The diamagnetism of the halogens

Proceedings - Mathematical Sciences -     

带簇及矩形带簇的推广

本将带簇和矩形带簇的概念推广到了n-带簇εn=[x^n 1=x^n]和n-矩形带簇εn=[x^nyx^n=x^n]，讨论了这两种半群簇之间的关系，特别对所有n-带簇的并集ε=∪εn和所有n-矩形带簇的并集Rε=∪Rεn进行了研究．     

Endomorphism monoids of bands

Communicated by Boris M. Schein     

All varieties of bands

The object of this paper is to describe the lattice of varieties of bands, and to present some related results. For example, each variety of bands is defined by exactly one identity from a countably infinite set of identities which we state explicitly. Two results concerning bands contained in an arbitrary variety of bands are stated.     

Relatively free bands

The operators h_n and i _n and their duals hmacr;_n and imacr;_n defined on the free semigroup X ⁺ for a nonempty set X by Gerhard and Petrich are proved here to be homomorphisms of onto certain subsets of X ⁺ with a suitable multiplication. From the work of the authors mentioned, these operators induce fully invariant congruences on X ⁺ corresponding to join irreducible varieties of bands if X is countably infinite. New operators on X ⁺ are defined by means of these operators which give homomorphisms in an analogous way and induce fully invariant congruences on X ⁺corresponding to all varieties of bands except for the variety of all bands, and some varieties of normal bands. The former of these was investigated by the mentioned authors and the latter must be treated differently. By means of the above operators we are able to characterize all cases of relatively free bands.     

Dualities for quasi-varieties of bands

We complete the characterization of finite bands admitting a natural duality, by showing that every finite normal band admits a natural duality. In particular we show that every finite normal band is finitely related.