On Ehresmann semigroups

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right étale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.

     

Algebras of Ehresmann semigroups and categories

E-Ehresmann semigroups are a commonly studied generalization of inverse semigroups. They are closely related to Ehresmann categories in the same way that inverse semigroups are related to inductive groupoids. We prove that under some finiteness condition, the semigroup algebra of an E-Ehresmann semigroup is isomorphic to the category algebra of the corresponding Ehresmann category. This generalizes a result of Steinberg who proved this isomorphism for inverse semigroups and inductive groupoids and a result of Guo and Chen who proved it for ample semigroups. We also characterize E-Ehresmann semigroups whose corresponding Ehresmann category is an EI-category and give some natural examples.     

Fundamental regular semigroups with inverse transversals

Let C be a regular semigroup with an inverse transversal C° and let C be generated by its idempotents. Following W. D. Munn and T. E. Hall’s idea, in this paper, a fundamental regular semigroup T _C,C° with an inverse transversal T _C,C° ^° is constructed such that the following holds. For any regular semigroup S with an inverse transversal S° and 〈E(S)〉 = C, C° = C ∩ S°, there is a homomorphism φ from S to T _C,C° such that the kernel of φ is the maximum idempotent-separating congruence on S, and φ satisfies: (1) φ| _C is a homomorphism from C onto 〈E(T _C,C°)〉 ; (2) φ| _S° is a homomorphism from S° to T _C,C° °. In particular, S is fundamental if and only if S is isomorphic to a full subsemigroup of T _C,C°. Our fundamental regular semigroup T _C,C° is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation—although not the usual composition—is defined by means of composition.     

Fundamental semigroups having a band of idempotents

The construction by Hall of a fundamental orthodox semigroup W _B from a band B provides an important tool in the study of orthodox semigroups. We present here a semigroup S _B that plays the role of W _B for a class of semigroups having a band of idempotents B. Specifically, the semigroups we consider are weakly B-abundant and satisfy the congruence condition (C). Any orthodox semigroup S with E(S)=B lies in our class. On the other hand, if a semigroup S lies in our class, then S is Ehresmann if and only if B is a semilattice. The Hall semigroup W _B is a subsemigroup of S _B , as are the (weakly) idempotent connected semigroups V _B and U _B . We show how the structure of S _B can be used to extract information relating to arbitrary weakly B-abundant semigroups with (C). This work was carried out during a visit to Lisbon of the second author funded by the London Mathematical Society and while the first author was a member of project POCTI/0143/2003 of CAUL financed by FCT and FEDER.     

Automorphisms of Ehresmann connections

We prove that a diffeomorphism of a manifold with an Ehresmann connection is an automorphism of the Ehresmann connection, if and only if, it is a totally geodesic map (i.e., sends the geodesies, considered as parametrized curves, to geodesies) and preserves the strong torsion of the Ehresmann connection. This result generalizes and to some extent strengthens the classical theorem on the automorphisms of a D-manifold (manifold with covariant derivative). The second author is supported by the Hungarian Nat. Sci. Found. (OTKA), Grant No. NK68040.     

纯整Ehresmann半群的若干刻划

本文介绍纯整Ehresmann半群．纯整Ehresmann半群是一类特殊的U-半富足半群，我们给出了这类半群的若干刻划，并讨论了一些特殊的纯整Ehresmann半群．     

Ehresmann connections and feedforward neural networks

In this paper, we describe how Ehresmann connections can be used to study certain properties of feedforward neural networks. Essentially, we calculate a Lie group approximation to the structure of the inverse image set above a certain point in the output space and this structure can then be locally transported to the inverse image above a neighbouring point in the output space by means of an Ehresmann connection. This enables us to find a continuous approximation to the underlying topological structure of the data from discrete data pairs (input/output pairs).     

Quotient semigroups and extension semigroups

We discuss properties of quotient semigroup of abelian semigroup from the viewpoint of C ^*-algebra and apply them to a survey of extension semigroups. Certain interrelations among some equivalence relations of extensions are also considered.     

Invariant semigroups of orthodox semigroups

We consider, in a right inverse semigroupS with a multiplicative inverse transversalS ^o, the notion of anS ^o-invariant subsemigroup and use this to describe all the left amenable orders definable onS. The results obtained, together with their duals, are used to prove that ifS is an orthodox semigroup with a multiplicative inverse transversalS ^o, then every amenable order onS ^o can be extended to a unique amenable order onS. NATO Collaborative Research Grant 910765 is gratefully acknowledged. The second-named author also gratefully acknowledges support from the Calouste Gulbenkian Foundation, Lisbon.     

Preaffine semigroups and convex matrix semigroups

Algebraic conditions are given which guarantee that a semigroup on a subset of real topological vector space can be embedded in a convex matrix semigroup. We also study when the minimal ideal of a convex matrix semigroup will be convex.     

Embedding inverse semigroups in coset semigroups

The setK(G) of all cosets X of a group G, modulo all subgroups of G, forms an inverse semigroup under the multiplication X*Y=smallest coset that constains XY. In this note we show that each inverse semigroup S can be embedded in some coset semigroupK(G). This follows from a result which shows that symmetric inverse semigroups can be embedded in the coset semigroups of suitable symmetric groups. We also give necessary and sufficient conditions on an inverse semigroup S in order that it should be isomorphic to someK(G). This research was supported by a grant from the National Science Foundation.     

Preaffine semigroups and convex matrix semigroups

Algebraic conditions are given which guarantee that a semigroup on a subset of real topological vector space can be embedded in a convex matrix semigroup. We also study when the minimal ideal of a convex matrix semigroup will be convex.