A general theorem concerning semi-groups with certain finiteness conditions

For an arbitrary abstract theoretico-semi-group propertyθ, satisfying certain natural conditions, we describe (to within the structure of groups possessing the propertyθ) the structure of periodic subgroups with the propertyθ.     

A finiteness condition for semigroups which generalizes Green-Rees' theorem

LetS be a finitely generated semigroup. ThenS is finite if every finitely generated subgroup ofS is finite and, for some integerm≥1, for everym-tuplex ₁,x ₂,…x _m of elements ofS there exist an integeri: 1≤i≤m and an integer ρ>1 such that:x _i +1…x _m (x ₁ x ₂…x _m )^ρ=x _i +1…x _m x ₁…x _m . The proof of the result is a direct generalisation of the original one by Green and Rees for the casem=1.     

Frucht theorem for inverse semigroups

In the paper, the problem of representing a finite inverse semigroup by partial transformations of a graph is treated. The notions of weighted graph and its weighted partial isomorphisms are introduced. The main result is that any finite inverse semigroup is isomorphic to the semigroup of weighted partial isomorphisms of a weighted graph. This assertion is a natural generalization of the Frucht theorem for groups. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 246–251, February, 1997. This research was partially supported by the International Science Foundation under grant No. GSU 041049. Translated by A. I. Shtern     

On finiteness conditions for Rees matrix semigroups

Let T=[S; I; J; P] be a Rees matrix semigroup where S is a semigroup, I and J are index sets, and P is a J × I matrix with entries from S, and let U be the ideal generated by all the entries of P. If U has finite index in S, then we prove that T is periodic (locally finite) if and only if S is periodic (locally finite). Moreover, residual finiteness and having solvable word problem are investigated.     

A relation for general and inverse semigroups

The relation in the title is S defined by

A group-theoretical finiteness theorem

We start with the universal covering space of a closed n-manifold and with a tree of fundamental domains which zips it . Our result is that, between T and , is an intermediary object, , obtained by zipping, such that each fiber of p is finite and admits a section.      

An elementary Green imprimitivity theorem for inverse semigroups

Semigroup Forum - A Morita equivalence similar to that found by Green for crossed products by groups will be established for crossed products by inverse semigroups. More precisely, let S be an...     

A finiteness theorem for W-graphs

A W-graph for a Coxeter group W is a combinatorial structure that encodes a module for the group algebra of W, or more generally, a module for the associated Iwahori–Hecke algebra. Of special interest are the W-graphs that encode the action of the Hecke algebra on its Kazhdan–Lusztig basis, as well as the action on individual cells. In previous work, we isolated a few basic features common to the W-graphs in Kazhdan–Lusztig theory and used these to define the class of “admissible” W-graphs. The main result of this paper resolves one of the basic question about admissible W-graphs: there are only finitely many admissible W-cells (i.e., strongly connected admissible W-graphs) for each finite Coxeter group W. Ultimately, the finiteness depends only on the fact that admissible W-graphs have nonnegative integer edge weights. Indeed, we formulate a much more general finiteness theorem for “cells” in finite-dimensional algebras which in turn is fundamentally a finiteness theorem for nonnegative integer matrices satisfying a polynomial identity.     

Elliptic operators with general Wentzell boundary conditions,analytic semigroups and the angle concavity theorem

We prove a very general form of the Angle Concavity Theorem, which says that if (T (t)) defines a one parameter semigroup acting over various L^p spaces (over a fixed measure space), which is analytic in a sector of opening angle θ_p, then the maximal choice for θ_p is a concave function of 1 – 1/p. This and related results are applied to give improved estimates on the optimal L^p angle of ellipticity for a parabolic equation of the form ?u /?t = Au, where A is a uniformly elliptic second order partial differential operator with Wentzell or dynamic boundary conditions. Similar results are obtained for the higher order equation ?u /?t = (–1)^{m +l}A^mu, for all positive integers m.     

A finiteness condition for finitely generated semigroups

Communicated by G. Lallement     

Partial actions and an embedding theorem for inverse semigroups

Periodica Mathematica Hungarica - We give a simple construction involving partial actions which permits us to obtain an easy proof of a weakened version of L. O’Carroll’s...     

A finiteness theorem for isoparametric hypersurfaces

In this note we prove that for eachn there are only finitely many diffeomorphism classes of compact isoparametric hypersurfaces ofS ⁿ⁺¹ with four distinct principal curvatures.     

A finiteness theorem for primal extensions

A set W⊆V(G) is called homogeneous in a graph G if 2?|W|?|V(G)|-1, and N(x)?W=N(y)?W for each x,y∈W. A graph without homogeneous sets is called prime. A graph H is called a (primal) extension of a graph G if G is an induced subgraph of H, and H is a prime graph. An extension H of G is minimal if there are no extensions of G in the set ISub(H)?{H}. We denote by Ext(G) the set of all minimal extensions of a graph G.We investigate the following problem: find conditions under which Ext(G) is a finite set. The main result of Giakoumakis (Discrete Math. 177 (1997) 83-97) is the following sufficient condition.

Theorem. If every homogeneous set of G has exactly two vertices thenExt(G)is a finite set.