Catalan monoids,monoids of local endomorphisms,and their presentations

The Catalan monoid and partial Catalan monoid of a directed graph are introduced. Also introduced is the notion of a local endomorphism of a tree, and it is shown that the Catalan (resp. partial Catalan) monoid of a tree is simply its monoid of extensive local endomorphisms (resp. partial endomorphisms) of finite shift. The main results of this paper are presentations for the Catalan and partial Catalan monoids of a tree. Our presentation for the Catalan monoid of a tree is used to give an alternative proof for a result of Higgins. We also identify results of Aîzen?tat and Popova which give presentations for the Catalan monoid and partial Catalan monoid of a finite symmetric chain.     

Fixed points of endomorphisms of trace monoids

It is proved that the fixed point submonoid and the periodic point submonoid of a trace monoid endomorphism are always finitely generated. If the dependence alphabet is a transitive forest, it is proved that the set of regular fixed points of the (Scott) continuous extension of an endomorphism to real traces is Ω-rational for every endomorphism if and only if the monoid is a free product of free commutative monoids.     

Almost universal varieties of monoids

The constant mappings onto the unit form a zero subcategory of any category of monoid homomorphisms; a varietyV of monoids isalmost universal if every category of algebras is isomorphic to a class of all nonzero homomorphisms between members ofV. Almost universal monoid varieties are shown to be exactly those varieties containing all commutative monoids in which the identity xⁿyⁿ=(xy)ⁿ fails for every n>1. Almost universal varieties of monoids can also be characterized categorically as the varieties containing all groups with zero as one-object full subcategories.Presented by B. M. Schein.The support of NSERC is gratefully acknowledged.     

Limit varieties of J-trivial monoids

Semigroup Forum - We show that limit varieties of monoids recently discovered by Gusev, Zhang and Luo and their subvarieties are generated by monoids of the form $$M_\tau (W)$$ for certain...     

Polarized endomorphisms of complex normal varieties

It is shown that a complex normal projective variety has non-positive Kodaira dimension if it admits a non-isomorphic quasi-polarized endomorphism. The geometric structure of the variety is described by methods of equivariant lifting and fibrations.     

Invariant hypersurfaces of endomorphisms of projective varieties

We consider surjective endomorphisms f of degree >1 on projective manifolds X   of Picard number one and their f⁻¹$f^{-1}$-stable hypersurfaces V, and show that V   is rationally chain connected. Also given is an optimal upper bound for the number of f⁻¹$f^{-1}$-stable prime divisors on (not necessarily smooth) projective varieties.     

Galois coverings and endomorphisms of projective varieties

We prove that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild conditions. This results is applied to the study of ramified endomorphisms of Fano manifolds with b ₂ = 1. It is conjectured that is the only Fano manifold admitting an endomorphism of degree d ≥ 2, and we verify this conjecture in several cases. An important ingredient is a generalization of a theorem of Andreatta–Wisniewski, characterizing projective space via the existence of an ample subsheaf in the tangent bundle. Marian Aprodu was supported in part by a Humboldt Research Fellowship and a Humboldt Return Fellowship. He expresses his special thanks to the Mathematical Institute of Bayreuth University for hospitality during the first stage of this work. Stefan Kebekus and Thomas Peternell were supported by the DFG-Schwerpunkt “Globale Methoden in der komplexen Geometrie” and the DFG-Forschergruppe “Classification of Algebraic Surfaces and Compact Complex Manifolds”. A part of this paper was worked out while Stefan Kebekus visited the Korea Institute for Advanced Study. He would like to thank Jun-Muk Hwang for the invitation.     

Endomorphism monoids in varieties of commutative semigroups

A concrete category is almost universal if its class of non-constant morphisms contains an isomorphic copy of every category of algebras as a full subcategory. This paper characterizes almost universal varieties of commutative semigroups. As a consequence we obtain that for every infinite cardinal κ there exists a commutative semigroup of cardinality κ such that it has exactly two endomorphisms, the identity endomorphism and a single constant endomorphism.     

On endomorphism monoids in varieties of bands

It will be proved that every non-trivial variety \({\mathbb{V}}\) of bands (idempotent semigroups) contains a proper generating class of non-isomorphic bands B such that B generates \({\mathbb{V}}\) and any band \({B\prime}\) having the same endomorphism monoid as B is isomorphic to B or to the opposite band B^op. Consequently, every sharply greater band variety has a sharply greater class of endomorphism monoids.     

Retracts and monomorphism monoids in semigroup varieties

If a semigroup varietyV contains the variety of commutative three-nilpotent semigroups, or if it is a variety of bands containing all semilattices, then, for anyA∈V and any left cancellative monoidM, there is a semigroupS∈V such thatA is a retract ofS andM is isomorphic to the monoid of all injective endomorphisms ofS.     

The varieties of languages corresponding to the varieties of finite band monoids

We describe the varieties of languages corresponding to the varieties of finite band monoids.     

Malcev products of unipotent monoids and varieties of bands

Let \(\mathcal{U}\) be the class of all unipotent monoids and \(\mathcal{B}\) the variety of all bands. We characterize the Malcev product \(\mathcal{U} \circ \mathcal{V}\) where \(\mathcal{V}\) is a subvariety of \(\mathcal{B}\) low in its lattice of subvarieties, \(\mathcal{B}\) itself and the subquasivariety \(\mathcal{S} \circ \mathcal{RB}\), where \(\mathcal{S}\) stands for semilattices and \(\mathcal{RB}\) for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation \(\widetilde{\mathcal{H}}= \widetilde{\mathcal{L}} \cap \widetilde{\mathcal{R}}\), where \(a\;\,\widetilde{\mathcal{L}}\;\,b\) if and only if a and b have the same idempotent right identities, and \(\widetilde{\mathcal{R}}\) is its dual.We also consider \((\mathcal{U} \circ \mathcal{RB}) \circ \mathcal{S}\) which provides the motivation for this study since \((\mathcal{G} \circ \mathcal{RB}) \circ \mathcal{S}\) coincides with completely regular semigroups, where \(\mathcal{G}\) is the variety of all groups. All this amounts to a generalization of the latter: \(\mathcal{U}\) instead of \(\mathcal{G}\).