects of the problem of determining the minimal dimension of an injective linear representation of a finite semigroup over a field. We outline some general techniques and results, and apply them to numerous examples. 1.
CATEGORIFICATION OF THE CATALAN MONOID
"... We construct a finitary additive 2-category whose Grothendieck ring is isomorphic to the semigroup algebra of the monoid of order-decreasing and order-preserving transformations of a finite chain. ..."
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We construct a finitary additive 2-category whose Grothendieck ring is isomorphic to the semigroup algebra of the monoid of order-decreasing and order-preserving transformations of a finite chain.
