Every semigroup is isomorphic to a transitive semigroup of binary relations

Every (finite) semigroup is isomorphic to a transitive semigroup of binary relations (on a finite set).

     

双循环半群上的同余关系

从双循环半群的同余关系出发,讨论了幂等元所在的同余类,证明了双循环半群上的一类同余ρd(d∈N)与其逆子半群之间的相互唯一确定关系,并对这种同余做成的集合进行了刻画,证明了这种同余做成的格与自然数集在某种偏序下做成的格同构,得到了一些有意义的结果.     

一类广义变换半群的格林关系

设X是一个全序集,E是X上的一个凸等价关系．令 OE（X）={f∈TE（X）：Ax,y∈X,x≤y→f（x）≤f（y））, 其中TE（X）是E-保持变换半群．对于取定的θ∈OE（X）,在OE（X）上定义运算fog=fθg,使OE（X）成为广义半群OE（X;θ）．对于有限全序集X上的凸等价关系E,本文刻画了广义半群OE（X;θ）的正则元,描述了这个半群的格林关系．     

Generators and relations of abelian semigroups and semigroup rings

The object of this paper is the study of the relations of finitely generated abelian semigroups. We give a new proof of the fact that each such semigroup S is finitely presented. Moreover, we show that the number of relations defining S is greater than or equal to the least number of generators of S minus the rank of the associated group of S. If equality holds, we say S is a complete intersection. The main part of this study is devoted to semigroups of natural numbers generated by 3 elements. These semigroups are complete intersections if and only if they are symmetric in the sense of R. Apéry [1]. This result applies to algebraic geometry: An affine space-curve C with the parametric equations x=t^a, y=t^b, z=t^c, a, b, c natural numbers with greatest common divisor 1, is a global idealtheoretic complete intersection, if and only if the semigroup S generated by a, b, c is symmetric.This paper forms part of the author's thesis, submitted at Lousiana State University.The writing of this paper was partially supported by NSF grant GP-6388 in which the author participated as a junior assistant at Purdue University.     

Contractivity properties of Ornstein-Uhlenbeck semigroup for general commutation relations

We study a certain class of von Neumann algebras generated by selfadjoint elements ω_i=a_i+a_i⁺, where a_i, a_i⁺ satisfy the general commutation relations:We assume that operator T for which the constants are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality were shown. In this paper we prove that T-Ornstein-Uhlenbeck semigroup U_t=Γ_T(e^−t), t>0 arising from the second quantization procedure is hyper- and ultracontractive. The optimal bounds for hypercontractivity are also discussed.This paper was partially supported by KBN grant no 2P03A00732 and also by RTN grant HPRN-CT-2002-00279.     

The Schützenberger group of anH-class in the semigroup of binary relations

K. A. Zaretskii has associated a lattice V(α) with each binary relation α, and he has shown that H_α is isomorphic with the group of all automorphisms of V(α) if H_α is a group. This result is extended in this paper by showing that for any binary relation α, the Schützenberger group Γ (H_α) is isomorphic with the group of all automorphisms of V(α).     

One-sided zeros of subsets of the semigroup of binary relations

The one-sided zeros of the elements and of the subsets of the semigroup of all binary relations on an arbitrary nonempty set, are described. For a finite set of zeros, formulas are obtained for their number.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 600–604, May, 1990.     

One-sided identities for subsets of the semigroup of binary relations

We describe one-sided identity elements and subsets of the semigroup of all binary relations on a nonempty set. We obtain formulas for the number of identities for a finite set.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1026–1031, August, 1990.     

On generating regular elements in the semigroup of binary relations

We prove the semigroup generated by four binary relations contains all regular binary relations.     

Covering in the set of principal ideals in the semigroup of binary relations

Communicated by Boris M. Schein     

The bracket semigroup of knots

The paper is devoted to a coding of links with marked point on an oriented component by means of regular bibracket structures, i.e., by some words in the alphabet (,),[,]. In this way we naturally obtain the semigroup of knots with concatenation as the semigroup operation, and with the equivalence classes modulo so-called “global relations” as elements. An important step in the construction of this semigroup is the coding of links with the help of so-calledd-diagrams. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 549–562, April, 2000.