On finitely presented,cancellative and commutative ordered monoids

We show that every finitely presented, cancellative and commutative ordered monoid is determined by a finitely generated and cancellative pseudoorder on the monoid (ℕ ⁿ ,+) for some positive integer n. Every cancellative pseudoorder on (ℕ ⁿ ,+) is determined by a submonoid of the group (ℤ ⁿ ,+), and we prove that the pseudoorder is finitely generated if and only if the submonoid is an affine monoid in ℤ ⁿ .     

Gaussian conditional independence relations have no finite complete characterization

We show that there can be no finite list of conditional independence relations which can be used to deduce all conditional independence implications among Gaussian random variables. To do this, we construct, for each n>3 a family of n conditional independence statements on n random variables which together imply that , and such that no subset have this same implication. The proof relies on binomial primary decomposition.     

Embeddings in finitely presented groups which preserve the center

V.N. Remeslennikov proposed in 1976 the following problem: is any countable abelian group a subgroup of the center of some finitely presented group? We prove that every finitely generated recursively presented group G is embeddable in a finitely presented group K such that the center of G coincide with that of K. We prove also that there exists a finitely presented group H with soluble word problem such that every countable abelian group is embeddable in the center of H. This gives a strong positive answer to the question raised by V.N. Remeslennikov.     

Finitely presented monoids with linear Dehn function need not have regular cross-sections

This paper shows that a finitely presented monoid with linear Dehn function need not have a regular cross-section, strengthening the previously-known result that such a monoid need not be presented by a finite complete string rewriting system, and contrasting with the fact that finitely presented groups with linear Dehn function always have regular cross-sections.     

On homeomorphisms of the plane which have no fixed points

Homeomorphisms of the planeR ² onto itself are studied, subject to the restriction that they should preserve the sense of orientation and have no fixed points. The results of this investigation are then applied to the problem of determining which homeomorphisms can be embedded in flows, i.e., in one-parameter subgroups of the full homeomorphism group of the plane. A “free mapping” ofR ² onto itself is defined to be a homeomorphismT, without fixed points, such thatC∩TC=0 impliesC∩T ⁿ C=0 for alln≠0 wheneverC is a compact connected subset ofR ². Free mappings turn out to be just those homeomorphisms ofR ² onto itself that preserve orientation and have no fixed points. A fundamental property of free mappingsis the fact that ifT is a free mapping andA is any compact subset ofR ² then\(\mathop U\limits_{ - \infty }^{ + \infty } T^n A\) does not meet some unbounded connected subsetB ofR ². The proof of this theorem is lengthy, and will appear elsewhere. The theorem can be weakened by adding the extra assumption thatT be embedded in a flow; the proof of this weakened version is much easier, and is included in the present article. It is found that for an arbitrary free mappingT there exists a natural partition of the plane into a collection of “fundamental regions”, with the property that ifT is embedded in a flow then each of the fundamental regions is invariant under the flow. An example is given of a free mapping whose fundamental regions are bad enough so that the mapping cannot be embedded in a flow. It is proved, on the other hand, that if a free mappingT has just one fundamental region thenT is equivalent to a translation, i.e., there is a homeomorphismU ofR ² onto itself such thatUTU ^?1 is just the translation(x, y)→(x+1, y). Indeed,T is equivalent to a translation if and only ifT has just one fundamental region.     

Equivariant, locally finite inverse representations with uniformly bounded zipping length, for arbitrary finitely presented groups

This is the first of a three parts paper providing full details for our previous announcement in Prepublications Orsay 2007-16, arXiv.org/abs/0711.3579. Here we prove the results stated in the title.     

From finite posets to chain complete posets having no infinite antichain

LetF denote the class of finite posets and letF ^* denote the larger class of chain complete posets which have no infinite antichain. We show that a variety of results which are known to hold for finite posets are also true for posets inF ^*.This paper was written while the first author was visiting the University of Calgary. Research supported by grants from the National Natural Science Foundation of China, and the National Education Committee of China for returning scholars from abroad.Research supported by NSERC grant #69-0982.     

A chain complete poset with no infinite antichain has a finite core

It is well known that dismantling a finite posetP leads to a retract, called the core ofP, which has the fixed-point property if and only ifP itself has this property. The PT-order, or passing through order, of a posetP is the quasi order defined onP so thatab holds if and only if every maximal chain ofP which passes througha also passes throughb. This leads to a generalization of the dismantling procedure which works for arbitrary chain complete posets which have no infinite antichain. We prove that such a poset also has a finite core, i.e. a finite retract which reflects the fixed-point property forP.This research was written while the first author was visiting the University of Calgary.Research supported by NSERC grant #69-0982.     

Properties of exponential series,the exponents of which have finite density

Conditions are presented for the summability by the Abel method of exponential series with complex exponents, having finite angular density at the noncorner points of the boundary of the convex polygonal domain of convergence of the series; conditions are established for the convergence of these series at the points indicated above.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 277–280, February, 1991.     

Finite groups in which no two distinct conjugacy classes have the same order

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