Combinatorially Factorizable Inverse Monoids

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Multilinear Equations in Free Inverse Monoids

We prove that finding solutions to the class of single multilinear equations in FIM(A) is decidable; while finding solutions to the class of systems of multilinear equations in FIM(A) is undecidable.     

Representations of Inverse Monoids by Partial Automorphisms

It is shown that any inverse semigroup of endomorphisms of an object in a properly (E, M)-structured category admitting intersections may be embedded in an inverse monoid of partial automorphisms between retracts of that object. It follows that every inverse monoid is isomorphic with an inverse monoid of all partial automorphisms between [non-trival] retracts of some object of any [almost] algebraically universal and properly (E, M)-structured category with intersections; in particular, of an [almost] algebraically universal and finitely complete category with arbitrary intersections. Several examples are given.     

On Incidence Algebras of Combinatorial Inverse Monoids

We study the incidence algebra of the reduced standard division category of a combinatorial bisimple inverse monoid [with (E(S), ≤) locally finite], and we describe semigroups of poset type (i.e., a combinatorial inverse semigroup for which the corresponding Möbius category is a poset) as being combinatorial strict inverse semigroups. Up to isomorphism, the only Möbius-division categories are the reduced standard division categories of combinatorial inverse monoids.     

含幺逆半群的半格的同余对

本文利用一族含幺逆半群的同余对刻画了其半格的同余对,并给出了含幺逆半群半格的正规同余对族的格与其标准同余对的格之间的同构关系.     

含幺逆半群的半格上的同余格

本文证明了含幺逆半群Sα（α∈Y）的半格的容许同余格同构于半格S上同余格的一个子格。     

Coset Decomposition of Positively Self-Conjugate Inverse Submonoids of Polycyclic Monoids

Abstract. We obtain several new results about the coding of a monoid by an appropriate submonoid of a polycyclic monoid. In particular, we characterize groups, periodic monoids, and right cancellative monoids in terms of the coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids.     

右（左）逆半群的半直积及圈积

给出了两个幺半群的半直积及圈积为右（左）逆半群的充分必要条件，从而推广了[2]中两个幺半群的半幺直积和圈积为逆半群的充分必要条件．     

On a Class of Factorizable Inverse Monoids Associated with Braid Groups

We prove that a constructible nilpotent-by-abelian group G can be obtained from a polycyclic group by forming d successive properly ascending HNN-extensions if and only if d is the dimension of the linear subspace of Hom(G, R) spanned by the geometric invariant Σ¹ (G, Z) ^c . We also obtain a result on the finiteness properties “type FP _m ” of certain subgroups of G     

Correction of Two Errors in ‘Topological Properties of Inverse Limits of Free Abelian Monoids’

No abstract. December 8, 1999     

Skew Monoidal Monoids

Skew monoidal categories are monoidal categories with non-invertible “coherence” morphisms. As shown in a previous article, bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod-R in which the unit object is R_R. This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present article, we study the one-object case: skew monoidal monoids (SMMs). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. These bialgebroids are submonoids of their own base and are rank 1 free over the base on the source side. We give various equivalent definitions of SMM, study the structure of their (co)module categories, and discuss the possible closed and Hopf structures on a SMM.     

Monoids and decay

We propose some formalization of the concept of critical decay number and describe the class of models with this number at most 4 (i.e., every object is decomposable which consists of four or more elements).     

Ideal Divisibility Monoids

Inspired by the monograph of Larsen/McCarthy, [26], in [10] and [11] the author started a series of articles concerning abstract multiplicative ideal theory along the problem lines of [26]. In this paper we turn to multiplicative lattices having the left Priifer property, that is to m-lattices satisfying the implication a¹ + … + a_n ? B ? a₁ +… + a_n ¦? B or even the multiplication property A ? B ? A ¦B, respectively. Clearly, studying such structures includes studying substructures of d-semigroups.     

Multiplier Hopf Monoids

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele’s definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved — in an appropriate, multiplier-valued sense — which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved — which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.     

Partial Actions of Monoids

We investigate partial monoid actions, in the sense of Megrelishvili and Schroder [12]. These are equivalent to a class of premorphisms, which we call strong premorphisms. We describe two distinct methods for constructing a monoid action from a partial monoid action: the expansion method provides a generalisation of a result of Kellendonk and Lawson [10] in the group case, whilst the approach via globalisation extends results of both [12] and [10].